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Arbitrage pricing of several new exotic options the partial tunnel and Get-Out options
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Full Text

ARBITRAGE PRICING OF SEVERAL NEW EXOTIC OPTIONS: THE PARTIAL
TUNNEL AND GET-OUT OPTIONS

By

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1999

To my brother

ACKNOWLEDGMENTS

I would especially like to thank my family for their unconditional love and support. I thank my advisor for introducing me to this field and guiding me through my research, each of my committee members for their input and all of my friends who reminded me that social life is still a big part of life even when you are working on a Ph.D. I would also like to express my appreciation to Jim Cohen at William R. Hough & Co. for the internship and consulting opportunities through which I have gained invaluable experience.

page

ACKNOW LEDGM ENTS ........................................................................................... iii

ABSTRACT ................................................................................................................ vi

CHAPTERS

1 INTRODU CTION .................................................................................................. 1

1.1 Background ................................................................................................. 1
1.2 Arbitrage Pricing ......................................................................................... 2
1 3 The Discrete Case ......................................................................................... 2
1.3.1 The M odel ........................................................................................ 2
1.3.2 The M artingale M easure ................................................................... 4
1.4 The Continuous Case .................................................................................. 6
1.4.1 The Black-Scholes M odel ................................................................. 6
1.4.2 The M artingale M easure ................................................................... 7
1.4.3 An Example ........................................................................................ 8

2 RECEN T RESULTS ............................................................................................. 9

2.1 Barrier Options ............................................................................................. 9
2.2 Partial Barrier Options ................................................................................. 11
2.3 Double Barrier Options .............................................................................. 12

3 SOME NEW RESULTS ON BROWNIAN MOTION ......................................... 14

4 PARTIAL TUNNEL OPTION S ............................................................................ 27

4.1 Description ................................................................................................... 27
4.2 Rem oving the D rift ...................................................................................... 27
4.3 Partial Tunnel Options of Type I ................................................................... 29
4.3.1 Partial Tunnel Call Option of Type I ........................... 29
4.3.2 Partial Tunnel Put Option of Type I ............................................... 31
4.4 Partial Tunnel Options of Type II ............................................................... 33
4.4.1 Partial Tunnel Call Option of Type II ............................................. 33
4.4.2 Partial Tunnel Put Option of Type II ................................................ 34

5 ANALYSIS OF THE PARTIAL TUNNEL OPTIONS PRICING
FORM ULAS ...................................................................................................... 36

5.1 Partial Tunnel Options Pricing Formulas as Extensions to Existing
Form ulas ..................................................................................................... 37
5.1.1 Limits of Partial Tunnel Options Prices as Expectations ................... 37
5.1.2 Limits of Partial Tunnel Options Formulas ...................................... 38
5.2 Num erical Results ....................................................................................... 42
5.2.1 The Partial Tunnel Call Option: Type I .......................................... 42
5.2.2 The Partial Tunnel Call Option: Type II ........................................... 45

6 THE GET-OUT OPTION .................................................................................. 48

6.1 Description ................................................................................................... 48
6.2 The M ultidim ensional Black-Scholes Setup ............................................... 48
6.2.1 The M odel ....................................................................................... 48
6.2.2 The M artingale M easure ................................................................. 49
6.2.3 The Payoff ....................................................................................... 50
6.3 Pricing ........................................................................................................ 51
6.3.1 The Independent Case ...................................................................... 51
6.3.2 The Dependent Case ........................................................................ 55

APPENDIX: EXPLICIT OPTIONS' PRICING FORMULAS ................................... 77

REFEREN CES ............................................................................................................ 82

BIOGRAPHICAL SKETCH .................................................................................... 83

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy ARBITRAGE PRICING OF SEVERAL NEW EXOTIC OPTIONS: THE PARTIAL TUNNEL AND GET-OUT OPTIONS By

December 1999

Chairman: Dr. Joseph Glover
Major Department: Mathematics

We begin with an introduction to the classic Black-Scholes option pricing model in chapter 1. This explains the method by which stochastic processes are used to obtain arbitrage prices for options. Chapter 2 follows with recent results in the field and touches on how some of these results will be extended in chapter 4.

Chapter 3 provides some new results on Brownian motion, which are of great use in chapter 4. Chapter 4 begins by introducing the first new class of options, the partial barrier tunnel options. It then proceeds to price these options using the results from chapter 3. Chapter 5 is a follow-up of chapter 4. It first shows how the results of chapter

4 generalize those in the existing literature and then provides numerical results and an analysis that illustrates how changing the parameters affects the price.

Chapter 6 introduces another new option, the Get-Out option. This option depends on two underlying securities whereas all of the aforementioned options depend only on one. The goal again is to derive an expression for the arbitrage price of this option. We separate the pricing into two cases. The first case assumes that the two underlying securities are independent and therefore do not require joint distributions for the processes. The second case, however, assumes that they are correlated. Consequently, the pricing is not as straightforward and requires joint distributions of the processes and some of their functionals.

CHAPTER I
INTRODUCTION

1.1 Background

The financial markets are growing more rapidly now than ever before. Investors at all levels are using financial products to hedge their positions, that is, to help reduce their risk. As a result, the derivatives market has grown as well. One of the most basic and earliest financial instruments used for hedging is the standard European call option on stock. This call option gives its owner the right, but not the obligation, to purchase a security at a prespecified price and time in the future. The prespecified price is called the strike price and the prespecified time is the expiry. A put option is similar in nature but gives the holder the right to sell instead of buy. American options are very similar to European options but they allow the owner to exercise the option at any point up until expiry, whereas European options may only be exercised at expiry. From here on we will work with only European options.

Suppose one holds a long position in a particular stock. Purchasing a put option on the same stock would guarantee this investor the ability to sell the stock at the strike price. The downside loss would then be reduced. Determining prices for options similar to these but with additional stipulations is the point of this work.

1.2 Arbitrage Pricing

One can see that the price of a standard call option must be less than or equal to the current price of the underlying stock. If this were not the case, one could sell the call option and buy the stock. Since the option costs more, this investor would also have some cash left over. Furthermore, at expiration only two things can occur. If the call option were exercised, he would only be left with the cash left from the original transaction. If it is not exercised, then he could simply sell the stock back into the market and have even more money. In particular, if the call option is priced higher than the stock, then there are opportunities to make money without any initial investment or risk. Thus, if one were to purchase a call option, he would not pay more than the current stock price. Arguments similar to this dictate ranges in which prices must fall. Definition 1.2.1 An arbitrage opportunity is a situation in which one may make a profit with no initial investment and without taking any risk.

If two exchanges carry the same security but at different prices, then an individual could simultaneously purchase the security at the lower price and sell at the higher price. This is an example of an arbitrage opportunity. Next we will see how arbitrage can be used to price a call option.

1.3 The Discrete Case

1.3.1 The Model

We begin with the simplest model available for a stock, the single step binomial model. This model assumes that a stock either goes up or down by certain percentages over the lifetime of the option [1]. For simplicity we will assume that we have zero interest

rates here. Denote by S, S., Sd and k the initial price, the price if the stock goes up, the price if it goes down and the strike price.

su

so

Sd

Figure 1.3: Discrete model

If the stock goes up, then the value of the call option is (S. -k)+ =max(S, -k,O). (1.1) If it goes down then it is worth (Sd-k)+ = max(Sd-k,0). (1.2) One could also choose to create a portfolio of the stock and a cash bond valued at 1 dollar so that the net worth of the portfolio would be the same as the value of the call option at expiry. To see how to do this we start with x units of stock and y units of bond. Depending upon whether the stock goes up or down, x and y must satisfy the equations x.SS + y I=(S. -k)+ (1.3)

x.- Sd + yl I=(Sd- k)+

in order to replicate the payoff of the call option. We can solve this system for x and y so that the portfolio has the same terminal value as the call option. The cost of composing

such a portfolio would then have to be the appropriate price for the call option. Any other price would lead to an arbitrage opportunity. Definition 1.3.1 The arbitrage price for an option is the value of the portfolio that replicates the payoff of the option.

1.3.2 The Martingale Measure

Notice that we did not use any probability in the preceding arbitrage arguments. However, pricing the call option in this manner does indeed induce a probability measure. Let C be the price of the call option and ST be the price of the stock at expiry. We would like a probability measure P under which C is the expected value of the possible outcomes. That is, P must satisfy

C=E[(ST -k) p -(S.-k)+ + p. -k)+ (1.4) where P and p2 are the probabilities of the stock going up (ST = S.) or down (ST = Sd ) under P. Since p, and P2 must also satisfy P + P2 = 1, it is clear that there is a unique solution for p, and p2. Moreover, the relationship (Sd - k) < C _ (S,, - k) must hold in order to prevent arbitrage and this ensures that both p, and p2 are nonnegative. The probability measure P induced in this manner has an additional implication. We must first clarify a couple of concepts. We first introduce the more general discrete process S = {S, }'=o defined on a sample space 92. The process S may branch up or down, as in the single step model, at each step. The sample space 92 is composed of all possible trajectories of the process. This more general model is known as the Cox-Ross-Rubinstein model [9].

Definition 1.3.2 The filtration generated by the discrete process S = (S }jo is the collection .-s = {_, }=0 of a -algebras Y, = a-(S, j <_ i) generated by the process.

Definition 1.3.3 A discrete process S = {S, }o is a martingale with respect to a measure P and itsfiltration .Fs = {, }2=0 if EIS, I < ci for all i and E[Si IJ] = S, for all i < j.

Under this measure, the stock price turns out to be a martingale. In general, the interest rates are not equal to zero. When interest rates are nonzero, we must adjust for the time value of money. In particular, receiving x dollars at a future time t is equivalent to purchasing e-rx worth of a bond with interest rate r now. Note that e-r x invested in this bond will payoff erl (e-rtx) = x dollars at time t. So the value of receiving x dollars at a future time I is worth e-rtx dollars now. This is what we call a discounting factor. So in the case of nonzero interest rates, the discounted stock price process, e -rtSt , will be a martingale under this measure.

Definition 1.3.4 A martingale measure is a measure under which the discounted stock price is a martingale.

Once we have a martingale measure, we may price other options by simply taking the expected value of their discounted payoff under this measure. Theorem 1.3.4 The arbitrage price C for an option is given by C = E' [X] where P is the martingale measure and X is the discounted payoff of the option.

We omit the proof of this theorem as it may be found in [9]. This reduces the problem of pricing options to finding the martingale measure. Once the martingale measure is found, we may simply take the expected value of the discounted payoff to obtain the arbitrage price of any option.

1.4 The Continuous Case

1.4.1 The Black-Scholes Model

We now move to a better model that allows for more than just two possible

outcomes for the stock price. In fact, this model allows the stock price to take any positive value. A stochastic process is used to model the stock price and an exponential function (deterministic) is used for the bond price [1].

The Black-Scholes model for options pricing is the most widely accepted and used model in the field [3]. Despite some of its unrealistic assumptions such as continuous time trading and no transaction costs, the model has persisted in both academia and industry since its birth in 1973. We first fix a probability space (R., F, P) on which we define a standard one-dimensional Brownian motion W,. Denote by ., = a (W,;0 < s < t) the o- - algebra generated by W,. The basic Black-Scholes model is set forth by the following stochastic differential equations

dS, = cSW, +,Sdt (1.5) dB, = rBdi

where S, denotes the stock price process, B, the bond price and -, ,u and r are constants representing the volatility, drift and interest rate, respectively. Corresponding to each

option, there is a payoff that is dependent upon what the underlying security does by the expiry time T.

Definition 1.4.1 A contingent claim X is an FT measurable random variable. A contingent claim is used to describe the payoff of an option. Thus the contingent claim for a standard European call option is X = (ST - k) (=1.4.2 The Martingale Measure

Just as in the discrete case, we need only find a martingale measure under which we can compute option prices by simply taking the expected values. Though not as simple as solving a system of linear equations as in the discrete case, we are guaranteed a martingale measure by the well-known Cameron-Martin-Girsanov theorem [7]. Under this measure the discounted stock price process is a martingale. In other words, if we let Z , be the discounted stock price process, then we have dZ, = oZfd , where W, is ZB,

a Brownian motion under the martingale measure. Thus we may price options by taking their expected value under this measure. Since this measure always exists in our setting, we will assume that the measure P we use to price options is the martingale measure. Furthermore, we will drop the Wk, notation and use W, as a Brownian motion under the martingale measure P. An important consequence of using the martingale measure is that the original drift term u drops out. In order for the discounted stock price process to be a

-2
martingale, the drift term in the process must be r - a- [7]. Since the original drift term
2

is irrelevant, we choose to start with the martingale measure and avoid it altogether.

1.4.3 An Example

Take a standard European call option on a stock with volatility o, drift ,u. Let k and T be the strike price and expiry, respectively. Let P be the martingale measure. Under this measure we have

dS, = aSdW + r Sdt (1.6) where W is a Brownian motion under P. Note that the drift p is irrelevant since we are using the martingale measure. The solution to this stochastic differential equation is given by

S, = oepa,+r-C t (1.7) We need only compute C = EP [e- (ST - k)+ " We use the superscript to emphasize that this expected value is taken under the martingale measure P. This expression depends only on the marginal distribution of \$T. After making appropriate substitutions we may express the price C in terms of the normal cumulative distribution function as

C = SON (X-( (1.8)

where N (.) is the normal cumulative distribution function (i.e. N(x) = P(X _ x) where X is a standard normal random variable). Again notice that the drift term 'u does not appear in this formula.

CHAPTER 2
RECENT RESULTS

2.1 Barrier Options

One of the first modifications that can be made to the basic call or put option is the addition of a barrier. A barrier is a prespecified price level, which is monitored in order to see whether or not the underlying stock hits the barrier during the course of the option. These new options, called barrier options, come in two types: knock-out and knock-in [9]. A knock-out barrier option would be the same as a regular option, but with the added stipulation that the stock price must not hit the barrier. If the stock price hits the barrier during the course of the option, then the option automatically expires with zero payoff.

:i.

0 T

Figure 2.1: Barrier option

On the other hand, a knock-in option has zero payoff by default. The option's payoff takes place only if the stock price hits the barrier during the lifetime of the option. For example, the payoff of a knock-out barrier call option could be of the form I{S,;thV1E[O,T]} (ST - k)+ (2.1)

where h is the barrier level. In this example the barrier is placed below the initial stock price, that is h < S.. This type is commonly referred to as a down and out option. Pricing such an option depends not only on the marginal distribution of ST, but also on whether or not the stock price hits the barrier. Depending upon whether the barrier is below or above the current stock price, the joint density for the pair (ST, mT) or (ST ,M,) is necessary [12]. Here m, and M, denote the running minimum and maximum of the process W as given by

m, = inf W
O 0
These densities are readily found in most texts on stochastic processes including [7] and [10]. The density is given by

P(W, dr, M, edy) - x) exp (2) ddy. (2.3) The barrier option price admits a representation by a linear combination of cumulative normal density functions [9].

Barrier options have a variety of applications in the markets. Consider the case where one has a future cash flow in another currency [2]. The exchange rate affects the value of this cash flow relative to the investor. A down and out call option on the exchange rate would enable him to hedge his position. If the exchange rate fell sharply, then the call option would be of little use anymore and the option get knocked out. Thus the introduction of the barrier would make the call option less expensive for the investor and still suit his needs.

2.2 Partial Barrier Options

A further generalization of the barrier option is the partial barrier [4]. The only

difference between a partial barrier option and a regular barrier option is that the barrier is not monitored for the entire lifetime of the option. The monitoring period would either start at time zero and end at some time before expiry or it would begin at some time before expiry and end at the expiry. Notice that for the latter case, the boundary could be hit from above or below.

0 T

Figure 2.2: Partial barrier option

These two cases are distinguished and priced separately in [4]. The payoff for a partial barrier option where the monitoring period starts at time zero is lIS,#hVEt[O,t1 ]} (ST - k)+ (2.4) where t, < T. Conditional expectations and the Markov property are used in [4] to compute the expectation of each payoff. The prices are expressed as linear combinations of bivariate and univariate normal cumulative distribution functions. The bivariate cumulative distribution function is defined as N(X, Y, p) = P(X < x, Y < y) where X and Y are normal random variables with correlation coefficient p. A more direct

approach for several of the computations above is found in [2]. The joint density for the pair (B, M, ) where s # t is derived and then used to price a knock-out call option where the partial barrier begins at time zero. Not surprisingly, the joint density is a linear combination of bivariate and univariate normal density functions.

Partial barrier options may be of use both individually or when packaged with

other options. The nature of this option allows one to compensate for changes in volatility over the lifetime of the option. One may expect moderate volatility in the near future but increased volatility soon after. In this case, one could choose a partial barrier option with the barrier only monitored in the near future. This would serve to decrease the cost of the option and still serve to hedge or speculate an underlying security. One may also package barrier and partial barrier options together to create ladder options [4]. A ladder option is just like a regular call or put option except for that it locks in intrinsic value when certain barriers are reached.

2.3 Double Barrier Options

Another modification made to the standard barrier option is the addition of a second barrier. Naturally, the two barriers are on opposite sides of the stock price. Although standard and partial barrier options use flat barriers, double barrier options where the barriers may be curved are treated in [8]. In order to price such an option, the density must reflect that the stock has not hit either of the barriers. They use an extension of Levy's density for Brownian motion confined to an interval [a,b] (i.e. a Brownian motion for which a < mt < M, < b). This density is based on repeated applications of the

reflection principle of Brownian motion and is written as an infinite sum [10]. The density is

1(x+2j(b-a))2 (x+2j(b-a)-2b)2
P(A)= ,._X je -e jdc (2.5)

where A = (a < mt < M1 < b,2 e I c [a, b]}. Their results indicate that the convergence of this series is sufficiently rapid to make it amenable to numerical implementations. Despite the generality of their results with curved boundaries, we will confine ourselves to double barrier options in which the barriers are flat. In what follows, we will refer to these types of barrier options as tunnel options.

Just like other barrier options, double barrier options may be used to speculate on the perceived volatility for an asset. There are many different ways to compute volatility, none of which are universally accepted [5]. Consider an investor who expects the volatility to be less from implied volatility of a given call option. He could sell the call option and purchase the same option with an appropriate double barrier. If he is right, then both options will be worth the same at expiry and will thus leave him with no obligations. He will profit by the amount equal to the difference in the options' prices. Double barrier options have been developed and traded by several Tokyo banks including Fuji Bank, Sanwa Bank and Nippon Credit Bank [8].

CHAPTER 3
SOME NEW RESULTS ON BROWNIAN MOTION

This chapter presents some new results on several functionals of Brownian motion. These results will be of great use in chapter 4. The corollaries following the main theorems provide densities for a Brownian motion at time T confined to a tunnel during a fixed period. The fixed period will either start at time zero and end at some time t before time T or will begin at some time t and end at time T. We use several lemmas to conceal tedious computations and obstacles in the proofs of main theorems. Lemma 3.1 Fix the values x1, x2 and y1 and define the function h(c, d) by

h(c,d) = I f f exr x . (3.1) 2;r~xt(J-t X, A 2t 2(T -t) Then

h(c, d) = exp,-2 - dd[N / - (a-x2 (3.2)

where Np (.,.) denotes the bivariate normal distribution function with correlation

coefficient p = , a=ct-d and /3=cT-d.

- e xy y-(x + d)2 (y _-) -T Proof: Let f(x,y) = cy - ). We first factor out the quantity -t 2t 2(T -t) 2(T- t)

fromf so that we have

15

-T___ 2(T-t) cy+ 2(T-t) (y - x)2 + 2(T - t)(x+d)2 (3.3)
f(x,y) 2(T- t) T T 2(T-t) T 2( Expansion, cancellation, grouping like terms and simplification yields x__ 2 +2d(1I-LJ 2x y 2 ~2(~)~ 2 f(Xy) -T - - (3.4)
yxy-2(T -t) t T T 2t

We would now like to express the quantity 1-) 2xy y2 - 2c(T - t)y

1'T(3.5)
tT T

in the form

(x-a)2 2(x-aXy-) (yfl)2 Y. (3.6) t T T In order to solve for the constants a, 8l and y we first equate the coefficients of x and y terms from (3.5) and (3.6). This leads to the system of equations 2d1IJt)= 2a 26 t T (3.7)

-2c(T-t) 2,+ 2a
T T T

Solving this system yields a ct - d
6= cT-d" (3.8) We next set the constants from both expressions equal and obtain a2 2afl 2
' T 3 T

So we may now rewrite (3.4) as

-[(xa)2 2(x-a)(y-f1) (y-f1)21 c2
f(xy) = + +y -- (3.10)
2( t) t T T 2t

where a,,fiandy are defined as in (3.8) and (3.9). We take the y term outside of the brackets so that

__ _ _ _ _ _ __ _ _ _ _ _ _ -T
fXy) -T (x - a)' 2(x - a)(y - fl) + (y - l)2+ (T
fxy-2(T - t) It T T 2(T - t)

d2
21

(3.11)

-T
We now simplify the term y 2(T-1)

d2 d . Substituting the values for a and P3 from (3.8)
2t

into (3.9) for y, the above expression becomes

-T
2(T - 1)

d2 21

(3.12)

After expanding and simplifying (3.12) we have

-T
2(T -t)y

c2T
- -cd.
2

(3.13)

Substituting (3.13) into (3.11) yields

(x-a)2 2(x-a)(y-j)+(y1+J)2] c2T
t T T 2

Set 9- = 4 =, -y = -and p = j-. Then we have S1-p2

-T and P 2(T - t) o-,ay

so that (3.14) becomes

c2 _T a2 2a+ P2 C 2 2t 2(T - t) T T)2t

T ((at-d)2 2(ct-d)(cT-d) (+ T-d)_2(T-t) t T T

f(x,y) = - -T
2(T - t)

(3.14)

1
T

f(x, y) = - 1 [( )2 2p(x-a)(y -,) + /)2 +c2T _cd.
2 1-p 2 T,,CrY CY 2

Recall that h(c, d) = 1C tei y,(x + 2

(- t2) dydr. Thus we

have

h~cd) 1 X2 0
h~~c,~ d)J exp(f(x,y))dydc
2ff t(-t1) XY
e2 ) _& e2)i72[ - x1a2 2pXxa'aXy-j) Y2 j
e_ 2_ X2_2_ o.,ay a|y

(3.16)

Making the substitutions F = a x
ax

= = P Y and switching the limits of integration
Ory

yields (3.2). LI Lemma 3.2 The density

1 * exp (x+2j(b-a))2 F (x+2j(b-a)-2b)2I
L ,[2 7t j-eCP[2t2tJ convergent on the interval [a, b].

Proof: If a series of the form 3exp [.(x + 2(b - a))2J y~e~p2t

converges uniformly on the interval [a, b], then a series of the form

Sexp (x + 2j(b-a)- 2b)2j Ix 2tI

is uniformly

(3.17)

(3.18)

(3.15)

will also converge uniformly on the same interval. To see this, first recall that a < 0 < b. Then observe that

(x + 2j(b - a))2 (x + 2(j + 1)(b - a) - 2b)2
2t 2t for large positive values ofj and (x + 2j(b - a))2 > (x + 2j(b - a) - 2b)2 2t 2t for large negative values ofj. So the convergence of (3.17) ensures that (3.18) will also converge. Thus we need only show that a series of the form (3.17) converges uniformly on the interval [a, b]. However, this is equivalent to the one sided series

I exp (x + 2j(b-a))2 (3.19) j-1 2t

converging uniformly on the same interval. Now we choose an integer N large enough that j 2! N implies x + 2j(b - a) > 0. We then observe that for j __ N we have

ex[ (x + 2j(b -a))2 j< exp (a + 2j(b- a))2] Vx c [a,b]. (3.20)
ex1 2t 2t Now the series

++0 exp 1 (3.21) jiI 2t

clearly converges. If we define = sup ~Ze x + 2j(b - a)) (322 M , = sup exp (3.22) xela,bl j-1 2t'

then we have

lim M, = lim sup eexp a+2j(b a,

_ lr +exp (a+2j(b-a))2 L .. 2t (3.23)

The inequality in (3.23) is due to (3.20). The limit goes to zero since the series converges. (3.23) is a necessary and sufficient condition for the uniform convergence of (3.19) [11]. This proves the lemma. Z

Theorem 3.3 Let A = {a < m , < b,BT > y,} where t < T. Then we have

where h(., .) is given by (3.2), cj = 2j(b - a) and d, = 2j(b - a) - 2b. Proof: Let A, = {a _ m, - M, !5 b} and A2 = {BT > yj ) so that A = A, -A2. Then we have

Using a simple property of expectations, we take an inner expectation conditioned on tso that (3.25) becomes

E[E[eXr 1A1A2 (3.26) We now note that the random variable IA, is , -measurable. Thus we can take IA, outside of the inner conditional expectation so that (3.26) becomes E[IA, - E[eBr "lA. t ]. (3.27)

Next we rewrite (3.27) using the shift operator 01 and the Markov property of Brownian motion. This yields

E [1lA, 'E e' r "1A2 I ]]= E [1A , E [(eA T- "2 ) ï¿½ 0t , -" (3.28) =EIIA] .EBt [e ABT-f. A]]1

where A2 = {BT., t y1 } and Ez [-] denotes the expectation for the process started at z instead of the origin. We may compute the inner expectation as

E"2 = El2 r(TA-yI fexp(2y).exp 2(T-- t ay, (3.29) which is a function of B,. So we define the function f by

f(x) = E' [e 4r_. i=2( exp(Ay) - exp (Y d( (3.30) A2y, 2(T - t) and write (3.28) as

E[lA, "f(Bl)]. (3.31) Recall that A1 = {a _< ni_ M, < b}. So we compute (3.31) as E[lA "f(Bt)] = fk(x). f(x)dx (3.32)
a
where

o Kxp (x + 2j(b-a))2 exp (x + 2j(b-a) - 2b)2 (3
2 -4 k L 21 2t

Set c, = 2j(b - a) and d. = 2j(b - a) - 2b. Then we may express (3.32) as

b 1 +0 x ( X + exp l fx X ) ( . 4
V --, L K 21 2t .fxd. (34

Substituting f(x) back into (3.34) and pulling the constants to the front of the summation we obtain

e 2 exp 2 exp [ ( 2 ]]x
S(3.35) fexp(Ay)'expi (-)2(T- t) where I = [y1, oo). Since f(x) is bounded on the interval [a, b], we use lemma 3.2 to interchange the infinite limit with the outer integral. We may also combine the integrands to obtain

+0f b exp[(x + -C)2 exp [ L . (3.36) exp(y)"exp(- 2(T- t) 99dy Distributing the exponentials on the right yields

[+00__ b([xp(X +c )2 A (y -X),
) I exp 2 2(T - t)
j O2r " t T 0.I -"(3.37)

exp [Ay 2 2- We now have a doubly infinite series where the terms are of the form h(A,c j) -h(A,aj ) (3.38)

where h(-, .) is defined as in (3.2). We use lemma 3.1 to complete the proof LI Corollary 3.4 P(a <_mt y)= (h(o,c,)-h(O,d,)) for t

(cd)= 1 Y (x+d)2 (Y - X) 2vd . (3.39) Remark: This function differs from h(c, d) only by the limits of integration on they variable.

Lemma 3.5 The function h(c, d) may be expressed as

4(c, d) =expf C T~cd[Njj2-a ,yI-Ii8)_N(x-a yl1I (3.40) where p = J , a=ct-d and /3=cT-d.

Proof: This proof is the same as the proof of lemma 3.1 except for making the last substitutions

x-a and - (3.41) instead of

Sa-x and f y (3.42) ax 07y

11

Lemma 3.6 Define the function g(c,d) by

1 2t 2- (x +d)2 (y_ X)2dy dx. g~~)=2;rvt(-T-t) f fp~ 2t 2(T -t 1) ' Then

c cd)[ N,(X2-a Y2- ) _Np xl-a 'y -

{Np (x2-a Y1lTf3IJ

- NP x, -ay,-p6
,f_ /1f

where Np (.,.) denotes the bivariate normal distribution function with correlation

coefficient p = J , a =ct-d and P =cT-d.

Proof: This proof is immediate since g(c, d) is the difference of two functions having the X Y2
form h(c,d) or h(c,d). In particular, any integral of the form J (.)dydx may be x] Y1

X2 Y2 X2 Y]
written as f f (.)dydx -f f (-)dy dr so that the right hand side is the difference of two

XI --W

X1 --W

functions of the form h(c,d). LI Theorem 3.7 Define w(c,d) by

w(c,d)= expC T _ cd x

y(x2+ -a y2-3J

{N..

Cx2+d-a y1-fl
1- 1 i1)

- NP(x, +d-a y2 -IJ Np(x+d-a ay1-/

(3.43) (3.44)

(3.45)

and set B ={a< B. < b Vu E [t,T],BT E I} where t < T and I = [yl,y2] c: [a,b]. Then we have

E[eB' "1+]: 'B(w(>,c,)-w(,d. (3.46) where c= 2j(b - a) and dj = 2j(b - a) - 2b.

Proof: Just as in theorem 3.3 we use basic properties of conditional expectation and the Markov property of Brownian motion to obtain E ,[er 1]= 1,E[E [e B ] =E[E [(eB7, 1, )01.F]] (3.47) E E[EBt [eAT- . 1i,]]

where b = {a < B. _ b Vu c [0, T -1],BT., c I}. We may compute the inner expectation as

EB [eB'-' . I,] = flk(Y- B,). exp(Ay)dy (3.48) where

k(x) :2[(T - t) [exp 2(T -t) expr (x + 2j(b-a) - 2b)1(3.49) 2-;rT - ) 2(T- t)2(T -t) Define the function f(x) by

f(x) = Ex [le -,. 1,] = k(y- x),exp(Ay)dy. (3.50)

We may then compute the entire expectation from (3.47) as

E[EBt [eBT-'. l]] = E[f (Bt)]

f f(x)exp- 4*

(3.51)

We now substitute the function f(x) into (3.51) to obtain

(3.52)

at fJIk(y - x)' exp(Ay)dy exp-21d.

and the density k(x) into (3.52) which becomes

b

l ;;22;rJ 2ir(T - t)

(3.53)

exp(Ay)4y. exp x 2)dx

Now f(x) is bounded so we may use the uniform convergence of the series to move the summation outside of the integral signs. After simplifying (3.53) becomes

+ b
2I jj-I I
M 27-.oo0

y - X + cj2
2(T -t)

x2
21

(3.54)

exp Ay -

2(T- t)

Define

1 X2 Y2 r 2 (y -x + ) iv(c, d) = 2f It exp cy2t (T-) yd

and recall that

cd) 1 Y exp cy +d)2 (-2 dydx .
244, T-t) y,2 2(T -t)

(3.55)

(3.56)

Notice that we could translate the x variable in i(c,d) and rewrite (3.55) as

i1(cd) xf-dY2 exp cy -2' jic . (3.57)
2;" t(x/ - t) x- y, 2t 2(T - t) We appeal to lemma 3.4 to see that iiv(c,d) may be written as

w(c, d) =exp - cdl x

N , /7d- 'y2 - _N.0 xl a Y2f~- (3.58)

{N(x2 -d- ,Y-fl -N,,(xl-d - -,8

where p=J i' a = ct -d and fl = cT-d. But then iiv(c,d) =w(c,d) so (3.54) becomes

Ec _ [w(A, c,) -w(A,dj)]. (3.59)
j--.ao

This completes the proof LI
+00
Corollary 3.8 P(a __B, and I = [y, y2 ] c [a, b].

Proof. This is immediate from theorem 3.6. 0

CHAPTER 4
PARTIAL TUNNEL OPTIONS
4.1 Description

A partial tunnel option is precisely what its name indicates. An option of this type does not monitor the barrier for the entire life of the option. Just as with a partial barrier option, the monitoring period either starts at time zero and ends at a certain time t1 before expiry or starts at a certain non-initial time 1I and ends at expiry. We distinguish these two cases as Type I and Type II options, respectively. This chapter will give explicit pricing formulas for call and put options of both Type I and Type II in terms of cumulative bivariate distribution functions.

4.2 Removing the Drift

Recall from chapter 1 that the price of an option is simply the expectation of its discounted payoff with respect to the martingale measure P. The payoffs from each of the partial tunnel options depend on the distribution of the stock price process

S, = S. e (4.1) where W, is a Brownian motion under P. All of the densities we have derived in chapter

3 and most that are found in the literature are based on Brownian motion with no drift term. However, the Brownian motion that appears in the stock price process is always coupled with a drift term. Observe that

0-2.

In particular, the process we are dealing with is Brownian motion with drift [r - . In order to use densities for different functionals of Brownian motion we must remove the drift term. This can readily be done using the Cameron-Martin-Girsanov theorem as we did in chapter 1. We implicitly define the probability measure Q, under which the process

=W - r t (4.3) is a Brownian motion, via the Radon-Nikodym derivative dP 3 2
- = exp 9WT --,UT~

= exp((" -pt)- .2 T) (4.4) = exp (,U P 12J2T
12

where

P : r- 0,(4.5) 0- 2

Observe that the quantity p defined in (4.5) is not the original drift for the stock price process as in chapter 1. Here p is the drift of the stock price process under the martingale measure P. Now the arbitrage price for an option is found by taking the expectation under the martingale measure P. The expectation under the measure P is not equal to the expectation under the measure Q in general. However, the Radon-

Nikodym derivative serves to relate the two expectations. In particular, let X ( FT. Then we have

EP [XI = EQ [ X]. (4.6) Thus the arbitrage price for an option may be computed as an expectation under Q of the discounted payoff multiplied by the Radon-Nikodym derivative in (4.4). In this manner we will be able to use the densities for Brownian motion with no drift.

4.3 Partial Tunnel Options of Type I

4.3.1 Partial Tunnel Call Option of Type I

A partial tunnel call option of type I (PTCO-I) has the payoff function

X = l{aS.
VIc = EP [e-rT. {a
where the second equality is from (4.6). Under the new measure Q we have SSeawt+ "-a2 t(49 St =S e [.9
=SecrWt

Define A = { Su 5 b Vu E [0,t] }. Using (4.9) we rewrite A as A={a A ( a_< .! b Vu EO[t,l }1] (4.10)

where i - and/ . Substituting - from (4.4) and A from (4.10) we y cdQ

may express the price from (4.8) as

ViC = EQ exp ( T I 2j2erT (S{T

Lemma 4.1 The price VIc of the PTCO-I is given by

V1C = e S.EQ Fe(P+C)PT 51JP. 9V[,

=kE I'"~ T ISEa;P:5 VUE[Oj],W7-E[k,0)
_kE[~' { # v Otl# ï¿½ )] (4.12)

ln(-J

where k
a"

Proof: Since

{ST - k > 0) {= k (}

VIC =E={ r 1 u[/, ])}'W E ) (ST -k)]. (4.14) Next we break this up into two separate expectations so that Vc = EQ FexpPT - l Tl e . V1f- 1 (4.15)
Rl[ exp (P 9)T n 2T erT 1{ nt yiedsk Replacing ST with the process in (4.9) and factoring out constants yields

r1 2TF .
VIc = e-rT-1T Is. EQ e(4+)WT "{ :5. Vu[O,t ],;ri[c,)}
-KEL e (4.16)

This proves the lemma. 0

Theorem 4.2 The price VIc of the PTCO-I is given by VIC = Z e [S(h(zucr,cj)-h(lu+ad))-k.(h(, c))-h(, dj))] (4.17)
jz---O

where

ci = 2(b -6), dj = 2(b -6) - 2b, p I a=ct, -d and /J=cT-d. Proof: From lemma 4.1 we have

VIC = e-rTA2T s. EQ L e(p+)T "{<.< VuE[0,i,TE[k,,Oo)}]

F ] , (4.19) ek E 'LeWT {a< WV. VUE[O.1l,WkTE[k,0)}

Applying theorem 3.3 to each of the expectations completes the proof. U The price Vjc of the PTCO-I, as well as the price of each of the other partial tunnel options, is written out explicitly in the appendix.

4.3.2 Partial Tunnel Put Option of Type I

Computations similar to those above can be used to show that the arbitrage price V' for the partial tunnel put option of type I (PTPO-I) can be expressed as

rT-1 2TC F1
VI =e- r -Jr k.EQ IePWT.1 , I!;Vu [0, ],Wr (.,k)
L I . (4.20)
-S'EQ [e(+)Wr ï¿½ l VuE[O,t,],WTE(_C,k)}] " 4 Theorem 4.4 The price VP of the PTPO-I is given by
V, p e-rT-I 2T (,,;-,,, - (,+ o ,,,, ] ,2:
2u~ [k=(u j-(ud))S((u rc)kuad)] (4.21) where

J(c'd) =exp ( 2 N -a kp l]l 'ti-a ' (4.22)

c, =2(b-i), d =2(b-i)-2b, p ,a=ct,-d and /3=cT-d. Proof: We have

VIP = e r T !k. E9 eWT 1. | Jt -SL" < u [, ,, ._ } (4.23)

- S.EQ [e (P+U)PP . U[~jT(-0k

Using the same methods as in proof of theorem 3.3 we can see that

where A = {5< m,, < M,, < ', BT E (-0, y,1}. Using (4.24) to evaluate (4.23) completes

the proof W

Remark: The only differences between the valuation of the PTCO-I and the PTPO-I are the payoffs and the limits of integration. This is what motivated the introduction of the function h(c, d).

4.4 Partial Tunnel Options: Type II

4.4.1 Partial Tunnel Call Option of Type II

The same type of computations can be used to show that the arbitrage price VC for the partial tunnel call option of type II (PTCO-I1) can be expressed as

1-rT- e(IZ+t)WT 1
VIC = e- ï¿½l (S-l{, (uaT1<, W<:9 VuE[t,,T],WrE[k,C0)}

(4.25)

Theorem 4.5 The price Vjc of the PTCO-ll is given by

e-rT- -2T
II c e2 [S.-(w(,u + a, cj)- w(pu+ a, d)) - k -(w~p, c,) - w~p, dj))j (4.26) where

w(c, d) =exp(>f2 T cd){Np,( +d-a b-flJN 6+d-a b-fl)

_[N + d a , Sfl ]N ,(4.27)

8 = max(fi,k), c1 = 2(/ -6), d, = 2(/ -ai)- 2b, a = ct -d and fl =cT- d.

Proof: We have

VIC = e-rT-l/ T s E Q le( ,U+ )T Vu r[t, , ;k Eï¿½, )}.
L . (4.28)

-k. E [ APT I

We do not know whether 6 or d <_k. This is the reason 8 = max(a,k) is introduced. We may then rewrite (4.28) with 9 yielding

r 1 2 T e1ï¿½OW
V C = e-rT - T I . EQ . (4.29)

- k.: U [etTWr '{a , We then apply theorem 3.6 to evaluate the expectations which yields (4.26). This completes the proof LI

4.4.2 Partial Tunnel Put Option of Type I

The same type of computations can be used to show that the arbitrage price VP for the partial tunnel put option of type II (PTPO-II) can be expressed as
S= -rT-l/ 2T k.E e/t'.l _< ._/ u[t.]fr( ]}

erT~P T E" 1 Lk E 1ePWT (4.30)

-S'EQ [e(P+a)Wr ï¿½1{< ,5 Vut,,r],WIE(,i}]J ) Theorem 4.6 The price VI of the PTPO-l1 is given by

Vh = er2 k. (w(,uc,)- w(udj)) -S. (w(u +acj)- w(p +crdj))j (4.3 1) where w(c, d) is as defined as

w(c,,d) = pxp icd N ' (4.32)

_[N, +- -,6)N- (+- i- (4.32

/ = min(), c), = 2(/b-6), di = 2(b -i)-2b, a=ct -d and 63=cT-d. Proof: We have

- S EQ [e(u+ ' l<

Again we do not know whether k__/b or k. So we introduce V = min(b, k) and rewrite (4.33) as

V II = e k " E Q Ie " 6< 1k <:5 V U [f al.,,rT ( ., ]

- S' EQ [e(l+a'T ï¿½{a< P < VUE[tl,T],WTE(a,]}]" . We then apply theorem 3.6 to evaluate the expectations which yields (4.31). This

completes the proof 0

CHAPTER 5
ANALYSIS OF THE PARTIAL TUNNEL OPTIONS PRICING FORMULAS

The first part of this chapter shows how the partial tunnel options pricing formulas generalize those of the existing options pricing formulas. Two methods are used, both of which are straightforward limiting procedures. In particular, we take the limit of a partial tunnel pricing formula as the upper and lower barriers go to infinity and zero, respectively. We also fix the upper and lower barriers and take the limit as the monitoring time goes to zero. In the case of Type I partial tunnel options, both of these limits turn out to be the pricing formulas for the standard options. However, a subtlety arises in the latter limit for partial tunnel options of Type II. We first revert back to the original expressions for the options prices, as expectations of payoffs. This will greatly simplify the task as we take limits. Second, we take the limit of the new pricing formula given in chapter 4 as an infinite series. Although this requires much more detail and cumbersome analysis, we will see the Black-Scholes formula emerge as the limit of the PTCO-I price as the monitoring period t1 approaches zero. The second part of the chapter provides numerical results and explains the patterns found in the results.

5.1 Partial Tunnel Options Pricing Formulas as Extensions of Existing Formulas

5.1.1 Limits of Partial Tunnel Options Prices as Expectations

Recall that the arbitrage price Vc of a PTCO-I is given by

VC = EP [e-rT ." {a S, b VuE[O,tj]} (ST - k)- (5.1) Clearly we have

0 l{asS.Sb Vu0o,]} (ST - k)(.
(5.2)
_ (S, - k)'

Moreover,

lim l{aSIbVu[Ot]} ( -k)+ = l{a
liml I o (S k)+ =(ST- -k). (5.4) Since these limits are increasing and l{a 0 (5.5) we may use the Monotone Convergence Theorem (twice) to obtain

lim E [l S][(ST- k)+= EP [(ST-k)+. (5.6)
a--+ I a S b VUE[Ou
b--co

That is, the limit of the price of a PTCO-I option as the barriers go off to oo and 0 is the price of a standard call option. Remark: The first application of the Monotone Convergence Theorem shows that the partial tunnel option price converges to that of a partial barrier down and out call as the

upper barrier goes to oo. Since the limits can be taken in either order, we also see that the partial tunnel option price converges to that of a partial barrier up and out call as the lower barrier goes to 0.

Another way to see (5.6) is to take the limit as the length of the monitoring period, t1, goes to zero. In particular, we have

,olim ISbvU[t]} (ST -k)+ = (ST -k)+. (5.7) Since this is also an increasing limit and we have (5.5), we may again apply the Monotone Convergence Theorem to obtain (5.6).

Remark: Clearly we could repeat the same analysis for the PTPO-I. However, in the case of the Type H partial tunnel options, taking the limit as t, approaches T does not converge to the price of a standard option. In this case the payoffs are always bounded This is due to the fact that all of the terms in the limit are uniformly bounded No matter how close t is to T, the barriers still exist at time T and therefore impose a bound on the payoffs for call and put options. They do however converge to the prices for call-like and put-like options whose payoffs are given by X = I {a
5.1.2 Limits of Partial Tunnel Options Formulas

In this section we will focus on the PTCO-I. In particular, we will take the limit of the pricing formula obtained in chapter 4 as the length of the monitoring period t approaches zero. The formula is expressed as a doubly infinite series. We will see that all of the terms except for the central term, the term corresponding to j = 0, will approach

zero and therefore drop out from the formula. The resulting limit is the classic BlackScholes formula for a call option. Note that we use the uniform convergence from chapter

3 to interchange the infinite sum with the limit.

The formula for the PTCO-I found in the appendix is given by

c F -(t+ï¿½')cj [N (/u + a)1 - c -6 (:u+ c)T- c,

(p+a)t,-c,-b (u+a)T-c.-r.l

-e-(p~ad [N )t1 - di ( + a)T -d, -k
-e vIY

_NP Y~d b, (u+a)T-d-k.]

e-rT k ï¿½ e /ucJ N p f --ct i ' "T-P k J

N,, utj-c,- , pT- c,-i]

[Nt- j f, - JT k J]
-e - d j N P p l - d j - ii p T - d y - )
-e- ~J[N', . (5.8)

-Np 1 P~u - -d, -ak4 - ]

In a
where =5-"
weeci 2(b -i), dj =2(b - ii) - 2b and p =J .Recall that d S and

In b
= Since aS wehave i 0. First we consider the
ar

following quantities

-ci - a,
-Ci - b,
-dj - 6 59

-d

for j # 0. In particular, we will show that each is either positive or negative according to whether j is positive or negative. We substitute cj = 2(b - 6) and d= 2(b - 6) - 2b into (5.9) so that we have
-cj - di = -2j(/ - 6i) - 6i,

-cj -b = -2j(b -)-b, (5.10)
-d - 5 = -2j(b -a) + 2b - i
-dg -b =-2j(b- i) + 2b- b. Simplifying yields

-ci -6 = (2j - 1)d - 2jb,
-ci - b = 2ja - (2j + 1)b, (.1
-dj - 6 = (2j -1),i- (2j - E)b

-dj -b = 2jf - (2j - 2)b. Observe that if j > 0 then all of the quantities in (5.11) are negative. On the other hand, if j < 0 then all of the quantities are positive. These observations should be evident from the fact that 6 < 0 and /, > 0. Now consider the difference I(+ur)t-cj,-at (p+u)T-c -ï¿½

-Np (U + o)t - C) -/ a((, + o-)T - c, --(.2
S(5.12)
-Nj +(P ~J c, (+a)T - c -

For j > 0, the first argument in each bivariate normal distribution function is negative for sufficiently small values of tj. As t, tends to zero each of these arguments goes to -oo. Thus each distribution function, and hence the entire expression, will go to zero. For j < 0, the first arguments are both positive and therefore tend to +oo. Each bivariate distribution function then converges to a (univariate) normal distribution. Since the second arguments in each bivariate distribution function are the same, the difference approaches zero. The same argument shows that each of the three other difference expressions go to zero as well. This leaves us with the term corresponding to j = 0. This term simplifies to

S(+a)( ) Nou+ /)Tk '

- (p+ oj -b (,u+ u)T

-N)((u +)t1 +2b- , (p + )T + 2b-k

-e- IN N 1,

JJ (5.13)
NJp J +2b-b (pT+2b-k)1

Eight bivariate normal distribution functions appear in this term. We use the same analysis as we did above. Recall that i < 0 and b > 0. Note also that 2b -6 > 0. As t, goes to zero, the first and fifth bivariate distribution functions converge to a (univariate) normal. The second and sixth each go to zero. The four remaining distribution functions each go to 1 and therefore cancel each other out. Thus the first term is simply +'a)T- k rTk./J T -k " (5.14)

Replacing u = -r - and k - we may simplify (5.14) to recover the classic o" 2 - 7

Black-Scholes formula.

5.2 Numerical Results

In this section we examine some features of the PTCO-I and PTCO-lI. Naturally, the PTPO-I and PTPO-II will share many of these features, so they will be left out. For each of the partial tunnel call options, we will numerically illustrate the analytic results of section 5. 1. We will also examine the effects that the volatility and the length of the monitoring period have on the price of each option.

5.2.1 The Partial Tunnel Call Option: Type I

Consider a PTCO-I where we have the following parameters: initial stock price S = 55, volatility a = .2, expiry T = 1, monitoring time 1 = .5, strike price k = 65, lower barrier a = 40, upper barrier b = 80 and interest rate r = .06. We will first show that as the upper and lower barriers of this option approach infinity and zero, respectively,

the price of the option will approach that of the standard call option with initial stock price S = 55, volatility cr = .2, expiry T = 1, strike price k = 65 interest rate r = .06. The appropriate price for this standard call option is \$2.166. We price the same PTCO-I with upper barriers ranging from 80 to 120 and lower barriers ranging from 40 to 5. The results are presented in figure 5.1. Observe that the price is the same as the Black-Scholes price when the lower and upper barriers have been moved to 40 and 120, respectively.

(40,120, 2.166)

lower4 50
80 upper

Figure 5.1: Varying upper and lower barriers for a PTCO-I

We now consider the case above but let the length of the monitoring time t, go to zero. In order to illustrate the effect that this has on the option price, we will use the same parameters as above except that the upper and lower barriers will be 70 and 50, respectively. We observe the prices with t ranging from 0 to 1. The results are displayed in figure 5.2. Observe once again that the price approaches the standard Black-Scholes price as tj approaches zero.

price 1
0.8
0.6.
0.4
0.2
0.2 .o4 0.6 '.8 1
length of the monitoring period

Figure 5.2: Varying the length of the monitoring period for a PTCO-I

We now observe the affect that the volatility or and the length of the monitoring period t, have on the price. In the case of the standard call option, increasing the volatility increases the price of the option. However, in the case of the partial tunnel options, increasing the volatility only increases the price up to a point. After this point, the probability of hitting a barrier and having a zero payoff is so high that increasing the volatility only decreases the chance of not hitting a barrier and having a positive payoff Thus increasing the volatility after this point only serves to decrease the price. This phenomenon is present in both Type I and Type II partial tunnel options.

The length of the monitoring period is also a factor in the price. For partial tunnel options of Type I the monitoring period begins at time zero and ends at some time t1. Thus the length of the monitoring period is simply t,. Since increasing t only increases the chance of getting a zero payoff, we see that the price will decrease as we increase t.

Here we take the case where the intial stock price is S = 20, o- = .18, r = .06, T = 2, the upper barrier is 45, the lower barrier is 15 and the strike price isk = 25. Figure 5.3 plots the values of the prices as the volatility ranges from .05 to .95 and the monitoring time ranges from 0 to 2.

5.
4
price 3
2

0.5 0.4 0 1
volatility "2 0. 1 '
monitoring period

Figure 5.3: Varying or and t, for a PTCO-I

Notice that as the length of the monitoring period t gets closer to zero, the price appears to increase as a- increases. This occurs as a result of the PTCO-I converging to a regular call option where the price always increases as volatility increases. This phenomenon will not occur in options of Type II.

5.2.2 The Partial Tunnel Call Option: Type II

Consider a PTCO-II where we have the following parameters: initial stock price S = 55, volatility or =.2, expiry T = 1, monitoring time tI =.5, strike price k = 65, lower barrier a = 40, upper barrier b = 80 and interest rate r = .06. We show that as the upper and lower barriers of this option approach infinity and zero, respectively, the price

of the option will approach that of the standard call option with the same parameters. We will not be taking the limit as the length of the monitoring time goes to zero for the reasons mentioned in the above remark. Recall that the appropriate price for this standard call option is \$2.166. We price the same PTCO-lI with upper barriers ranging from 80 to 125 and lower barriers ranging from 55 to 25. The results are presented in figure 5.4. Observe that the price is the same (to at least three digits of accuracy) as the BlackScholes price when the lower and upper barriers have been moved to 25 and 125, respectively.

(25,125,2.166)

1.5
price 1
0,5 120

iO0upper

lower

Figure 5.4: Varying upper and lower barriers for a PTCO-II

We again observe the effect that the volatility a and the length of the monitoring period have on the price options. Now for options of Type II, the monitoring period begins at t and ends at time T. Therefore the length of the monitoring period in this case is T - ti. As mentioned above, increasing the volatility only increases the price up to a point and then the price begins to decrease. We now look at the the case where the intial stock price is S = 55, or = .2, r = .06, T = 2, the upper barrier is 100, the lower barrier is

30 and the strike price is k = 65. The volatility ranges from .05 to .30 and the length of the monitoring period ranges from 0 to 2. These results are presented in figure 5.5 below. Note that the axis for the monitoring period represents the length of the monitoring period T - t1 which was t in the case of the Type I option above.

monitoring period

Figure 5.5: Varying a and T - tj for a PTCO-II

Observe the plot where the length of the monitoring period T - 1, gets closer to zero. As a increases, the price still only increases to a point and then decreases. This phenomenon is different from that of the PTCO-I where the price appeared to strictly increase as a- increased for small values of t,.

CHAPTER 6
THE GET-OUT OPTION

6.1 Description

The get-out option is based on a strategy that will get one out of the market when a certain trigger is initiated. Consider a bullish investor who, for some reason, cannot monitor the market and wishes to implement a strategy that would exit him from further exposure if one of his holdings were to go below a certain level. The get-out option depends upon two underlying securities and its payoff is determined by whether or not these securities go below a certain level during the lifetime of the option. There are three possible occurrences:

1. Stock 1 hits a lower barrier a and triggers the payoff of an exercised call on
Stock 2 before time T.
2. Stock 2 hits a lower barrier b and triggers the payoff of an exercised call on
Stock 1 before time T.
3. Neither stock hits its respective barrier before the expiry time T and the
option returns the payoffs of call options on both stocks at time T.

This option is different from the options of previous chapters in that it depends on two underlying stocks. As a result, the Black-Scholes setup from chapter 1 will no longer suffice. We introduce a generalization of this model that will enable us to price this new option appropriately.

6.2 The Multidimensional Black-Scholes Setup

6.2.1 The Model

We must now use the multidimensional (2 dimensional) Black-Scholes Model [9]. In this model we have the filtered probability space (., ', P) in which the processes

dS = Su(6.1) dB, = rBdt

are defined. Here S, is a (2xl) vector of stock price processes, u, is a (2xl) vector of drifts, Yt is the volatility (2x2) matrix and Wt is a multidimensional (2xl) Brownian motion under the probability measure P. We also have the natural filtration _,, that is the filtration generated by the 2 dimensional process S,. Definition 6.1 A process r for which r, ï¿½ 1 - u, = Z, ï¿½ y, is called the market price for risk.

The existence of this process in conjunction with Girsanov's Theorem gives rise to a martingale measure for our model. Recall that the martingale measure is simply the measure under which the discounted stock price process is a martingale. Since the stock price process is 2 dimensional, this implies that each component of the process ertSt is a martingale. For simplicity we consider the case where u, = U, It = Y is nonsingular and r = r. In particular, the drift, volatility and interest rate do not depend on time. Under these assumptions it is clear that the process y indeed exists and is given by Y = E-1 (r- I - u). This model together with these assumptions is referred to as the classic Black-Scholes model.

6.2.2 The Martingale Measure

As before, we begin with the martingale measure P. Under this measure we have J(6.2)

for i=1,2 where V' denotes the ith row of the matrix Z. In particular, the discounted stock price process is a martingale under P. The solution to this stochastic differential equation is given by

S,=S'exp i-Wt+ r -12 + (6.3) when I is given by

(tG21 21. (6.4) We may write out (6.3) as

S,'= S'explahlWi'+ U,2W (r a12 +U2jj(6.5)

6.2.3 The Payoff

The discounted payoff X for the get-out option can be expressed as the sum of

three exclusive payoffs that correspond to the three possible outcomes set forth in 5.1. Set r, = inf {s >0: S" = a}, r2 = inf {s >0: S, = b} and define the following sets accordingly:

A 1={r1 <2; r, 1< T}
A2={r2 T; 2 T}

Clearly we have P (A A2 A3 ) = 1 and P (4 rA,) = 0 for i j. So the discounted

payoff X may be expressed as

+142 (e-r2 -.(slr .k)+ (6.7)

Our goal now is to price this option. Just as before, we need only take the expectation of X under the martingale measure P.

6.3 Pricing

6.3.1 The Independent Case

This is the simpler of the two cases. For this case we assume that the two stock price processes are independent. Consequently, the non-diagonal entries in the volatility matrix 1 must be zero. Otherwise the processes would both contain the same nonzero Brownian motion terms and therefore be correlated. If we denote the diagonal entries of the volatility matrix by a1 = all and a-2 = -22 then we may write the stock price processes as

S= = S' exp(-;wi +pit) (6.8)

2
where , = r - Or' for i=1,2. Recall from (6.6) and (6.7) that we will be taking the
2

expectation of

l= , (e-rT (s - k,)+

where

A1 ={T1 A = 12 < { l ; r2 < T}. (6.10) A3 ={(r, T; r2 tT}

We begin with the first term. We wish to compute E[14 (eTI' . (02 _kjj+ (6.11) under the martingale measure P. Recall that the process S is defined as = S2 exp(U2W: + i2u). (6.12) Define the processes Xu by

X =u;,Wi +pAu for i=1,2. (6.13) Then we have

S' =S' exp(Xu) fori=1,2. (6.14) Now write the set A, as A, = ArA, where Al= {r <2} and A"= {r < T}. If we have the density P(r c du, r, < r2, X2 c dy) then we will be able to express (6.11) as a Lebesgue integral. First note that A' = {m2 >In where m,2 is the running minimum for the process X2. Thus we may write

P(r e du,r < r2,X T Edy) = P(r, E du, mr > r,X r dy) (6.15) where In 1. Using methods in [12] and the independence we have

P(rl r du, m2 > b, X2 E dy)= P(rl E du, m2 >bX2 Edy)(
MiP(T1 d m >U 2(6.16) = P(r, E A).-P(mu > , x2 E dy)

53

The densities on the right hand side are found in [7]. The density P (X2 E dx, m,2 > b) is given by

1 x e(X --p 12/) (2b - x - p~u) 2 dc' (617)
2flU2U[ 2 u
exp 2u -exp r2 expu .

Recall that -, = inf {s >0 S = a}. We substitute S, = S' exp (X,) and rewrite this as T= inf {s > 0: S" =a}
-inf {s>0: S'exp(X.)=a =inf(s> 0: X" = 6} (6.18) =inf s>O:W' +AS-= }4 In(SIJ
where .5 Then we have the density
0G1
2

P(r1 r=du) exp du. (6.19)

So we may express (6.11) as

T +Go
f f e-- (S2ey -k2)+P(r, Edu, zI < r2,X2, Fdy) (6.20)
T +co
f f e-' (S2ey -k2 ) P(r- E du)P(X2 Edy, m2> )
0 -.0

We remove the (.)+ notation by changing the lower limit of integration of y to

in2 =In(L2) so that the quantity inside of (.) above is strictly positive. Then (6.20)

becomes

T +ao
f fe-- (S2eY -k2)+P(rl edu)P(X2 edy,m, >b). (6.21)
0Ok,

Notice that the second expectation is the same as the first with the roles of stock 1 and stock 2 reversed. Thus we can write it as

e-- (S'ey - k,)+ P(r, r=du)P (X.' cdy, m. > )

where/k =In S1- and d = In i a.

Lastly, we need to compute the value of the third expectation

= e-rT E[1A3 ((Sf -k) +(S -k )+)]

We now distribute the indicator function to each of the terms in the parentheses to compute each expectation. The first term is

(6.22) (6.23) (6.24)

Recall that A3 = {r, > T; r2 > T}. We may write A3 as A3 = A3 A3" where A3 = {T > T} and A3 = {r2 > T}. We break up the indicator function accordingly and express (6.24) as

e-rT E[1I (S k)+] (6.25)

We may factor this into two separate expectations using the independence so that (6.25) becomes

erT E[1A]"E[1A (S, -kl) (6.26) Note that we have

E[14.] = P (A,")
= P(r > T) (6.27) exp U2 ds
2 T

The last equality is given in [7]. Thus (6.26) is simply the price of a down and out barrier option multiplied by the probability in (6.27). This formula is given in [9]. The second term in the expectation (6.23) is the same with the roles reversed. Remark: Notice that the above analysis did not use the change of measure employed in earlier chapters to remove the drift. The reason for this was that the densities used already accounted for the processes with drift.

6.3.2 The Dependent Case

Just as before, under the martingale measure P the stock price processes are

S' = S' exp (f,f + U,2W + r - l22t for i=1,2 (6.28) where W,' and ,2 are standard Brownian motions. Moreover, we are still trying to compute the expectation (under P ) of the discounted payoff X where

+14 er2- Sr k 6.29)

and

A1(I ={1T; z2 _} Recall that r, = inf {s >0: S1 = a} and r2= inf {s2>0 S2 =b}.Thistimewewill remove the drift with a change of measure. Notice that we must remove the drift term from each of the processes in (6.28). We will change to a measure Q so that I = Wl+ (6.31) #2 = W2 + I21(631 where WI and t2 are standard Brownian motions under Q. However, the drift constants A and P2 must be chosen so that they satisfy

ailWt + O-i2W/ o- - ,ut) + 0,2 21- ) = I + 0'i2 _ 01lt - -,2AJ2t :O ,1lW/ + O,2'2 - (0.i4u - 0,2#2)t (6.32) U'-0,WI + o0,2W2 -(r - 2 0.]t for i= 1,2. In particular, A and P2 must satisfy the equations ï¿½'lzl 0. 2/a2 = r - O21 + a.22 2 2
2
2 2 (6.33)
0"21 + 0"22
0"21/41 +022/.12 = -r

57

Solving these equations for A and P2 yields

2r(022 oC12 '72, (-A1 + 2)+ 012 ("21 +U22)
2(0-110r22 0-120-21)

(6.34)
2r(011 - 021) -11 (021 2 +0 2)+0-21 (-2 + -2) = 2(r110-22 -0-120721)

So we define a new measure Q via the Radon-Nikodym derivative
- = exp(pj+hï¿½2dW2-T 2 . (6.35) dQ 2 ) Under the measure Q the stock price processes are S' = S' exp (a-A Wf" + aO2Wt2) for i=1,2 (6.36) where P, and P,2 are standard Brownian motions under Q. We may now compute the expectation of X from (6.29) as EP [XI=E [ d-] (6.37) Recall that r, = inf {s > 0: S = a). We may rewrite this as T = inf {s > 0: S' exp (a-CT1 + -,12W '2) = a) (6.38) = inf {s > 0 -11WI ) + 0-,2W2 } where = ln( Tl. Similarly, we may write r2 = inf {s > 0 a-21 k +a- 22kj2 =-b} where In (b).

We will make two substitutions, a translation and a rotation, so that we may use the joint densities found in [6] to compute the expectation in (6.37). Note first that the stopping times r, and r2 correspond geometrically to the first time a two dimensional Brownian motion W = , starting at the origin, first hits either one of the lines ZL1: -1iX +O'12Y =a
L2 . IX + 0712Y =-. (6.39) L2 0'21X + O22y=b

Note that d < 0 and b < 0 since a < S' and b < S2. So the intersection of these lines lies in the third quadrant. The point of intersection p = of these two lines is given by

o22 - a12
0110"22 - U12"21
(6.40)
Y Oulb - U216
0-I I"22 - al 2021

Denote the angle between the two lines as a (note that this is the obtuse angle containing the origin since both lines have negative slopes).

Figure 6.1: Original two lines in the plane

The densities in [6] are for a process that starts at specified point in the plane and then hits one of two lines. One of these lines is the x-axis. The other line goes through the origin and has a negative slope. The point at which the process starts must be located in the wedge that includes the first quadrant. Thus we will need to relocate the intersection of our two lines to the origin and then rotate the plane so that one of the lines coincides with the x-axis. In this manner we will be able to use the densities from [6].

We now make the first substitution

X, = 'k, (6.41) so that X, is a Brownian motion starting at -tb. This translation moves the intersection of the two lines to the origin. The slopes of line I and line 2 are m - al1 U12

and m2 - U21 respectively. We may suppose without loss of generality that m1 > m2.
U22

Note that we could simply switch the roles of stock 1 and stock 2 to accomplish this. However, the assumption that the slopes are not equal may not be as easy to see. If the slopes are equal then the stocks are perfectly correlated. Since this case is of no interest we may make the assumption that the slopes are indeed not equal.

Figure 6.2: Translated plane

The next substitution rotates the plane counter clockwise so that line 1 coincides with the x-axis. Since the slope of line 1 is mi - I , the angle that it makes between
612

itself and the positive x-axis is

0 = tan-i' - (6.42) U ï¿½12)

Note that we have 0 < 0 since - an < 0. Note also that the angle between line 2 and the U12

x-axis is also negative. Thus the angle is given by a = ir - tan-'- 0-U-21+t-I r -'11. (6.43) U-22 12)

We have a > - since both lines have negative slopes. Define a new process Y, by
2

Y, = R.ï¿½ -X, (6.44) where & is defined as

=( cos0 sin0(
R -sino ) (6.45) The matrix R-, will rotate the plane counterclockwise by an angle of -9 > 0. Thus Y, is a two dimensional Brownian motion starting at the point R-, - (-p) = -R_8p. Recall that the original process W, starts at the origin and the intersection of the two lines is in the third quadrant. Thus the translated and the rotated process Y, starts in either the first or second quadrant. Moreover, line 1 becomes the x-axis and line 2 creates an obtuse wedge that includes the first quadrant and a portion of the second quadrant. This comes from the fact that the original lines both have non-positive slopes. Note that one of the slopes, but

not both, may be zero. In this case the angle between line 1 and the positive x-axis is zero. Thus the rotation matrix above turns out to be the identity matrix in this case.

UX

Figure 6.3: Translated and rotated plane

Now the process Y may be expressed as

Y=&0 - X,

= &' K - R . p

P, -- - (, + R-0 p

= R_ - ., + P

Solving backward for , yields

(6.46) (6.47)

We now look at what the individual processes Wk and ,2 look like in terms of the process Y,. So we write out (6.47) as

,= R-I . Y +
C rcos9 snO Y
,sinO cos9 0 y2 yo) C cosO.Y,' -sin9 .Y2 +x0
"sin 0. Y1 +cos9-Y 2 +Y0

(6.48)

We define the functions f and g by f(x,y) = x. cosO - y. sin 0 + x0 (6.49) and

g(x, y) = x. sin 0 + y. cos0 + yo. (6.50) Then we have

_ y2)")(6.51)

This will help us write the increasingly large expressions more compactly. Now the stopping time r, defined as r1 = inf {s > 0: 011Wg's +012 = } (6.52) may be expressed as

rl =inf{s> 0: 011(cos .Y1-sin .Y2 x ,, (6.53) +U12 (sin0.Y,' +cos0-Y2 +yo) = i}" A simple calculation shows that

0.11x0 +a12Y0 =. (6.54) Thus we may write (6.53) as

T =inf {s> 0o,(cos0.Y -sinO.Y2)+Cr2(sinO.Y]I+coso.y2) = 01
f )y2 = 0) (6 -55) =inf {s > 0: (a11 COSO+U12 sin9O)Y.' +(012 cosO-a0" sinO)Y2 =.0}

Moreover, 9 = tan-', so we may compute the trigonometric functions sin 0 and k."1-2 }]

cos0 as

sinO = - 711
V/121 +Ua2
(6.56)
cosO -=
V/o2 +a02
Substituting these into (6.55) yields TI= inf {s >0: + o2.YS2=0} (6.57) Clearly the quantity oj21 + 12 is strictly positive, thus ri = inf {s>0"Y =0}. (6.58) This is precisely what we should expect. Recall that r, represents the first time that process Y, hits line 1. However, line 1 has been the translated and rotated so that it is the x-axis. Thus r, represents the first time that the process Y, hits the x-axis (i.e. the first time that Y2 = 0).

Our goal is to compuV the price E9 [p X} of the get-out'option. We first write this out explicitly as
EQdP X Q[dP( 14 ï¿½er'. 2_2

+lA3 (e-rT ((ST - k, )+ + (S - k2

The first term in this expectation is d-p ,j (e-rr''(c- k2) (6.60)

Under the measure Q we have S= S' exp(a,,l-' O2W2) for i=1,2 (6.61) and

dQ A W 2 (6.62)

where Pk' and P2 are standard Brownian motions. We now make the substitutions for k' and ,2 from (6.51). Thus (6.60) becomes
EQ [exp I , y '2 ) g (Y ' I + 2
EQ lexp P () - gfk T) T +P2 T T2 T)x . (6.63)

But r, = inf {s > 0: Y2 = 0) so Y2 = 0. Replacing this above yields

(6.64)

l -F11I .(S2ex~~~(~ )+2 (YI1o) _k2 )I

We use some simple properties of expectations to take an inner expectation conditioned on the sigma algebra FT,.rI. This yields

EQ .E~ f (S Y2 + P~u 2 '9 .f (y T yo) 2 g( )) k T)T II (6.65)

We take out the TT,,, measurable portion inside of the inner expectation. All that is left is the Radon-Nikodym derivative, which is a martingale. Recall that A = {zr < r2, r, < T}. In particular, r1 < T on this set so T A r, = r, on this set. Using the Optional Sampling theorem [7] for the process Y and the bounded stopping time T A r', (6.65) becomes

EQ IA, (e'' - (S' exp (a21 'f (Y1 ,o0) + 2, g (Y,o)) - k,) x

(6.66)

We again observe that Y. = 0 and substitute this into (6.66) to obtain

E Q 1 (e x p ( 2 1 f + 7 2 9 , 0) - k 2( 6 .6 7 )

exp (/A .f(Yr~!1 0) +, P2g(yry)- 2 , nj

Observe that the random variable inside of the expectation depends only on r, and Yr,
-1

So we define a function H (.,.) by

H(u,a) = -ru .(S2 exp (cr2,. f (a, 0)+a22. g(a,0))-k,)x

exp (pl f(a,0)+p2.g(a,O)- 2 +12)] (6.68)
2

Note that the ()+ notation has been removed. We may do this as long as we make sure that the quantity inside (.)+ is greater than or equal to zero. The following equivalent statements illustrate how to ensure this.

1) S2exp(a21 -f(a,O)o 22.g(a,0))-k 2 0

2)

3) a.21(acos9+Xo)+o22(asin9+yo)>lk2 =I

4) F2 *acosO+o22 .asinO+(o2Xo +o22yo) >2

5) CT21 a cos 0 + C22 a sin 0 +b _k-2

6) a(u21 cos0+ U22sin0)>_!k2-b

7) a C.12ï¿½21 -11--22 --k28) a1- aAo2
O12O2 - al 1F22

Statements 1-4 and 6 come from substituting and rearranging, 5 comes from a calculation similar to that of (6.54) and the coefficient on a in 6 is positive so we may divide both sides by it to get to 7.

Define the constant
r = max(k2 12 2 (6.69)
, = "22 max " 10"22

If k2> ,then we will only need to compute the expectation of H (rl, Y )over values of YrI which are greater than i2. If k2 < , then we will compute the expectation of H (rl, Yl) over all values of Y1 , i.e. values greater than or equal to zero. So the constant r is a lower bound for the values of Y.' for which we must take the expectation. Thus we may write (6.67) as EQ 1,11y -rl Yrl(6.70) But the joint density for the pair (,,I . I) when r1 < r2 given in [6] is

P (r edu, Y' Eda, r, 7_ a2+ro2in ( jro)ijl _0)] (6.71)
_ - exp 2 2 sin l2a u 2u yot a ) / where r ,r cos0 1. a =r - tan-' (_ 21 ) + tan' and IP is the modified
ijo sin 00 a2 a12) Bessel function of order /. Recall that

(X) 8 (x 2k! ' ] (6.72) where 17(a) = e-' t',-adt. Recall that a is simply the angle between the lines in which the process Y, starts. Thus we may express (6.70) as

T +oo
fJfH (u, a) P (r A, Y.' IE da, r, ,
0or
,( 2+<2

f r H(ua au exp 2u (6.73) S[ j "sn- I,, -- da du

This representation allows for its precise value to be calculated via standard numerical integration techniques as opposed to simulation.

This is only the first term in the expectation. Fortunately, the same

transformations enable us to use the densities in [6] to write out the expectations as Lebesgue integrals. The second expectation we must compute is E[1A2 (e -rr2".(SI 2- k,)) (6.74)

We again use the functions f and g defined in (6.49) and (6.50) so that we have

tP2) tg yy2) . (6.75)

We may then write (6.74) out as

EQ lexp A f(YT!YT),2' !TYT 2 T)J . (6.76)

We use the same method as in (6.66) so that (6.76) becomes 2 -r2
22 2 (6.77)
IA2(e - S' xp(a, f Yr , r2 + al2 9 Yr 11]

Recall that r2 =inf {s> 0: "2IW] + .22,2 =}.So 2= inf {s > 0 021;,sl + -22W2 =

=inf{s>0 0.2lf(Y1,y,2)+0.2.g y,,2)=

=inf {s >0: -21 (Y, cos0- Y2 sin0 +x,)+a22 (yt sin0 +Y2 cos0 + x0)=b} = inf {s>0: 0-21 (y1 cos0-Y 2 sin0+Xo)+o"22 (y] sin0 +Y,2 cos0 +x0)= b}
= inf {s > 0: (a-21 cos0 +o22 sin 0)YS, (6.78)

+(022 cos9--2, sin 0), + (0.21x0 +.22Y0) =/b}
= inf {s >0: (021 cosO+-22 sinO)Y' +(o-22cosO-021 sinO)Y2 +1b =b}

=inf{s >0: (021 cos0+0.22 sin0)Y' +(022 cos9-.21 sin0)Y2 = 0)

= 0 = a11021 +01l2a22.y2
inf {s > 0 " }110'22 - 07120'21

The coefficient of the Y, term is precisely the reciprocal of the slope of line 2 after the transformation. Define another constant M = 0110721 +'0.120-22 (6.79) 0110"22 -012021

so that

= infs>0:yl = 011021 +012022 .y2 L 0 11022 - 0712021 (6.80) =inf{s>0: YS, =m.Y2} Then we have

y1 M. y2 (6.81) r2 = .T2

Now we may rewrite (6.77) as

E2 Qex (A 2f(m .Y Y .gm 2 2
[rJ\2(M.~h O y2 2T
(6.82)
1 IeT2 h~~ ( My .i 2 M. yv2 2V~'

so that the random variable we are taking the expectation of depends only on r2 and y2
2

Define a function M (.,.) by

M(u, a) =exp (A1 f (m a,a) + p2 g(m a~a) - )X(.3
2 j. (6.83) (e-ru . ( ' exp(<,,l f (ma,a) + <12 ï¿½ g(m .a,a))-ki))

Then we must again find the lower bound r2 for the random variable y2 so that we may omit the (.)+ notation. Computations similar to those for r, yield

r2 = max k - 022 (6.84)

So we may express (6.82) as

E 42 JA21, > M~2, Yr (6.85) [ie2- n2M(Tj

Thus we only need the joint distribution for the pair (r2, Yr32) given r2 < r, But this is given in [61 by

y2 - < T'ï¿½= (- )
P(r, r=du,Y " l a, 12
XCos(a - 2) eIx L a2+r2.r OS(a;-r2)]X .(6.86)

J- ï¿½Cos (a i -/2) t

where = a0- . Note: If 4 is the point we get by reflecting p = ro Cin 0 about the sin sin aline y = tan (a). x, then 4:= r cos 0jï¿½. The cos( a- /2) factor appears as a result of
2 ro sin 0

projecting the process Y onto the y-axis to obtain the random variable Y2 Remark: The formula in [6] is given in polar coordinates and thus requires this additional transformation. Note that this was not necessary for the case of the first expectation. That is due to the fact that the polar coordinate radial value is identical to that of Y since Y1 was on the x-axis.

Using the density above we may express (6.82) as
E 1 1, -2 M ,Y2 )

rl A2lJyr 2IM( T2T)
T +o
=f f M(ua)P(12= dU,Y= I d,2 <1, )
or2
T+ KCos(a 'r2) a2 +r2 "cos2 (a- r/2) (6.87)
-JJM (u,a).= a a(x -=/)x ('f c2 au 2u ( COS2

[lj. sin prI ar j o ,2t~ da du

We need only compute the final expectation now. Just as with the first two terms, we will use the transformation so that we can use another density provided in [6]. Recall that we are computing

EQ (- ls (e-rT ((ST - kl )+ + (STr - k2)+ )]. (6.88) We break this up into two separate expectations

erT .E' [-I lA3 (ST- k1)+eI + e EQ [ A (ST -k2+. (6.89) We focus on the first expectation. First we remove the (.)+ notation with the use of an appropriate indicator function. This yields e-rT " EQ [ - A _14 k<,}(S j kl)] (6.90) Now we express the random variable in the expectation in terms of the transformed process Y,. The indicator function I s4_kj>0l is left as is but will be changed momentarily. This yields
EQ[expl.f(Y,Y)+ly 2.g(Y,Y2) I 2 +/i2TJx
EQl x ,-f T T 2 *9 T T2 T) X(6.91)

1AS{sI-kj0} (e-rT '(S' exp (all f (Y!, YT) + o12 g(YTY72 k) Again we simplify the notation by defining the function L (.,.) by
L(x,y)=e-r exp A.f(XY)+ P2.g(XY) 2 T _XT

2 . (6.92)
(S' exp (arl .*f (X,Y) + 12.g9(X, Y)) - kI)

Then we may write (6.91) as

E0 1A31{ o}L(YT, YT )I. (6.93) The random variable in this expectation depends only on the process YT when A3 = {-r __ T; r2 > T}. But we have this density as well from [6]. The density is given in polar coordinates as

P(r, > T,r2 > T,YT : dy)
2r K 2 2 +00 rA . i~ (rroi )d6. 4
-- exp : sin jx' sin J I I Idrd(.
Ta2T = a a j% T

Just as before, we need to examine the indicator function a little closer in order to determine the region over which we will integrate.
{S -k ,>_o}-{sex .(yl 2,)+Ul. (y2, y2))- k, )) )o}
(ST'- k,0)= ISp( all * . f(T 2 * ~ 9TT(6.95)

where In = kln as before. If we write (6.93) as a Lebesgue integral we have

a +C (6.96) =f 1BL(rcosO, rsin0)P(r, > T,r2 > T,Y EdrdO)
0 0

where B = {a f (r cos 0,r sin 0) + a. g (r cos 0, r sin 0) __1}. We may rewrite B as

B ={all (rcosO.cosO-rsinO.sinO+xo)

+0712 (rcoso sin9 +r sin cos0 +yo) 24, = {(r . a- cos9+ r.a2 sin 0)cos 0
+(r.'a2 cos09- rall sin0)sin0 +r(a1rxo +a12Yo) 2. } = {(r-a12 cosO-r-orl sin0)sin +6 >k)

= {r(ra12 cos0- 1Isin 0)sin 0 -> }

= {r~o-21 + 2 - sinoi>k1-a}

IH > a122 .sinï¿½J (6.97) Define the function q, (0\$) by q, (0)= max ,. ' (6.98)
+a1 u2 -sing 0

so that the lower bound on r for a fixed value of 0 is q, (b). The maximum between the quantity above and zero is used to ensure that r > 0 in the case when k, < a which is equivalent to k,

EQ IA31{S _-k,>o}L(YT,Y2)1

a +00 (6.99) =f f L(rcos 0,rsin0)P(r, > T,r2 > T,Y, Edrd-& )
0 q.(0)

We compute the second expectation in just the same manner. We write out the expectation in terms of the transformed process. Recall the second expectation is e-rrEQ --P IA (ST2- k2 ). (6.100)

Written out in terms of the transformed process Y, this looks like

E Q lexpf ( A f (Y+Y1?) p2 T ) T 2 T). (6.10 1)

Define the function U ( .,. ) by

U (x, y) =exp(A f(X, Y)+,P2.g9(X, Y) 2 T)Z Tx(612
~/2 T~(6.102) (e-rT . (S2 exp (a2, f (X,Y)+ 22 .9 (x,y ))- k2) Then (6. 101 ) becomes

Again the random variable depends only on the process Y, when A3 = {r T; r2 > T}. So we may express this in terms of the Lebesgue integral a +o (6.104) f f IEU(rcos#,rsino)P(r, > T,r2 > T,Y, e drd)
0 0

where E = {a2, f (r coso,rsin 0) + a22" g(rcoso, r sin #) >_ k2}. Calculations similar to those of (6.97) show that

E - (i2 - .a120'21 -'11--22 COSq + a12022 + all21 sin (6.105) UC11 + C12 Ia 1 + (12

Define the function q2 (') by

q2 (0) = max (k2-"K 12C"21 - 0'11O'22 Cos 0

2 2

(6.106)

+ a120'22 0711a2l sin , 0 01 1 +0a12

so that the lower bound on r for a fixed value of 0 is q2 (0). Then we may express (6.104) as

EQ I1A3 1{STk 42o}(YrYT )

a +co (6.107) f J U(rcoso,rsino)P(r, >T,r2 >T,Y Edrdo)
o q2(0)

This being the last necessary calculation, we may add up the quantities in (6.73), (6.87), (6.99) and (6.107) in order to obtain the price of the get-out option.

APPENDIX
EXPLICIT OPTIONS' PRICING FORMULAS The results for the partial tunnel options' prices from chapter 4 were expressed in terms of the functions h(c,d), h(c,d) and w(c,d) from chapter 3. Here we present the options' prices explicitly in terms of the bivariate normal distribution function. We first make some computations that will be used later.

Recall that u =I r - . So the quantity (ii + -)2 T becomes uk ( 2) 2

(,U + )2 T _(L2 +2p+a2 ) T
2 2 =P2T (20-J +U2) T
2 2
-- T- + 02T 2T a rj- 2 T

2 2 P2T (2r-_a2 +U2)T
2 2 T +rT
2

As a result of theorem 4.2 we have

we e r je 2 [S.(h(p+oc,)-h(p+ud,))-k.(h(pc,)-h(wdj))] (A.1)

where

h__d =_ ex_ --tj #vjT

(A.2)

c, =2(b - 6), dj = 2(i - 6) - 2b, p=jT a = c- d and 8 = cT - d. Substituting h(c, d) and simplifying yields

Vc = +XISI e-ucrj[N

-e-(p+a)dj

- e-rTk. re-PcJ

[N5

(,U + 09 - cj - i (u + -)T - cj
4TJ

-NP(/z~t)11 -cj -b ,(p+or)T-cJ-. [NP (( +a)t -d_-_ (p+cF)T-d-k

-N ,+cT)I-d-b (i+a)T-d-kjj

Cdu4-cJ--i 1uT-c
j, 'IT1

_,-ud 2v, ,fi , 4
-e ud[N. (lt, - d - ,T -d - /]]
it J[4 I I \f

l-Nj - p T - d - )

Theorem 4.4 gave us the price VP of the PTPO-I as

I= e - r .(uc,c)-L(,d j))-S.(L(+,cj)-,u+a,d ))] where
j=--

79

C ~d[,(- k- (_N -ak-fl

c, = 2(b-,, d = 2(b-)-2,, a =ct -d and 8 =cT-d. Substituting h(c, d) and simplifying yields
V
J e-rTk+-e-CNC N, .____-e-pdj[Npj , k -,ut .JN,( ,i-,T+d, - + d
-~ Ft -") , r ,
-(P+a [.I-( + ) +c, k-(u+ )T +c.

i_ -(+_)_, k-(p+)T+d,

Theorem 4.5 gave use the price Vk of the PTCO-II as , C e-rT-2 [S .(w(u +a,c,)- w(p + a, d))-k. (w(p,c,)- w(, d))]

where

W(C d =ex C2 _cd [ b +d-a b-fl 'i(a+d-a b-fl1 b+d-a S-/J' . +d-a 8-fl t- ,~ JFT N -SI 8=max(aik), c, = 2(/ -6), dj = 2(b -a)-2b, a=c1 -d and 8 =cT-d. Substituting w(c,d) and simplifying yields
C
~L~K Li~p/b+2c,-uztI b/I aT+cJJNaj2J/t -uTc)
[S. e-pa'c N, +2 , qT+ j A i+2 ,', -,' +" )
t F. "_ Ftv , ,FT
[NP b + 2c, -,ut 8 -uT cJ "( + 2c, - l't 8 -i.'T+ c,

~e-A'dj ~1N (+2d,-,'t, E-du'T+dj) Nrj+2dj-pt b u'T + di
-e k( CJ ( F~b 2 -t, ______ F____ ___F_I +2dj-,- 't, 8-t'T+dC1 2dc-jut, 8-p'T+dj
[ F- ,F j eI I [ b+2c-Ia" b-pTT+cj ) P b+2cd-Ut, -,u T + cd]

_[NtIcJI g- uT+d N - pt, djT~c,1. where /'=4i+2,+

Theorem 4.6 gave use the price VI' of the PTPO-II as
,,';:~~Nf t ,,r- ,.tu, w., - lw.+,,,-w ,.6)

where

w(c,d) = exxpC 2cd [NofFb ,+d-aI b+d-a d- - ji +d-a 6 -lJ]
NP- V Ft, VT ,N g , IT

y/= min(b,ki), c1 = 2(b-5), dj = 2(b-i)-2b, a =ct1 -d and 83 =cT-d. Substituting w(c,d) and simplifying yields

erT . (e I+2c-N> lt VI-'-T+c)} (j+2c-pt, y-it_ T+c__
[e- -Poo

_[Nb +2c,-:t _-_uT+cj

-N 6+2c

[d b+ 2dj-1tt y-pT+dy_ 6j+2d -ptj '-pT+dj /+2d, -p:t, b-pT+d,)_ 6+2d,-:t u -_T+d,_+ 2c -', V/-p'T+cj j ' +2c,-u't, - '+c,]
-S,je U [Cj+2[Nt J Np +t T

[N S+2c,- P't -PT + bN +c -pS- T+cj]

~e P dj N pt+ 2 J ~ P ' '

5j+ 2d, -pI , b-p'T+ d,

+2j-ptj ..FT~j NP

REFERENCES

[1] Baxter, Martin and Andrew Rennie, Financial Calculus, Cambridge University
Press, New York, 1996.

[2] Chuang, Chin-Shan, Joint distribution of Brownian motion and its maximum,
Statistics and Probability Letters, 28 (1996), 81-90.

[3] Duffie, Darrell, Dynamic Asset Pricing Theory, Princeton University Press,
Princeton, New Jersey, 1992.

[4] Heynen, Ronald and Harry Kat, Partial barrier options, Journal of Financial
Engineering, 3, no.3/4 (1994), 253-274.

[5] Hull, John C., Options, Futures, and Other Derivatives, Upper Saddle River
Prentice Hall, New Jersey, 1997.

[6] lyengar, Satish, Hitting lines with two-dimensional Brownian motion, SIAM Journal
of Applied Mathematics, 45, no.6 (1985), 983-989.

[7] Karatzas, loannis and Steven Shreve, Brownian Motion and Stochastic Calculus,
Springer, New York, 1997.

[8] Kunitomo, Naoto and Masayuki Ikeda, Pricing options with curved boundaries,
Mathematical Finance, 4, no. 2 (1992), 275-298.

[9] Musiela, Marek and Marek Rutkowski, Martingale Methods in Financial
Modelling, Springer, Berlin, 1998.

[10] Revuz, Daniel and MartinYor, Continuous Martingales and Brownian Motion,
Springer, New York, 1991.

[11] Rudin, Walter, Principles of Mathematical Analysis, McGraw Hill, Inc., New York,
1976.

[12] Shepp, Larry, Joint density of the maximum and its location for a Wiener process
with drift, Journal of Applied Probability, 16, (1979), 423-427.

BIOGRAPHICAL SKETCH

David Brask was born in St. Louis, Missouri. He moved to south Florida where he attended school from fourth through twelfth grade. After graduating from Deerfield Beach High School in 1991 he went to the University of Florida. He was admitted into graduate school early via the Mathematics Department's accelerated master's degree program. David received his bachelor's degree in 1995 for a double major in mathematics and statistics. The following year he completed his master's degree in mathematics with a specialization in applied mathematics. After studying image compression and processing for about one year, his research moved into the field of mathematical finance. Soon after changing focus, David had an internship at William R. Hough & Co., an investment banking firm, to gain experience in the financial field. He was then certain that this was the area in which he wished to do his research.

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philo phy.

( Jos ph ( over, Chairman
Pr essor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

Murali Rao
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philophy.

David Wilson
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philoshy.

Wa Horo'w(zso
Graduate Research Profess of Decision and Information Sciences

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

Mahendrarajah Nimalendran Associate Professor of Finance, Insurance and Real Estate

This dissertation was submitted to the Graduate Faculty of the Department of
Mathematics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy.

December 1999

Full Text

PAGE 1

ARBITRAGE PRICING OF SEVERAL NEW EXOTIC OPTIONS: THE PARTIAL TUNNEL AND GET-OUT OPTIONS By DAVID AARON BRASK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1999

PAGE 2

To my brother Â•f"' *> '*J }

PAGE 3

ACKNOWLEDGMENTS I would especially like to thank my family for their unconditional love and support. I thank my advisor for introducing me to this field and guiding me through my research, each of my committee members for their input and all of my friends who reminded me that social life is still a big part of life even when you are working on a Ph.D. I would also like to express my appreciation to Jim Cohen at William R. Hough & Co. for the internship and consulting opportunities through which I have gained invaluable experience.

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TABLE OF CONTENTS page ACKNOWLEDGMENTS iii ABSTRACT vi CHAPTERS 1 INTRODUCTION 1 1 . 1 Background 1 1.2 Arbitrage Pricing 2 1 .3 The Discrete Case 2 1.3.1 The Model 2 1.3.2 The Martingale Measure 4 1.4 The Continuous Case 6 1.4.1 The Black-Scholes Model 6 1.4.2 The Martingale Measure 7 1.4.3 An Example 8 2 RECENT RESULTS 9 2.1 Barrier Options 9 2.2 Partial Barrier Options 1 1 2.3 Double Barrier Options 12 3 SOME NEW RESULTS ON BROWNIAN MOTION 14 4 PARTIAL TUNNEL OPTIONS 27 4. 1 Description 27 4.2 Removing the Drift 27 4.3 Partial Tunnel Options of Type I 29 4.3 1 Partial Tunnel Call Option of Type 1 29 4.3.2 Partial Tunnel Put Option of Type 1 31 4.4 Partial Tunnel Options of Type II 33 4.4. 1 Partial Tunnel Call Option of Type II 33 4.4.2 Partial Tunnel Put Option of Type II 34 IV

PAGE 5

5 ANALYSIS OF THE PARTIAL TUNNEL OPTIONS PRICING FORMULAS 36 5. 1 Partial Tunnel Options Pricing Formulas as Extensions to Existing Formulas 37 5.1.1 Limits of Partial Tunnel Options Prices as Expectations 37 5.1.2 Limits of Partial Tunnel Options Formulas 38 5.2 Numerical Results 42 5.2. 1 The Partial Tunnel Call Option: Type I 42 5.2.2 The Partial Turmel Call Option: Type II 45 6 THE GET-OUT OPTION 48 6. 1 Description 48 6.2 The Multidimensional Black-Scholes Setup 48 6.2.1 The Model 48 6.2.2 The Martingale Measure 49 6.2.3 The Payoff 50 6.3 Pricing 51 6.3.1 The Independent Case 51 6.3.2 The Dependent Case 55 APPENDIX: EXPLICIT OPTIONS' PRICING FORMULAS 77 REFERENCES 82 BIOGRAPHICAL SKETCH 83 V

PAGE 6

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy ARBITRAGE PRICING OF SEVERAL NEW EXOTIC OPTIONS: THE PARTIAL TUNNEL AND GET-OUT OPTIONS By DAVID BRASK December 1999 Chairman: Dr. Joseph Glover Major Department: Mathematics We begin with an introduction to the classic Black-Scholes option pricing model in chapter 1 . This explains the method by which stochastic processes are used to obtain arbitrage prices for options. Chapter 2 follows with recent results in the field and touches on how some of these results will be extended in chapter 4. Chapter 3 provides some new results on Brownian motion, which are of great use in chapter 4. Chapter 4 begins by introducing the first new class of options, the partial barrier tunnel options. It then proceeds to price these options using the results fiÂ’om chapter 3. Chapter 5 is a follow-up of chapter 4. It first shows how the results of chapter 4 generalize those in the existing literature and then provides numerical results and an analysis that illustrates how changing the parameters affects the price. VI

PAGE 7

Chapter 6 introduces another new option, the Get-Out option. This option depends on two underlying securities whereas all of the aforementioned options depend only on one. The goal again is to derive an expression for the arbitrage price of this option. We separate the pricing into two cases. The first case assumes that the two underlying securities are independent and therefore do not require joint distributions for the processes. The second case, however, assumes that they are correlated. Consequently, the pricing is not as straightforward and requires joint distributions of the processes and some of their functionals. vii

PAGE 8

CHAPTER 1 INTRODUCTION 1.1 Background The financial markets are growing more rapidly now than ever before. Investors at all levels are using financial products to hedge their positions, that is, to help reduce their risk. As a result, the derivatives market has grown as well. One of the most basic and earliest financial instruments used for hedging is the standard European call option on stock. This call option gives its owner the right, but not the obligation, to purchase a security at a prespecified price and time in the future. The prespecified price is called the strike price and the prespecified time is the expiry. A put option is similar in nature but gives the holder the right to sell instead of buy. American options are very similar to European options but they allow the owner to exercise the option at any point up until expiry, whereas European options may only be exercised at expiry. From here on we will work with only European options. Suppose one holds a long position in a particular stock. Purchasing a put option on the same stock would guarantee this investor the ability to sell the stock at the strike price. The downside loss would then be reduced. Determining prices for options similar to these but with additional stipulations is the point of this work. 1

PAGE 9

2 1 .2 Arbitrage Pricing One can see that the price of a standard call option must be less than or equal to the current price of the underlying stock. If this were not the case, one could sell the call option and buy the stock. Since the option costs more, this investor would also have some cash left over. Furthermore, at expiration only two things can occur. If the call option were exercised, he would only be left with the cash left from the original transaction. If it is not exercised, then he could simply sell the stock back into the market and have even more money. In particular, if the call option is priced higher than the stock, then there are opportunities to make money without any initial investment or risk. Thus, if one were to purchase a call option, he would not pay more than the current stock price. Arguments similar to this dictate ranges in which prices must fall. Definition 1.2.1 An arbitrage opportunity is a situation in which one may make a profit with no initial investment and without taking any risk. If two exchanges carry the same security but at different prices, then an individual could simultaneously purchase the security at the lower price and sell at the higher price. This is an example of an arbitrage opportunity. Next we will see how arbitrage can be used to price a call option. 1.3 The Discrete Case 1.3.1 The Model We begin with the simplest model available for a stock, the single step binomial model. This model assumes that a stock either goes up or down by certain percentages over the lifetime of the option [1]. For simplicity we will assume that we have zero interest

PAGE 10

3 rates here. Denote by Sq,S^,Sj and it the initial price, the price if the stock goes up, the price if it goes down and the strike price. Figure 1.3: Discrete model If the stock goes up, then the value of the call option is -A:)" =max(^Â„ -A:,0). (1.1) If it goes down then it is worth ky = max(5^ -k,0). (1.2) One could also choose to create a portfolio of the stock and a cash bond valued at 1 dollar so that the net worth of the portfolio would be the same as the value of the call option at expiry. To see how to do this we start with x units of stock and y units of bond. Depending upon whether the stock goes up or down, x and y must satisfy the equations xS, +>;1 -ky (13) in order to replicate the payoff of the call option. We can solve this system for x and y so that the portfolio has the same terminal value as the call option. The cost of composing

PAGE 11

4 such a portfolio would then have to be the appropriate price for the call option. Any other price would lead to an arbitrage opportunity. Derinition 1.3.1 The arbitrage price for an option is the value of the portfolio that replicates the payoff of the option. 1.3.2 The Martingale Measure Notice that we did not use any probability in the preceding arbitrage arguments. However, pricing the call option in this manner does indeed induce a probability measure. Let C be the price of the call option and Sj be the price of the stock at expiry. We would like a probability measure P under which C is the expected value of the possible outcomes. That is, P must satisfy (1.4) where /?, and p.^. the probabilities of the stock going up = 5Â„) or down {St =Sj) under P Since /?, and p^ must also satisfy /?, + = 1 , it is clear that there is a unique solution for /?, and . Moreover, the relationship [Sj -kf
PAGE 12

5 Definition 1.3,2 The filtration generated by the discrete process S = {5, is the collection of a algebras f-a (5^ : j < ij generated by the process. Definition 1.3.3 A discrete process S = {5, is a martingale with respect to a measure P and its filtration if Â£Â’|5,| < oo for all i and = 5", for all i< j. Under this measure, the stock price turns out to be a martingale. In general, the interest rates are not equal to zero. When interest rates are nonzero, we must adjust for the time value of money. In particular, receiving x dollars at a future time t is equivalent to purchasing e~''Â‘x worth of a bond with interest rate r now. Note that invested in this bond will payoff e"' = x dollars at time t . So the value of receiving x dollars at a future time t is worth e~'Â’'x dollars now. This is what we call a discounting factor. So in the case of nonzero interest rates, the discounted stock price process, e~Â’^^'S, , will be a martingale under this measure. Definition 1.3.4 A martingale measure is a measure under which the discounted stock price is a martingale. Once we have a martingale measure, we may price other options by simply taking the expected value of their discounted payoff under this measure. Theorem 1.3.4 The arbitrage price C for an option is given by C = EÂ’^ [X] where P is the martingale measure and X is the discounted payoff of the option.

PAGE 13

6 We omit the proof of this theorem as it may be found in [9], This reduces the problem of pricing options to finding the martingale measure. Once the martingale measure is found, we may simply take the expected value of the discounted payoff to obtain the arbitrage price of any option. 1.4 The Continuous Case 1 .4. 1 The Black-Scholes Model We now move to a better model that allows for more than just two possible outcomes for the stock price. In fact, this model allows the stock price to take any positive value. A stochastic process is used to model the stock price and an exponential function (deterministic) is used for the bond price [1]. The Black-Scholes model for options pricing is the most widely accepted and used model in the field [3]. Despite some of its unrealistic assumptions such as continuous time trading and no transaction costs, the model has persisted in both academia and industry since its birth in 1973. We first fix a probability space on which we define a standard one-dimensional Brownian motion W, . Denote by T, =
PAGE 14

7 option, there is a payoff that is dependent upon what the underlying security does by the expiry time T . Definition 1.4.1 A contingent claim X is an J-j measurable random variable. A contingent claim is used to describe the payoff of an option. Thus the contingent claim for a standard European call option is X = (5;. e . 1.4,2 The Martingale Measure Just as in the discrete case, we need only find a martingale measure under which we can compute option prices by simply taking the expected values. Though not as simple as solving a system of linear equations as in the discrete case, we are guaranteed a martingale measure by the well-known Cameron-Martin-Girsanov theorem [7]. Under this measure the discounted stock price process is a martingale. In other words, if we let g ^ be the discounted stock price process, then we have dZ, = oZdW. where W is ' ' ' a Brownian motion under the martingale measure. Thus we may price options by taking their expected value under this measure. Since this measure always exists in our setting, we will assume that the measure P we use to price options is the martingale measure. Furthermore, we will drop the W, notation and use as a Brownian motion under the martingale measure P An important consequence of using the martingale measure is that the original drift term fi drops out. In order for the discounted stock price process to be a martingale, the drift term in the process must be r Â— [7]. Since the original drift term is irrelevant, we choose to start with the martingale measure and avoid it altogether.

PAGE 15

8 1.4.3 An Example Take a standard European call option on a stock with volatility cr , drift ^ . Let k and T be the strike price and expiry, respectively. Let P be the martingale measure. Under this measure we have fits, = T 1 r 1 f In Â— 2 . . k J + f 1 2^ r--a 1 2 J N T crVr V j ayff ^ J ( 1 . 8 ) where N (Â•) is the normal cumulative distribution function (i.e. N(x) = P(X < x) where A' is a standard normal random variable). Again notice that the drift term n does not appear in this formula.

PAGE 16

CHAPTER 2 RECENT RESULTS 2.1 Barrier Options One of the first modifications that can be made to the basic call or put option is the see whether or not the underlying stock hits the barrier during the course of the option. These new options, called barrier options, come in two types: knock-out and knock-in [9]. A knock-out barrier option would be the same as a regular option, but with the added stipulation that the stock price must not hit the barrier. If the stock price hits the barrier during the course of the option, then the option automatically expires with zero payoff. On the other hand, a knock-in option has zero payoff by default. The optionÂ’s payoff takes place only if the stock price hits the barrier during the lifetime of the option. For example, the payoff of a knock-out barrier call option could be of the form addition of a barrier. A barrier is a prespecified price level, which is monitored in order to h 0 T Figure 2. 1 : Barrier option (2.1) 9

PAGE 17

10 where h is the barrier level. In this example the barrier is placed below the initial stock price, that is h
PAGE 18

11 2,2 Partial Barrier Options A further generalization of the barrier option is the partial barrier [4], The only difference between a partial barrier option and a regular barrier option is that the barrier is not monitored for the entire lifetime of the option. The monitoring period would either start at time zero and end at some time before expiry or it would begin at some time before expiry and end at the expiry. Notice that for the latter case, the boundary could be hit from above or below. A h0 Figure 2.2: Partial barrier option These two cases are distinguished and priced separately in [4]. The payoff for a partial barrier option where the monitoring period starts at time zero is ^{S,^AV/Â€[0,/, ]} (2.4) where
PAGE 19

12 approach for several of the computations above is found in [2], The joint density for the pair ) where 5 r is derived and then used to price a knock-out call option where the partial barrier begins at time zero. Not surprisingly, the joint density is a linear combination of bivariate and univariate normal density functions. Partial barrier options may be of use both individually or when packaged with other options. The nature of this option allows one to compensate for changes in volatility over the lifetime of the option. One may expect moderate volatility in the near future but increased volatility soon after. In this case, one could choose a partial barrier option with the barrier only monitored in the near future. This would serve to decrease the cost of the option and still serve to hedge or speculate an underlying security. One may also package barrier and partial barrier options together to create ladder options [4]. A ladder option is just like a regular call or put option except for that it locks in intrinsic value when certain barriers are reached. 2.3 Double Barrier Options Another modification made to the standard barrier option is the addition of a second barrier. Naturally, the two barriers are on opposite sides of the stock price. Although standard and partial barrier options use flat barriers, double barrier options where the barriers may be curved are treated in [8]. In order to price such an option, the density must reflect that the stock has not hit either of the barriers. They use an extension of LevyÂ’s density for Brownian motion confined to an interval [a,b] (i.e. a Brownian motion for which a
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13 reflection principle of Brownian motion and is written as an infinite sum [10], The density is {x+2j{b-a)f It -e (x-v2j{b-a)-2b) 2t 2 N dx (2.5) where A < m, < M, < b,B, g 1 c [a, i]} , Their results indicate that the convergence of this series is sufficiently rapid to make it amenable to numerical implementations. Despite the generality of their results with curved boundaries, we will confine ourselves to double barrier options in which the barriers are flat. In what follows, we will refer to these types of barrier options as tunnel options. Just like other barrier options, double barrier options may be used to speculate on the perceived volatility for an asset. There are many different ways to compute volatility, none of which are universally accepted [5]. Consider an investor who expects the volatility to be less from implied volatility of a given call option. He could sell the call option and purchase the same option with an appropriate double barrier. If he is right, then both options will be worth the same at expiry and will thus leave him with no obligations. He will profit by the amount equal to the difference in the optionsÂ’ prices. Double barrier options have been developed and traded by several Tokyo banks including Fuji Bank, Sanwa Bank and Nippon Credit Bank [8].

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CHAPTERS SOME NEW RESULTS ON BROWNIAN MOTION This chapter presents some new results on several functionals of Brownian motion. These results will be of great use in chapter 4. The corollaries following the main theorems provide densities for a Brownian motion at time T confined to a tunnel during a fixed period. The fixed period will either start at time zero and end at some time t before time T or will begin at some time t and end at time T . We use several lemmas to conceal tedious computations and obstacles in the proofs of main theorems. Lemma 3.1 Fix the values x,,X 2 andy, and define the function h{c,d) by \2 A h{c,d) = 1 / f f exp cy(x + df (y x) 2t 2{T-t) dydx (3.1) Then h{c,d) = exp ( c^T cd j P \ P-yx It Â’ Vr \ V { ^-^2 P-yx '^ ^ /t Â’ 4f (3.2) where (Â•, Â•) denotes the bivariate normal distribution function with correlation coefficient p , a = ct-d and f -cT -d . (x-^-df Â—Y Proof: Let f{x,y) = cy ^ . We first factor out the quantity 2/ 2(7-0 2 ( 7-0 from / so that we have 14

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15 f(x,y) = -T 2(7-0 2(7-0 , 2(T-t){y-xf ^ 2{T-t){x + df T ^ T 2(7-0 7 2t Â• (3.3) Expansion, cancellation, grouping like terms and simplification yields -7 2(7 1) x^+2d ^-7 Tj 2xy ^ y -2c(T-t)y 7 T 2t -T7(3-4) We would now like to express the quantity r X + 2d 1-X V 2x;^ ^ y -2c(J-t)y T T (3.5) in the form {x-af 2(x-a)(y-P) , {y-pf , h h Y . t T T (3.6) In order to solve for the constants a, P and y we first equate the coefficients of x and y terms from (3.5) and (3.6). This leads to the system of equations 2d u t 2a 2P t T -2c{T-t) ^ 2p ^2a 7 ~ T T (3.7) Solving this system yields a -ct-d p^cT-d' (3.8) We next set the constants from both expressions equal and obtain a" 2ap p^ -y = Â— + 7Â— . / 7 7 (3.9)

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16 So we may now rewrite (3.4) as -T 2{T~t) {x-af 2{x-a\y-P) {y-pf , _ h h V t T T (3.10) 2t Â’ where a, >5 and/ are defined as in (3.8) and (3.9). We take the / term outside of the brackets so that 2{T 1) (,x-a) 2(x-a)(y-p) ^ jy-pf t T T -T (f H r 2{T-ty 2t (3.11) -T dP We now simplify the term Â— Â— Â— -/ . Substituting the values for a and p from (3.8) 2 ( i /) 2 / into (3.9) for y , the above expression becomes -T P -y-Â— = J f 0^/3 /?2 A Â„2 2 ( 7-0 2 / 2 ( 7-0 7 ay _ 2 ^ ^ p^ c Tt 2 ( 7-0 {ct-df 2{ct-d){cT-d) jcTd) t T 7 2 A 2t (3.12) After expanding and simplifying (3.12) we have -7 d^ PT -y = cd . 2 ( 7-0 2 (3.13) Substituting (3.13) into (3.11) yields fix,y) = -7 2 ( 7-0 {x-af 2{x-a){y-p) ^ {y~pf t T T Pt , ^ cd Set a -\ft, a \[f and /?=./Â— . Then we have Â— = V7 (3.14) Â— and P ^ 2 ( 7-0 T so that (3.14) becomes

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17 f{x,y) = -1 (x-g) 2p{x a){y p) ^ {y~pf O'." o-,or^ o-/ c'r . H cd . (3.15) Recall that +00 f ix+df J J Â•ti y, ^y 2t 2(T-t) J dydx . Thus we have 1 .J +00 6Y 6 c 27ryjt(T-t) nyh. JJ {x-af _ 2p{x-aXy-p) _^ jy-pf a/ CT^y CT Â„^ (3.16) dydx R y Making the substitutions x = , y = Â— Â— o. and switching the limits of integration yields (3.2). Lemma 3.2 The density k{x) = 1 \llKt ;=-cc exp (x + 2j{b a)f -exp {x + 2j{b-a)-2bfy 2t 2t . is uniformly convergent on the interval [a, b] . Proof: If a series of the form Z exp oo (x + 2j{b a)f 2t (3.17) converges uniformly on the interval [a,b], then a series of the form +00 Z e^P j Â— Â® {x + 2j{b-a)-2by 2t (3.18)

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18 will also converge uniformly on the same interval. To see this, first recall that a <0 -2t for large positive values of j and {x + 2j{b-a)f ^ {x + 2j{b-a)-2bf 2t ^ 2t for large negative values of j. So the convergence of (3.17) ensures that (3.18) will also converge. Thus we need only show that a series of the form (3.17) converges uniformly on the interval [a,Z>] . However, this is equivalent to the one sided series exp (x + 2j{_b a)f 2t (3.19) converging uniformly on the same interval. Now we choose an integer N large enough that j >N implies x + 2 j(b -a)> 0. We then observe that for j > N we have exp (x + 2jjb a)f 2t exp {a^2j{b-a)f 2t Vxe[a,Z>]. (3.20) Now the series exp (g + 2j{b a)f 2t (3.21) clearly converges. If we define r MÂ„ = sup jce(a,fc] exp 7=1 (x + 2y(Z>-g)) 2t (3.22) then we have

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19 f lim = lim sup Â„-*+C0 "-*Â«> ;ce[a,i] 2 ; exp ]-n (o + 2y(6 a)) 2 lA 2 / Jy < lim nÂ—^+oo ^exp y-n {a + 2j{b a) ) It 2 lA (3.23) = 0 The inequality in (3.23) is due to (3.20). The limit goes to zero since the series converges. (3.23) is a necessary and sufficient condition for the uniform convergence of (3. 19) [11]. This proves the lemma. Theorem 3.3 Let A = {a >'j } where t } so that A = A^r^A 2 Then we have e^^ 1 M = E A, 1 (3.25) Using a simple property of expectations, we take an inner expectation conditioned on so that (3.25) becomes E E (3.26) We now note that the random variable 1^^ is .7^ -measurable. Thus we can take 1^ outside of the inner conditional expectation so that (3.26) becomes Ia E ,XBt (3.27)

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20 Next we rewrite (3.27) using the shift operator 6, and the Markov property of Brownian motion. This yields = E 1. E e^\^ \-e, e^^-Â‘ 1 (3.28) where A 2 = {Bj_, > y'j } and E^ [Â•] denotes the expectation for the process started at z instead of the origin. We may compute the inner expectation as which is a function of B, . So we define the Sanction / by fix) = E and write (3.28) as 2{T-t) dy, (3.29) j +00 2 ( 7-0 dy (3.30) ^A,m) Recall that /4, = [a } . So we compute (3.3 1) as where k{x) = flnt (x + 2j{b a)f [x + 2j{b-a)-2b)'^ exp 2t -exp 2t (3.31) (3.32) (3.33) Set Cj = 2 j{b a) and dj 2 j{b -a)-2b . Then we may express (3.32) as

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21 J 1 +00 exp It exp 2t f{x)dx. (3.34) Substituting /(x) back into (3.34) and pulling the constants to the front of the summation we obtain 1 1 2n^t{T-t) jTL exp 2t exp (x + c/j' 2t \j exp(^>') Â• exp f ( \2 ^ (3.35) dydx 2{T-t) where / = [>^i,oo) . Since /(x) is bounded on the interval [a,b], we use lemma 3.2 to interchange the infinite limit with the outer integral. We may also combine the integrands to obtain J CO = 11 . 27i^t(T -t) exp 2t -exp 2 2t r r exp(A>') Â• exp V ilzA. HT-t) 2 dydx JJ (3.36) Distributing the exponentials on the right yields I b \ !L a exp 2t 2(7-0 exp 2t 2 ( 7-0 We now have a doubly infinite series where the terms are of the form dydx (3.37) h{X,Cj)-h(A,dj) (3.38)

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22 where /?(Â•, ) is defined as in (3.2). We use lemma 3.1 to complete the proof. +00 Corollary 3.4 P{a y^)= (h{Q,Cj)-h{Q,dj)) for t < T . Proof: This is immediate from theorem 3.3. Again we fix the constants Xj, Xj and>', and define a new function h(c,d) by 1 -*2 Â•*'1 Hc,d) = , f j exp c>^2 > 2t 2(7 -0 dydx. (3.39) Remark; This function differs from h{c,d) only by the limits of integration on they variable. Lemma 3.5 The function h{c,d) may be expressed as h(c,d) = exp ( 7 \ y cd 1 2 J P \ where p 17 = a=c,~ 7 ? ' 77 d and f -cT -d . ^2-Â« f^i-Â« y\-P^ 'Â’p y 4t Â’ Vr y (3.40) Proof: This proof is the same as the proof of lemma 3.1 except for making the last substitutions _ x-a . y-f X = and y Â— Â—
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23 2 -tj >2 = 1 f f exp 2n4I(T^){{ 2 A cy{x-^df (>'-y) It 2{T-t) dydx. (3.43) Then g{c,d) = ex^ ( c'^T r f f cd J LI [ % x^-a y^-p VF Â’ VF , -NÂ„ NÂ„ X2-a y,-p , VF Â’ Vf ^ r NÂ„ Xj-a y2~P >/F Â’ VF , ^1 -g Ti -P Â’ VF , (3.44) where (Â•, Â•) denotes the bivariate normal distribution function with correlation coefficient p , a = ct-d and f = cT -d . Proof: This proof is immediate since g{c,d) is the difference of two functions having the , Â‘2 >Â’2 form h{c,d) or h{c,d) In particular, any integral of the form 1 1 {^(fydx may be *1 yi ^2 >2 written as J" J ( )dydx -j j" (^-^dydx so that the right hand side is the difference of two functions of the form h(c,d) . Theorem 3.7 Define w{c,d) by w{c,d) = exp TV, cd \ '' xj +d-a f Xj +d-a y^-p^ V VF Â’ VF y V VF Â’ VF y X 2 +d-a 2izZl_Ar ( +d-a y,~P VF Â’ VF VF Â’ VF (3.45)

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24 andsetB-{a -a) -2b. Proof: Just as in theorem 3.3 we use basic properties of conditional expectation and the Markov property of Brownian motion to obtain = Â£ = E (3.47) where B {a < B^ < 6 Vw g [0, 7 t],Bj_, e /} . We may compute the inner expectation as (3.48) where k{x) 1 +00 (x + 2j(b a)f exp {x + 2jib-a)-2bfl ^27T(T-t)jÂ±L 2(7-0 2(7-0 (3.49) Define the function / (x) by f{x) = E'' e^^-' = ^jk{y x)Qx^[Xy)dy . (3.50) We may then compute the entire expectation from (3.47) as

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25 Â£[Â£''[Â£-Â‘Â«'-.1j]] = Â£[/(5,)] We now substitute the function f{x) into (3.51) to obtain 1 * 1 1 / ~ (^y) ^ and the density A:(x) into (3.52) which becomes vÂ’2^ dx (3.51) v'20 dx (3.52) II; yj2n{T 1) J exp 2 A 2(7Â’-/) &x\)[Xy)dy exp exp 2(7Â’-/) 2 ^ (3.53) vÂ’2^ dx Now / (x) is bounded so we may use the uniform convergence of the series to move the summation outside of the integral signs. After simplifying (3.53) becomes [y-x + c^f x^ ,._oo 27iyjt{T 6 f Jl, a exp V Ay-2(7Â’-/) 2/ exp Ay2 (y-x + dj) 2(7Â’-/) 2/ (3.54) dydx Define w{c,d) = 2nyjt{T -t) JC, Vj I Jexp ^i y> x^ (y-x + d) cy ^ 2 ^ 2/ 2(7Â’ /) dydx (3.55) and recall that g(c,^7) = 1 yi (x + ^)^ 2n^tiT1 ) J J Â®"^P 'i >Â’i 2/ 2(7Â’-/) J ^7va!)c. (3.56)

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26 Notice that we could translate the x variable in w(c,d) and rewrite (3.55) as 2 ^2-Â‘i yi w(c,d) = f fexp cy{x + df (>^-x) 2 A 2t 2 ( 7-0 cfydx . (3.57) We appeal to lemma 3.4 to see that w(c,d) may be written as w{c,d) = exp cd -d-a -p /t Â’ Vr V y ^1 -d-a y^ -p yft Â’ Vr jj V X 2 -d-a yi~P]_j^ (x^-d-a y^-p^ Â‘''p ' Vr V .Tt Â’ ^/7 y (3.58) where p Jy , a = ct-d and P -cT -d . But then w(c,d) = w(c,d) so (3.54) becomes +
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CHAPTER 4 PARTIAL TUNNEL OPTIONS 4,1 Description A partial tunnel option is precisely what its name indicates. An option of this type does not monitor the barrier for the entire life of the option. Just as with a partial barrier option, the monitoring period either starts at time zero and ends at a certain time before expiry or starts at a certain non-initial time /j and ends at expiry. We distinguish these two cases as Type I and Type II options, respectively. This chapter will give explicit pricing formulas for call and put options of both Type I and Type II in terms of cumulative bivariate distribution functions. 4.2 Removing the Drift Recall from chapter 1 that the price of an option is simply the expectation of its discounted payoff with respect to the martingale measure P . The payoffs from each of the partial tunnel options depend on the distribution of the stock price process S,=Se aW,A r-(4.1) where W, is a Brownian motion under P . All of the densities we have derived in chapter 3 and most that are found in the literature are based on Brownian motion with no drift term. However, the Brownian motion that appears in the stock price process is always coupled with a drift term. Observe that 27

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28 ^ G^ ( 1 r ^ 2 t = G w,+Â— 1 r 2 t ) C7fV,+ In particular, the process we are dealing with is Brownian motion with drift Â— a (4.2) r In order to use densities for different functionals of Brownian motion we must remove the drift term. This can readily be done using the Cameron-Martin-Girsanov theorem as we did in chapter 1. We implicitly define the probability measure Q , under which the process W^=W,+a rt (4.3) is a Brownian motion, via the Radon-Nikodym derivative dP ^ ^ ^ 2 = exp|^//(lT7= exp nWj Â— fi^T (4.4) where 1 G r (4.5) Observe that the quantity n defined in (4.5) is not the original drift for the stock price process as in chapter 1 . Here fx is the drift of the stock price process under the martingale measure P . Now the arbitrage price for an option is found by taking the expectation under the martingale measure P . The expectation under the measure P is not equal to the expectation under the measure Q in general. However, the Radon-

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29 Nikodym derivative serves to relate the two expectations. In particular, let X . Then we have [X] = E9 (4.6) Thus the arbitrage price for an option may be computed as an expectation under Q of the discounted payoff multiplied by the Radon-Nikodym derivative in (4.4). In this manner we will be able to use the densities for Brownian motion with no drift. 4.3 Partial Tunnel Options of Type I 4 3.1 Partial Tunnel Call Option of Type I A partial tunnel call option of type I (PTCO-I) has the payoff function where k is the strike price and the barriers are a and b . Thus the arbitrage price of a PTCO-I is given by = E^ dP (4.8) where the second equality is from (4.6). Under the new measure Q we have S, = S e = Se aWi+ (4.9) aiV, Define A-{a< S^
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30 In where a = Â— In s and b Â— . Substituting from (4.4) and A from (4.10) we (T dQ may express the price from (4.8) as exp f ~ \ ^ ^-rT ^a0}={Wj. >yt} = \Wr^[k,^)] exp pW,-^p^T 2 y Â• L Vue[0,/,]| Next we break this up into two separate expectations so that (4.13) Â• (4.14) ( 1 N, exp pW, p^T \ 2 -E'^ f exp pW, \ Â• 1 \a
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31 V S-E^ Â•1 Vwe[0,/,].Pr7.e[i^,oo)| -k-EP This proves the lemma. Theorem 4.2 The price of the PTCO-I is given by (4.16) +00 Â—rTÂ—Â—iTT 2 \^s\h{p + a,Cj)-h(,p + a,dj))-k-[h{p,Cj)-h(,p,dj))'^ (4.17) /Â— 00 where h{c,d) = exp f c^T ,] N a-b /3-k -N 1 Â•Q 1 1 2 J p Â•'''p (4.18) Cj 2{b -d), dj = 2{b -d)-2b , p = Jy , a = ctj -d and f-cT-d Proof: From lemma 4. 1 we have V Â•1 -k-EP a < Wy^b Vu6[0,/|],^rj.e[^,Â«)| VMe[0,/,],#j.e[^,oo)| (4.19) Applying theorem 3.3 to each of the expectations completes the proof The price of the PTCO-I, as well as the price of each of the other partial tunnel options, is written out explicitly in the appendix. 4.3.2 Partial Tunnel Put Option of Type I Computations similar to those above can be used to show that the arbitrage price Vf for the partial tunnel put option of type I (PTPO-I) can be expressed as

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32 -rT-\n^T ^ Vf=e k-E^ ^a Â—rTÂ—Â—jET r / -. ^ \j^\Kf^^^j)-h{p,dj)^-S-(h{p + 'i]|. Using (4.24) to evaluate (4.23) completes the proof

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33 Remark: The only differences between the valuation of the PTCO-I and the PTPO-I are the payoffs and the limits of integration. This is what motivated the introduction of the function h{c,d). 4.4 Partial Tunnel Options: Type II 4 4. 1 Partial Tunnel Call Option of Type II The same type of computations can be used to show that the arbitrage price for the partial tunnel call option of type II (PTCO-II) can be expressed as S-E^ VÂ«6[r, j'],irj.e[fc,cxD)| -k-E^ eP^T . I (4.25) Jy Theorem 4.5 The price of the PTCO-II is given by ^ -rT-^fpT = Z ^ ^ \s [Hp + ^.Cj)~w{p + G,dj))-k-{w{p,Cj)-w{p,dj))'^{A 26 ) where w{c,d) = exp r cd Nn 2 p V ^ y V L V b +d-a bp IT'^ \ f ~ a + d-a bp E Â’~JT, b+d-a S~P ylh Â’~VF .X .(4.27) -NÂ„ d+d-a 5p yjr,'y[f 5 = max(a,^), c^ -2(b-d), dj = 2(b -d)-2b , a = ct^ -d and p = cT-d .

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34 Proof: We have -k-EP Â•1 Vm 6[/, jÂ’],fTj.e[it,ao)| (4.28) We do not know whether k
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35 w{c,d) exp -cd b +d -a y/ P p VL V
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CHAPTER 5 ANALYSIS OF THE PARTIAL TUNNEL OPTIONS PRICING FORMULAS The first part of this chapter shows how the partial tunnel options pricing formulas generalize those of the existing options pricing formulas. Two methods are used, both of which are straightforward limiting procedures. In particular, we take the limit of a partial tunnel pricing formula as the upper and lower barriers go to infinity and zero, respectively. We also fix the upper and lower barriers and take the limit as the monitoring time goes to zero. In the case of Type I partial tunnel options, both of these limits turn out to be the pricing formulas for the standard options. However, a subtlety arises in the latter limit for partial tunnel options of Type II. We first revert back to the original expressions for the options prices, as expectations of payoffs. This will greatly simplify the task as we take limits. Second, we take the limit of the new pricing formula given in chapter 4 as an infinite series. Although this requires much more detail and cumbersome analysis, we will see the BlackScholes formula emerge as the limit of the PTCO-I price as the monitoring period /j approaches zero. The second part of the chapter provides numerical results and explains the patterns found in the results. 36

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37 5, 1 Partial Tunnel Options Pricing Formulas as Extensions of Existing Formulas 5.11 Limits of Partial Tunnel Options Prices as Expectations Recall that the arbitrage price of a PTCO-I is given by ^-rT (5.1) Clearly we have < (5^ Arf Moreover, and (5.4) Since these limits are increasing and ^{a
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38 upper barrier goes to cc . Since the limits can be taken in either order, we also see that the partial tunnel option price converges to that of a partial barrier up and out call as the lower barrier goes to 0. Another way to see (5.6) is to take the limit as the length of the monitoring period, /j , goes to zero. In particular, we have =('^7-"^) Â• (5-7) Since this is also an increasing limit and we have (5.5), we may again apply the Monotone Convergence Theorem to obtain (5.6). Remark: Clearly we could repeat the same analysis for the PTPO-I. However, in the case of the Type 11 partial tunnel options, taking the limit as t^ approaches T does not converge to the price of a standard option. In this case the payoffs are always bounded. This is due to the fact that all of the terms in the limit are uniformly bounded. No matter how close /, is to T , the barriers still exist at time T and therefore impose a bound on the payoffs for call and put options. They do however converge to the prices for call-like and put-like options whose payoffs are given by X = 1^^^^ [Sj k)* and 5 . 1 .2 Limits of Partial Tunnel Options Formulas In this section we will focus on the PTCO-I. In particular, we will take the limit of the pricing formula obtained in chapter 4 as the length of the monitoring period /j approaches zero. The formula is expressed as a doubly infinite series. We will see that all of the terms except for the central term, the term corresponding to j = 0, will approach

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39 zero and therefore drop out from the formula. The resulting limit is the classic BlackScholes formula for a call option. Note that we use the uniform convergence from chapter 3 to interchange the infinite sum with the limit. The formula for the PTCO-I found in the appendix is given by y=-oo S -{p+a)c. e ^ f K V (z^ + o-x -Cj -a {^ + a)T-Cj -k ^ ^lh Â’ Vt ^ (/u + a)ti-Cj -b {/j + (j)T Cj Vf -(/i+CT)^Z Â— ^ J (ju + a)ti-dj a (/j + a)T -dj -ic^ -NÂ„ + -dj-b ip + (T)T-dj-P^'^ e-^'^k ( -pc e Â•' N pt\ -Cj -d pT -Cj -ic^ p -N^ A Â’ -Mdj ^ dj-d fiT -dj ~ic^^ Vr yj -N. Htx -d^-b dj k T^ Â’ 7T >1 (5.8) 1 ^ In Â— where 2(b -d), dj = 2{b -d)-2b and p = .1Â— . Recall that d = Â— Â— and '' r a InC . b Since a < S and b> S we have d <0 and b >0 . First we consider the o following quantities

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40 -dj-b for 7 0 . In particular, we will show that each is either positive or negative according to whether j is positive or negative. We substitute = 2{b a) and dj = 2(b -a) -2b into (5.9) so that we have -Cj a = -2j{b-d)-d, -Cj -b = -2j{b-a)-b, -dj a = -2 j(b -a) + 2b a -dj b = -2 j{b -d) + 2b -b. (5.10) Simplifying yields -Cj d = {2j-\)a-2jb, -Cj-b = 2jd {2j + ])b, -dj-a = (2j-\)a-(2j-2)b (5.11) -dj-b = 2ja-i2j-2)b. Observe that if j >0 then all of the quantities in (5. 1 1) are negative. On the other hand, if j <0 then all of the quantities are positive. These observations should be evident from the fact that a <0 and b > 0 . Now consider the difference ^ + (j)T Cj ^ -b {f^+a)T -Cj -ic^ (5.12)

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41 For j > 0, the first argument in each bivariate normal distribution fimction is negative for sufficiently small values of . As /j tends to zero each of these arguments goes to -oo , Thus each distribution function, and hence the entire expression, will go to zero. For j <0, the first arguments are both positive and therefore tend to -foo . Each bivariate distribution function then converges to a (univariate) normal distribution. Since the second arguments in each bivariate distribution function are the same, the difference approaches zero. The same argument shows that each of the three other difference expressions go to zero as well. This leaves us with the term corresponding to j = 0. This term simplifies to f p V V (/i + cr)r, a {^ + g)T -k A Vr -NÂ„ \^ + (j)t^ -b {^ + a)T-k'^ -e -{/u+cr){-2b) + +2b-a {/u + cr)T + 2b k^^ J) -N^ + +2b-b {ju + a)T + 2b A -e-^^k. f p V V a juT -ic^ A -Jt -N.. ^ /jli b fjJ -ic^ Â’ -It -e -Mi-2b) ^^t^+2b-a fiT + 2b-ii^^ -N. J) ^ /i/, -\-2b-b fiT -\-2b-k'^ A Vf (5.13)

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42 Eight bivariate normal distribution functions appear in this term. We use the same analysis as we did above. Recall that a < 0 and b>0 . Note also that 2b-a>0 . As tj goes to zero, the first and fifth bivariate distribution functions converge to a (univariate) normal. The second and sixth each go to zero. The four remaining distribution functions each go to 1 and therefore cancel each other out. Thus the first term is simply S-N {/j.+g)T -k vr Â• N f MT-k^ y Â•Jf J ( 5 . 14 ) Replacing /^ = Â— a ( -2 ^ In r \ and k V we may simplify (5.14) to recover the classic Black-Scholes formula. 5.2 Numerical Results In this section we examine some features of the PTCO-I and PTCO-II. Naturally, the PTPO-I and PTPO-II will share many of these features, so they will be left out. For each of the partial tunnel call options, we will numerically illustrate the analytic results of section 5 1 We will also examine the effects that the volatility and the length of the monitoring period have on the price of each option. 5.2.1 The Partial Tunnel Call Option: Type I Consider a PTCO-I where we have the following parameters; initial stock price 6Â’ = 55, volatility cr = .2, expiry T = 1, monitoring time /, = .5 , strike price k -6S, lower barrier a = 40, upper barrier ^ = 80 and interest rate r = .06 . We will first show that as the upper and lower barriers of this option approach infinity and zero, respectively.

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43 the price of the option will approach that of the standard call option with initial stock price 6Â’ = 55 , volatility a = .2 , expiry T = \, strike price k = 65 interest rate r = .06 . The appropriate price for this standard call option is \$2.166 . We price the same PTCO-I with upper barriers ranging from 80 to 120 and lower barriers ranging from 40 to 5. The results are presented in figure 5.1. Observe that the price is the same as the Black-Scholes price when the lower and upper barriers have been moved to 40 and 120 , respectively. We now consider the case above but let the length of the monitoring time go to zero In order to illustrate the effect that this has on the option price, we will use the same parameters as above except that the upper and lower barriers will be 70 and 50 , respectively. We observe the prices with /, ranging from 0 to 1 . The results are displayed in figure 5.2. Observe once again that the price approaches the standard Black-Scholes price as approaches zero. ( 40 , 120 , 2 . 166 ) pnce Figure 5.1: Varying upper and lower barriers for a PTCO-I

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44 Figure 5.2: Varying the length of the monitoring period for a PTCO-I We now observe the affect that the volatility a and the length of the monitoring period t, have on the price. In the case of the standard call option, increasing the volatility increases the price of the option. However, in the case of the partial tunnel options, increasing the volatility only increases the price up to a point. After this point, the probability of hitting a barrier and having a zero payoff is so high that increasing the volatility only decreases the chance of not hitting a barrier and having a positive payoff. Thus increasing the volatility after this point only serves to decrease the price. This phenomenon is present in both Type I and Type II partial tunnel options. The length of the monitoring period is also a factor in the price. For partial tunnel options of Type I the monitoring period begins at time zero and ends at some time . Thus the length of the monitoring period is simply /, . Since increasing only increases the chance of getting a zero payoff, we see that the price will decrease as we increase .

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45 Here we take the case where the intial stock price is = 20 , cr = . 1 8 , r = .06, 7 = 2 , the upper barrier is 45, the lower barrier is 15 and the strike price isyt = 25 . Figure 5.3 plots the values of the prices as the volatility ranges from .05 to .95 and the monitoring time ranges from 0 to 2. 5 4 price 3 2 1 Figure 5.3: Varying cr and /j foraPTCO-I Notice that as the length of the monitoring period gets closer to zero, the price appears to increase as
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46 of the option will approach that of the standard call option with the same parameters. We will not be taking the limit as the length of the monitoring time goes to zero for the reasons mentioned in the above remark. Recall that the appropriate price for this standard call option is \$2. 166 . We price the same PTCO-II with upper barriers ranging from 80 to 125 and lower barriers ranging from 55 to 25. The results are presented in figure 5.4. Observe that the price is the same (to at least three digits of accuracy) as the BlackScholes price when the lower and upper barriers have been moved to 25 and 125, respectively. We again observe the effect that the volatility cr and the length of the monitoring period have on the price options. Now for options of Type II, the monitoring period begins at /, and ends at time T . Therefore the length of the monitoring period in this case is r /j . As mentioned above, increasing the volatility only increases the price up to a point and then the price begins to decrease. We now look at the the case where the intial stock price is ^ = 55 , a = ,2 , r = .06, T = 2 , the upper barrier is 100, the lower barrier is ( 25 , 125 , 2 . 166 ) 1 . 5 : price ^ : Figure 5.4: Varying upper and lower barriers for a PTCO-II

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47 30 and the strike price is ^ = 65 . The volatility ranges from .05 to .30 and the length of the monitoring period ranges from 0 to 2. These results are presented in figure 5.5 below. Note that the axis for the monitoring period represents the length of the monitoring period r 1, which was /j in the case of the Type I option above. Figure 5.5: Varying a and T-ti for aPTCO-II Observe the plot where the length of the monitoring period T gets closer to zero. As a increases, the price still only increases to a point and then decreases. This phenomenon is different from that of the PTCO-I where the price appeared to strictly increase as a increased for small values of .

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CHAPTER 6 THE GET-OUT OPTION 61 Description The get-out option is based on a strategy that will get one out of the market when a certain trigger is initiated. Consider a bullish investor who, for some reason, cannot monitor the market and wishes to implement a strategy that would exit him from further exposure if one of his holdings were to go below a certain level. The get-out option depends upon two underlying securities and its payoff is determined by whether or not these securities go below a certain level during the lifetime of the option. There are three possible occurrences; 1 . Stock 1 hits a lower barrier a and triggers the payoff of an exercised call on Stock 2 before time T. 2. Stock 2 hits a lower barrier b and triggers the payoff of an exercised call on Stock 1 before time T. 3. Neither stock hits its respective barrier before the expiry time r and the option returns the payoffs of call options on both stocks at time T. This option is different from the options of previous chapters in that it depends on two underlying stocks. As a result, the Black-Scholes setup from chapter 1 will no longer suffice. We introduce a generalization of this model that will enable us to price this new option appropriately. 6.2 The Multidimensional Black-Scholes Setup 6 21 The Model We must now use the multidimensional (2 dimensional) Black-Scholes Model [9]. In this model we have the filtered probability space (Q, T, P) in which the processes 48

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49 dS,=S,{u,dt + i:,-dW,) dB, = VfBjdt ( 6 . 1 ) are defined. Here S, is a (2x1) vector of stock price processes, u, is a (2x1) vector of drifts, I, is the volatility (2x2) matrix and W, is a multidimensional (2x1) Brownian motion under the probability measure P . We also have the natural filtration that is the filtration generated by the 2 dimensional process S, . Definition 6.1 A process y for which r, Â• 1 m, = Z, Â• y, is called the market price for risk. The existence of this process in conjunction with GirsanovÂ’s Theorem gives rise to a martingale measure for our model. Recall that the martingale measure is simply the measure under which the discounted stock price process is a martingale. Since the stock price process is 2 dimensional, this implies that each component of the process e~Â’''S, is a martingale For simplicity we consider the case where u, = u, Z, = Z is nonsingular and r, = /Â• . In particular, the drift, volatility and interest rate do not depend on time. Under these assumptions it is clear that the process y indeed exists and is given by y IT^r -\-u) . This model together with these assumptions is referred to as the classic Black-Scholes model. 6.2.2 The Martingale Measure As before, we begin with the martingale measure P . Under this measure we have 'si'" (6.2)

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50 for i=l,2 where Z' denotes the i*** row of the matrix Z . In particular, the discounted stock price process is a martingale under P . The solution to this stochastic differential equation is given by f f 2 2 > \ = S' exp Z' Â• W, + _ 0-,l +CT/2 t 1 ^ ) y (6.3) when Z is given by V *^21 (6.4) We may write out (6.3) as S', = S' exp f f 2 2 > aX + 0-,i +0-,-2 t 2 J V K y (6.5) 6.2 3 The Payoff The discounted payoff X for the get-out option can be expressed as the sum of three exclusive payoffs that correspond to the three possible outcomes set forth in 5. 1 . Set r, = inf ^5 > 0 : S] = ^|, ^2 = inf > 0 : S^ = and define the following sets accordingly: A^ ={r, T, T2>T} Clearly we have P [A^ A 2
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51 (6.7) Our goal now is to price this option. Just as before, we need only take the expectation of X under the martingale measure P . 6.3 Pricing 6.3, 1 The Independent Case This is the simpler of the two cases. For this case we assume that the two stock price processes are independent. Consequently, the non-diagonal entries in the volatility matrix Z must be zero. Otherwise the processes would both contain the same nonzero Brownian motion terms and therefore be correlated. If we denote the diagonal entries of the volatility matrix by ct, = cr^^ and CTj = CTjj then we may write the stock price processes as S; exp(o-,JF/ ( 6 . 8 ) where //, = r for i-1,2. Recall from (6.6) and (6.7) that we will be taking the expectation of (6.9) where

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52 A ={^1 <^ 2 ', ^\< T ] A ={^2 <^I> ^2 <^j 4 = {r, >7; T 2 >r} ( 6 . 10 ) We begin with the first term. We wish to compute E Â“ f ^ / \ + \ '4 e ' 1 1 ( 6 . 11 ) under the martingale measure P . Recall that the process Sl is defined as Si = expfo-jlf;^ + //2Â«) Define the processes by + n-u for i=l, 2 . (6.12) Then we have SI =SÂ‘ exp(A'') for i= 1,2. (6.13) (6.14) Now write the set ^ as 4 = A'r^A" where 4' = {r, < 72 } and 4'= {rj < Tj. If we have the density P(tj g du, < T 2 ,XI^ e (fy) then we will be able to express ( 6 . 1 1 ) as a Lebesgue integral. First note that 4 ' = > In ^ | where mf is the running minimum for the process Xf . Thus we may write P(t, g du, r, < T 2 , X^ &dy)^ P(r, g du, m\ >h,X] Gdy) (6.1 5) where ^ = In VO y Using methods in [12] and the independence we have P(r, G du, > b, X]^ ^dy)= P(r, g du, ml >b,XlG dy) = P(r, G du) P(ml >b,XlG dy) (6.16)

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53 The densities on the right hand side are found in [7], The density p(^Xl &dx,ml> is given by P(xUdx,ml>b) = 1 TUJ^U exp 2<7^U exp V y exp Icr^u dx (6.17) Recall that r, inf ^5 > 0 : S] = urj We substitute 5"] = exp^X] ^ and rewrite this as T^ = inf ^5 > 0 : 5"] = a| = inf > 0 : exp^Zj) = a| = inf ^5 > 0 : Xl = a| = inf|5>0: IT/ +^s = a[ (6.18) In where a = vÂ«y Then we have the density P(r, e du) = \a-^u] \d\ 1 J 1 V 2u du (6.19) So we may express (6. 1 1) as T +00 = j J e"Â™ ) P(r, G du, r, < Tj, e dy) (6.20) 0 Â—00 T +00 = I j ) P(r, G du)P(^X^ edy,ml> 0 Â—00

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54 We remove the (Â•)'^ notation by changing the lower limit of integration of y to k-, = In so that the quantity inside of ()'" above is strictly positive. Then (6.20) becomes T +00 J j e Â™ (S^e^ ^ 2 )^ P(Ti Â€ du)P{^Xl e dy,ml > b) . (6.21) 0 Notice that the second expectation is the same as the first with the roles of stock 1 and stock 2 reversed. Thus we can write it as T +00 1 J P(t2 g du)P(^Xl ecfy,ml> a) (6.22) 0 t, where A:, = In fir ^ and <5 = In I S' J Lastly, we need to compute the value of the third expectation = E |(*^r ^ 1 ) + {Sf A^2 ) (6.23) We now distribute the indicator function to each of the terms in the parentheses to compute each expectation. The first term is E % {s'r-k,y (6.24) Recall that /I 3 = {r, > T; r 2 > T} . We may write ^3 as n AÂ” where ^3 {^1 > and A^ {^"2 > T). We break up the indicator function accordingly and express (6.24) as

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55 -rT E ^ (^r ~^i) (6.25) We may factor this into two separate expectations using the independence so that (6.25) becomes Note that we have e -rT (6.26) E = p{Ai) = P(z,>T) (6.27) Jl T exp v2 ^ b-^S 2s ds \ y The last equality is given in [7]. Thus (6.26) is simply the price of a down and out barrier option multiplied by the probability in (6.27). This formula is given in [9]. The second term in the expectation (6.23) is the same with the roles reversed. Remark: Notice that the above analysis did not use the change of measure employed in earlier chapters to remove the drift. The reason for this was that the densities used already accounted for the processes with drift. 6 3 2 The Dependent Case Just as before, under the martingale measure/* the stock price processes are S[ S' exp r 2 , 2 ^ ^ q~,i +^,2 for i=L2 (6.28) where JT/ and fV,^ are standard Brownian motions . Moreover, we are still trying to compute the expectation (under P) of the discounted payoff N where

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56 ^ = 1 A -n. K ^2) and +\ (42 -*.)1 (6.29) +^A, 1 + M 1 Â•)) A = {2Â’l TIV ^a} and T 2 = inf > 0 : . This time we will remove the drift with a change of measure. Notice that we must remove the drift term from each of the processes in (6.28). We will change to a measure Q so that where W,^ and are standard Brownian motions under Q . However, the drift constants ju^ and jU 2 must be chosen so that they satisfy MiÂ‘) + o-,2 M2*) = o-,,#; + a,2W,^ o-,,//,/ = ^,1^/ + a, 2^2)^ (6.31) (6.32) r ^2 , _2 ^ for i-1,2. In particular, and ^2 satisfy the equations ^ 12/^2 ~ ^ 2 2 <^11 <^12 2 2 CT21 +<^22 (6.33) + ^ 22/^2 ~ ^ 2

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57 Solving these equations for //, and //j yields 2 r {022 cT\2 ) ^22 ^12 ) ^12 i^2i *Â“ *^22 ) i i i L 2(cTi,Cr22 ~0 'j2(T2j) (6.34) M 2 = _ 2 /(gn 0-21 ) 1 (q2^ + 0-22 ) + 0-21 (o-U + 0-12 ) 2( 0 : S] = Â« j We may rewrite this as r, = inf > 0 : 5Â’Â’ exp {^ 0 : CFnWj + ct, 2 ^/ = (6.38) where a = In . Similarly, we may write Tj = inf |5 > 0 : g 2 i^i +^ 22 ^^ = where b = \n r h\

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58 We will make two substitutions, a translation and a rotation, so that we may use the joint densities found in [6] to compute the expectation in (6.37). Note first that the stopping times rj and correspond geometrically to the first time a two dimensional Brownian motion W, , starting at the origin, first hits either one of the lines Lj : o'iiX + o'i2y L^. CT2iX + a22y = b (6.39) Note that a <0 and b <0 since a
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59 The densities in [6] are for a process that starts at specified point in the plane and then hits one of two lines. One of these lines is the x-axis. The other line goes through the origin and has a negative slope. The point at which the process starts must be located in the wedge that includes the first quadrant. Thus we will need to relocate the intersection of our two lines to the origin and then rotate the plane so that one of the lines coincides with the x-axis. In this manner we will be able to use the densities from [6]. We now make the first substitution ^.=W,-p (6.41) so that Xj is a Brownian motion starting at -p . This translation moves the intersection of the two lines to the origin. The slopes of line 1 and line 2 are /w, = W 2 . 022 Note that we could simply switch the roles of stock 1 and stock 2 to accomplish this. However, the assumption that the slopes are not equal may not be as easy to see. If the slopes are equal then the stocks are perfectly correlated. Since this case is of no interest we may make the assumption that the slopes are indeed not equal. Figure 6.2: Translated plane

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60 The next substitution rotates the plane counter clockwise so that line 1 coincides with the x-axis. Since the slope of line 1 is /Wi = 11, the angle that it makes between '12 itself and the positive x-axis is 6 = tan -1 Â’-'11 V ^12 J (6.42) Note that we have ^ < 0 since Â— ll< 0 . Note also that the angle between line 2 and the '12 x-axis is also negative. Thus the angle is given by a n tan' CT 21 -i-tan * f-Â— 1 V *^22 J 1 *^12 y (6.43) n We have a>-^ since both lines have negative slopes. Define a new process Y, by y.=R-o-x, (6.44) where R_n is defined as R-e^ cos 9 sin^'' -sin^ cos^ (6.45) The matrix R_(^ will rotate the plane counterclockwise by an angle of -9 > 0 . Thus 7, is a two dimensional Brownian motion starting at the point R_g (-p) = -R_gp . Recall that the original process W, starts at the origin and the intersection of the two lines is in the third quadrant. Thus the translated and the rotated process Y, starts in either the first or second quadrant. Moreover, line 1 becomes the x-axis and line 2 creates an obtuse wedge that includes the first quadrant and a portion of the second quadrant. This comes from the fact that the original lines both have non-positive slopes. Note that one of the slopes, but

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61 not both, may be zero. In this case the angle between line 1 and the positive x-axis is zero. Thus the rotation matrix above turns out to be the identity matrix in this case. Figure 6.3: Translated and rotated plane Now the process 7, may be expressed as Y,=R.e-X, = R-9-[W,-p) . (6.46) = R-e-W, -R_ep Solving backward for W, yields W,=RZl{Y,+R,p) = R-ey,^p (6.47) We now look at what the individual processes and look like in terms of the process 7, . So we write out (6 47) as W,=RZl-Y,^p -sin^'' fv' ^ Â•* J + fx ^ cos^ ^ 7^ Vi ^ .yoy ^cos07/ sin ^ Â• 7,^ + ^ ^sin ^ Â• 7/ + cos 9-Y,^ (6.48)

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62 We define the functions / and g by f(x,y) = xcos^j/-sin^ + Xo and ^(x, >^) = X Â• sin ^ + >^ Â• cos^ + >^o . (6.49) (6.50) Then we have w,= (6.52) (6.51) This will help us write the increasingly large expressions more compactly. Now the stopping time r, defined as = inf ^5 > 0 : ctjjIF/ + may be expressed as Tj = inf |5 > 0 : CTjj (cos^ Â• Y' sin ^ Â• 7/ + Xq) +0: a,, (cos6Â» 7/ sin (9 Â• 7/ ) + <7,2 (sin 6>-7/ +cos(9-7/) = oj (6.53) = inf|5>0. (cr,, cos +<7,2 sin 6>) 7/ + (cr, 2 cos (9 -a,, sin (9) 7/ = o| Â• (6.55) Moreover, d = tan' V 0-12 SO we may compute the trigonometric functions sin 0 and cos^ as

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63 sin^ = cosff = -cr. II +0-12 '12 yj^n + 0 -] (6.56) Substituting these into (6.55) yields T, = inf > 0 : Â• 7/ = o| (6.57) Clearly the quantity + a ^2 is strictly positive, thus r, = inf {5 > 0 : 7/ = o} . (6.58) This is precisely what we should expect. Recall that r, represents the first time that process 7, hits line 1 . However, line 1 has been the translated and rotated so that it is the x-axis. Thus r, represents the first time that the process 7, hits the x-axis (i.e. the first time that 7/ = 0 ). Our goal is to compute the price this out explicitly as dQ X, of the get-out option. We first write dQ X dP f dQ i Â•(??, -* 2 )* (6.59) + ] / / 1 \ / \ + ((sj-*,) +(j T ~ ^7) )))] The first term in this expectation is dQ -n. 'K ^ 2 ) (6.60)

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64 Under the measure Q we have S\ = 5' exp^CTjjfr/ + 0 : = o| so = 0 . Replacing this above yields 1^" exp (o2 , / , o) + 0-22 Â• ^ (y/j , o)j ^ 2 ) 'A ~f~y\ e ' (6.64) We use some simple properties of expectations to take an inner expectation conditioned on the sigma algebra . This yields E^ E f exp V .2 , ..2 ^ ^.f[Yl,Y,^) + M2g(yrJT)-^^^^T exp (cT2i / (y/j , o) + CT22 Â• g (y/j , 0 jj ^2 1 Er r ./ aTi (6.65)

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65 We take out the measurable portion inside of the inner expectation. All that is left is the Radon-Nikodym derivative, which is a martingale. Recall that A = {^1 < ^ 2 . ^1 < 7Â’} . In particular,
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66 So we define a function H (,Â•) by H{u,a) = e (5^exp((T2i f {a, 0 ) + 0^2 g{a,Q)) ^ 2 , 2 ^ exp f{a, 0 )^ ^g{a,Qi)^ 9 I ^ (6.68) Note that the (-)^ notation has been removed. We may do this as long as we make sure that the quantity inside ( )^ is greater than or equal to zero. The following equivalent statements illustrate how to ensure this. 1) 5^exp(CT2, /(a[,0) + cr22-g(a,0))-^2 ^0 2) 0-2, /(a,0) + o-22-^(a,0)>A2 =ln (h.] U'J 3) CT21 (acos^ + Xo) + CT22(asin^ + ^o)>^2 = 4) a2i a cos 9 + CT22 Â• a sin ^ + (ct2,Xq + CT22>'o ) ^2 5) (T21 * Cl cos ^ "1Â“ ^'2,'^ * ^ Sin 0 b ^ h-2 6) a(cT2i cos^ + CT22 sin^) > A:2 i 7) a f ^ ^12^21 ~ *^11^22 > k., b o\ ^ I 2 ^ 8) a> yjcTu + C7i2 (Ti2
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67 Define the constant Tj = max k^-b yCTijCTji CTjjCr22 +0-12,0 (6.69) If ^2 > ^ , then we will only need to compute the expectation of H j over values of which are greater than ^2 If ^2 ^ > then we will compute the expectation of H j over all values of Yj^ , i.e. values greater than or equal to zero. So the constant is a lower bound for the values of Y^^ for which we must take the expectation. Thus we may write (6.67) as h i. . But the joint density for the pair ) when r, < T 2 given in [6] is &da,T^
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68 T +00 \ \ H {u,a)p(x^ e duJl^ eda,r^< Tj) r +00 r 2 , 2 ^ a +/-n ' 0 /: a au (6.73) z y -0 7 -sin . a / ^a/^ V ' y dadu This representation allows for its precise value to be calculated via standard numerical integration techniques as opposed to simulation. This is only the first term in the expectation. Fortunately, the same transformations enable us to use the densities in [6] to write out the expectations as Lebesgue integrals. The second expectation we must compute is (6.74) We again use the functions / and g defined in (6.49) and (6.50) so that we have W = fwy ^y.'.y.^). (6.75) We may then write (6.74) out as exp [s' exp(aÂ„ (6.76) We use the same method as in (6.66) so that (6.76) becomes exp 2,2 ^ Ml +P2 , ~T~ \ ^ y f ^ 1 / / \ / \\ \+V e 2 .|^iexp|crÂ„ (6.77)

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69 Recall that = inf ^5 > 0 : + ctjjPF/ = i j . So Tj = inf ^5 > 0 : ( 72 iW^ + 0Â’22^/ = = inf {5 > 0 : Â• / (n* , 7/ ) + Â• g { y^ J^) = b] = inf > 0 : CT21 (7/ cos^-7/ sin ^ +0-22 (j",* sin^ +7/ cos^ + Xq) = = inf ^5 > 0 : (T21 (7/ cos ^-7/ sin ^ + Xq ) + <722 ( 7/ sin^ +7/ cos^ +Xq) = = inf |5 > 0 : ( 0 : (a-21 cos 6 + O22 sin &) 7/ + (a22 cos 6 (J21 sin 9 ) Yf +b = = inf > 0 ; (a2i cos^ + a 22 sin 6 )Yl + [a 22 cos 9 0-21 sin 9 )Y^ = o| = inf|5>0: 7; = ^11^21+^120-22 .^4 I O'! 1 0-22 Â“CTj 20-21 J The coefficient of the 7/ term is precisely the reciprocal of the slope of line 2 after the transformation. Define another constant ^ _ 0^110^21 +0^120-22 0"llO-22 ~0 -i2CT2i (6.79) so that r2 = inf |5>0: 7/ = . yA I O11CT22 cTi20-2i J (6.80) = inf{5>0: 7 ; =/w-7/} Then we have (6.81) Now we may rewrite (6.77) as

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70 exp exp (aÂ„ Â• / (m Â• , i " r2 ) + 0-12 Â• ^ (^ Â• >^4 Â’ )) " ^ ) -, (6.82) so that the random variable we are taking the expectation of depends only on Tj and 7^^ Define a function M (Â• , ) by 2 , ..2 ^ M(w,a) = exp //, f{m-a,a) + fi^-g{m a,a)^ u [e exp(o-Â„ Â•/("Â» a,a) + ai 2 .g(/w a,a))-^i)) (6.83) Then we must again find the lower bound rj for the random variable so that we may omit the (-)^ notation. Computations similar to those for /j yield r X 2 = max k^-a I 2 2 Â” +cr,2 ,0 (6.84) So we may express (6.82) as (6.85) Thus we only need the joint distribution for the pair ^X 2 , Y^^ j given r 2 < r, . But this is given in [6] by

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71 p(t^ sdu,Y^^ Gda,T2
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72 We need only compute the final expectation now. Just as with the first two terms, we will use the transformation so that we can use another density provided in [6], Recall that we are computing E9 ( 6 . 88 ) We break this up into two separate expectations e -rT + e~^'^E9 (6.89) We focus on the first expectation. First we remove the ( Â• Y notation with the use of an appropriate indicator function. This yields -rT (6.90) Now we express the random variable in the expectation in terms of the transformed process F, The indicator function is left as is but will be changed momentarily. This yields exp 1 v2\ M\ tA r X ) + c^i2 )) ^i)j Again we simplify the notation by defining the function by (6.91) .2 , ..2 L{x,y) = e "^exp //, f [x,y) + g{x,y)~ ^ Y x V 2 j (692) (^' exp(o-Â„ Â•/(x,>') + a,2 Then we may write (6.91) as

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73 1.1 (6.93) The random variable in this expectation depends only on the process Yj. when ^3 = {r, > T; ^2 > r} . But we have this density as well from [6], The density is given in polar coordinates as P(r, > T,T 2 > TJj &dy) 2r Ta exp f 2 , 2 ^ r + /Â•Â„ 2T jnT,T 2 > T,Y, e drdO) (6.96) 0 0 where Â•/(/'cos^,/-sin(9) + CT ,2 Â•g(rcos^,/-sin<9) > Ajj. We may rewrite B as

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74 B = {o-,j (rcos^ Â• cos^ -r sin^ -sin^ + Xg) +c7,2 {r cos ^ sin ^ + r sin (f> cos d +yo)>ky = {(/ Â• cTji cos G + r + {r a ,2 cos ^ r Â• <7ji sin sin ^ +r(c7Â„Xo+a,2>'o)^^i} = {('Â•ctj 2 cos^ r c 7,, sin^)sin^ + 5 > = \r{(Ji 2 cos^-cr,j sin^)sin^ >^i -a^ sin ^ a| r > (^1 <5) sin^ Define the function (^) by (6.97) ^1 ( is (^) . The maximum between the quantity above and zero is used to ensure that r > 0 in the case when k^
,rsin)P[T^>T,T 2 >T,Y,Gdrdif>) 0 9,(Â«l) (6.99) We compute the second expectation in just the same manner. We write out the expectation in terms of the transformed process. Recall the second expectation is [dP ,9 , Â€ 19* 1 T ~ ^2) (6.100)

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75 Written out in terms of the transformed process Y, this looks like exp '4 Hfc, 20 | (5' Â«>Â‘P K Â• / {rr . iV' ) + <^22 Â• Â« (n-Â‘. }?)) *2 )) Define the fijnction (/(,) by ( 6 . 101 ) 2 , ..2 A ^/(x,>;) = exp //, /(x,jy) + ^.g(x,y)-^. '^^ T y (t? exp ( 0 2 , f{x,y) + a^ S {x, y)) k^)"j ( 6 . 102 ) Then (6. 101) becomes (6.103) Again the random variable depends only on the process Y, when = {r, >T; x^> T] So we may express this in terms of the Lebesgue integral U ^ou = J J l^f/(rcos^zi,rsin^) 2 P(r, >T,z^> T,Y, & dr d) 00 ' (6.104) where Â£ {a,, Â• /(/-cos,/-sin^) + a ,2 Â• g(/-cos^,r sin^) > ^Â’ 2 }. Calculations similar those of (6.97) show that to E = \h-b) r \ yl^u +^i2 J COS(p + r \ ^ 12^22 Wn + o -,2 sin^ (6.105)

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76 Define the function q.^ (Â•) by q 2 {) = max (*>*) + (7,2 is ^2 (^) Â• Then we may express (6. 104) as a .00 (6.107) = J I U [r cos (!>,r sin (!>)P{t^>T, V 2 >T,Y, & dr d(f>) 0 q2{) This being the last necessary calculation, we may add up the quantities in (6.73), (6.87), (6.99) and (6 . 107) in order to obtain the price of the get-out option.

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in APPENDIX EXPLICIT OPTIONS' PRICING FORMULAS The results for the partial tunnel optionsÂ’ prices from chapter 4 were expressed i terms of the functions h(c,d), h(c,d) and w(c,d) from chapter 3. Here we present the optionsÂ’ prices explicitly in terms of the bivariate normal distribution function. We first make some computations that will be used later. Recall that = Â— a r V 2; So the quantity ^ becomes 2 (/i + cr)^r +2o/i +CT^)r 2 2 + f 1 f 2 > 2a r a + a^ V a \ 2 J y T 2 2 _ ^ [2r-a^ " 2 ^ 2 u^T = ^ Â— + /-T As a result of theorem 4.2 we have )-h{n + a,dj))-k[h{n, ) h{n, dj ))] (A. 1 ) -rT-^ n-T y=-cc where 77

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78 h{c,d) = exp -cd ^ a-h p-k ^ -N^ ^ a-d p~k^ Â’ Vr yj Cj 2{b a) , dj = 2{b -a)-2b , p = , a=ct^-d and P = cT -d . h{c,d) and simplifying yields (p + cr)/] Cj-d (p + a)T -Cj-k (A.2) Substituting +00 S ( -{ti+(T)c e ' 1 N 1 V >/T -A, {fu + a)t^ c j h {^ + a)T-Cj-k^ A Vf -e -(p+' -A. (p + {fi + a)T-dj-k r\ 7f Jy A. p/[ a fiT -Cj -ic^ 7T Â’ y? -A. p/, -Cj -b juT -Cj -k A Â’ ^ . A, fit^-dj-d i^T-dj-k^^ \ -N. yy ^ /2t^-d -b /uT -d -ic jy . A Â’ ^ Theorem 4.4 gave us the price of the PTPO-I as K [k^, Cj )K m, dj )) *^ Â• (Km + (T,Cj)-kM + dj ))] where

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79 h{c,d) = exp -cd ^ b-a k~P^ -N^ ^ d-a k Â’ VF 7J Cj = 2(6 a ) , dj = 2(6 -d)-2b , p = , a = ct^-d and P = cT-d Substituting h(c,d) and simplifying yields N '^b-pt,+Cj ic-pt,+Cj^ ^a-pT + Cj ic-pt,+Cj^ ^ \F\ 'Tt ^ ^p < 'Tt -S.{. -Hd 9 J -iH+a)c^ N. -N_ NÂ„ ^ bpt^+dj ic ///, + dj ^ Jr, Â’ ^/^ ^ b {p + u)t, + Cj ic ~{p + a)T + Cj^ '' a pT + dj ic pt, + dj ^ ^F^ Â’ 'Tt yj -NÂ„ ^ d-{p + cr)r, +Cj ic ~{p + a)T + c. ^ yj -{p+cj)d > J ^ b ~{p + a)t,+dj ic ~{p + a)T +dj^ 7F -iV. a-{p + cr)t, +dj ic ~{p + a)T + dyj Theorem 4,5 gave use the price of the PTCO-II as J^Â—
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80 w{c,d) = exp ( c^T ^ f r cd 1 2 j V P \ b +d -a bP A ^b+d-a S-p^ -N^ -N. ^d + d-a b~P^ n/^ Â’ Vr ^d+d~a S~p 4h Â’Â”VT 7 ^ Â’ Vf 5 = max(a,^), = 2{b a), dj = 2(b-d)-2b , a = ct^-d and P = cT -d Substituting w(c,d) and simplifying yields K? = sf e ^ N b + 2 c j pl^ b pT + c. N d + 2 c j pt^ b pT + c. ^ V V P P N. e -fd'dj VL A'Â„ yV ^ 6 + 2 c j /y'/, ^'T + c ^ 6 + Id. b /iT + d . . k ^ J ^ b + Id^ /i'/, ^ pT + dj ^ r -N. ^ z' -N V? -A' -Â’T , -e k iV i + 2c^ /i f, b fdT +c a + 2 Cj ~ pt^ 5 pT + Cj y Â’ yfr a + 2 d. pt^ b pT + d^ \ k ^ . ^a + 2 dj-pt^ 5 pT + d ~1^Â’ 7 F yj )) -N a + 2 Cj b /jT + Cj^ e -Md, x/F yj d + 2 c^~ S nT + c. N VL N b + 2 dj b fjT + d b + 2 dj /v/| S pT + d ^ ' Vf ^ f d + 2d S /jT + d ^ -N ^ ' where /u' = p+cr . Theorem 4.6 gave use the price Fjf of the PTPO-II as VF yj yy = z ^ -rT-WT k ( ) w{p, d ^ )) .S Â• (w^/u +a,c^)w{^ + a, d ^ ))] where

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81 w{c,d) = exp r y y cd 1 2 J V P \ b + d-a y/ P ^ b + d ~a d~ 1 T'^ -N. ^ d + d-a y/ Â•J/F ^ yy 6 Â’r r -/i'c e Â•'j V N. b + 2c j //'t, y/ /i'T + Cj Vt, Â’ VF -At, d + 2c^ /i't, ^4/^ //T + Cj Â’ 7 F yj N. ^b + 2 c -n't, b-n'T + ck ^ ^ ^ At Â’ ^/F A e -/i't^, y r At. 6 + 2i/^ /i't, n'T + d^ p r -N. f z At, b + Idj n't^ b n'T + dj 7^ yf -At, a + 2c j /i't, b n'T + Cj Â’ VF 5 + 2dj /i't, y/ n'T + <3?^ F ~jr , a + 2 dj n't\ b n'T + dj ^ . 7^ ' JJi

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REFERENCES [1] Baxter, Martin and Andrew Rennie, Financial Calculus, Cambridge University Press, New York, 1996. [2] Chuang, Chin-Shan, Joint distribution of Brownian motion and its maximum. Statistics and Probability Letters, 28 ( 1 996), 8 1 -90, [3] DufFie, Darrell, Dynamic Asset Pricing Theory, Princeton University Press, Princeton, New Jersey, 1992. [4] Heynen, Ronald and Harry Kat, Partial barrier options, Journal of Financial Engineering, 3, no. 3/4 (1994), 253-274. [5] Hull, John C., Options, Futures, and Other Derivatives, Upper Saddle River Prentice Hall, New Jersey, 1997. [6] Iyengar, Satish, Hitting lines with two-dimensional Brownian motion, SIAM Journal of Applied Mathematics, 45, no. 6 (1985), 983-989. [7] Karatzas, loannis and Steven Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, 1997. [8] Kunitomo, Naoto and Masayuki Ikeda, Pricing options with curved boundaries. Mathematical Finance, 4, no. 2 (1992), 275-298. [9] Musiela, Marek and Marek Rutkowski, Martingale Methods in Financial Modelling, Springer, Berlin, 1998. [ 1 0] Revuz, Daniel and Martin Y or. Continuous Martingales and Brownian Motion, Springer, New York, 1991. [11] Rudin, Walter, Principles of Mathematical Analysis, McGraw Hill, Inc., New York, 1976. [ 1 2] Shepp, Larry, Joint density of the maximum and its location for a Wiener process with drift, Journal of Applied Probability, 16, (1979), 423-427. 82

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BIOGRAPHICAL SKETCH David Brask was bom in St. Louis, Missouri. He moved to south Florida where he attended school from fourth through twelfth grade. After graduating from Deerfield Beach High School in 1991 he went to the University of Florida. He was admitted into graduate school early via the Mathematics DepartmentÂ’s accelerated masterÂ’s degree program. David received his bachelorÂ’s degree in 1995 for a double major in mathematics and statistics. The following year he completed his masterÂ’s degree in mathematics with a specialization in applied mathematics. After studying image compression and processing for about one year, his research moved into the field of mathematical finance. Soon after changing focus, David had an internship at William R. Hough & Co., an investment banking firm, to gain experience in the financial field. He was then certain that this was the area in which he wished to do his research. 83

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