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## Material Information- Title:
- The risk-spread option in a potential theoretic framework
- Creator:
- Swearingen, Michael C. (
*Dissertant*) Glover, Joseph (*Thesis advisor*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 2000
- Copyright Date:
- 2000
- Language:
- English
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- ix, 97 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Arbitrage ( jstor )
Cauchy problem ( jstor ) Finance ( jstor ) Financial bonds ( jstor ) Financial portfolios ( jstor ) Interest rates ( jstor ) Investors ( jstor ) Martingales ( jstor ) Mathematics ( jstor ) Prices ( jstor ) Dissertations, Academic -- Mathematics -- UF ( lcsh ) Mathematics thesis, Ph. D ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Abstract:
- A fixed-income economy, which includes defaultable securities, is developed through a potential theoretic approach to modeling the spot rate of interest. Under the assumption of an arbitrage free market, the riskless and risky state-price densities are used as inputs to generate the respective spot rates in a Markovian setting. The riskless state-price density is simply the discounted conditional expectation of the derivative of the martingale measure Q with respect to the reference probability P associated with the underlying Markov process X subscript t. The risky state-price density is an original modification of its riskless counterpart. If the time to default is modeled as the first jump in a generalized Poisson process with intensity lambda subscript t = lambda (X subscript t), then the risky state-price density is defined as the discounted conditional expectation of the derivative of the forward martingale measure F with respect to P. However, the discounting is done with respect to the default intensity lambda rather than the riskless spot rate. Furthermore, it is revealed through the resulting expression for the risky bond price, that the default intensity lambda is the risk spread between the riskless and risky spot rates. The main example used to illustrate this procedure is the well-known Ornstein-Uhlenbeck process from which a Cox-Ingersoll-Ross model of both spot rates is derived. In addition to computing bond prices with this example, a Cauchy problem for an originally designed option on the risk spread is derived through the Feynmann-Kac Theorem. A series solution is then developed using a modern potential theoretic version of the classical parametrix method for parabolic partial differential equations. ( , )
- Subject:
- KEYWORDS: interest rate derivative, potential theory, state-price density, fixed-income finance, stochastic processes
- Thesis:
- Thesis (Ph. D.)--University of Florida, 2000.
- Bibliography:
- Includes bibliographical references (leaves 94-95).
- Additional Physical Form:
- Also available on the World Wide Web; PDF reader required.
- General Note:
- Printout.
- General Note:
- Vita.
- Statement of Responsibility:
- by Michael C. Swearingen.
## Record Information- Source Institution:
- University of Florida
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- Copyright Michael C. Swearingen. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 45825348 ( oclc )
002639260 ( alephbibnum ) ANA6086 ( notis )
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THE RISK-SPREAD OPTION IN A POTENTIAL THEORETIC FRAMEWORK By MICHAEL C. SWEARINGEN 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 2000 Copyright 2000 by Michael C. Swearingen To My Late Father Thank You for a Lifetime of Encouragement Wish You Were Here PREFACE Participants in fixed-income markets trade in various securities whose value ultimately depends upon a particular rate of interest at which a financial entity is willing to issue debt. The financial entities involved may be governments or corporations, which determines whether the interest rate is riskless or risky, respectively. The risk involved in this context is the credit risk associated with the possibility of default on a corporate bond. Since there is no credit risk with government bonds, it is reasonable to assume that the risky interest rate should always be greater than the riskless interest rate. But, what happens when there is inflation? Does the difference between these interest rates, or risk spread, remain the same as the government rate rises? This raises the issue of hedging against the inflation risk associated with corporate bonds. The main result of this dissertation is the development and pricing of an originally designed interest rate derivative which shall be known as the risk-spread option. This option may be used by investors to hedge away the risk associated with the difference between the government riskless interest rate and that of a corporate bond. The potential theoretic approach to this pricing problem is general enough to generate various stochastic models of both the riskless and risky interest rates. Moreover, it provides a model that is analogous to physical systems that employ potential theory; thus, the physics of fixed-income finance is illuminated revealing a mathematical structure behind economic intuitions. The first section of Chapter 1 establishes the basic definitions and assumptions in fixed-income finance and arbitrage pricing theory that will be used throughout this work. Sections 2 and 3 lay down a general potential theoretic framework in which to evaluate interest rate derivatives. As an example of the procedure developed in these sections, Section 4 uses an Ornstein-Uhlenbeck process to derive models for the risky and riskless interest rates as well as bond prices. In Section 5, the risk-spread option is introduced by means of a discrete example. Chapter 2 contains the main theoretical work necessary to represent the price of the risk-spread option as the solution to a Cauchy problem. In Chapter 3, the Fourier and Laplace transforms are used to represent the solution in a more tractable form. Also, it is shown how to hedge the risk-spread option using a portfolio of riskless and risky bonds. In the final section of Chapter 3, the graphs of the riskless and risky yield curves are displayed for various parameters. A summary of the results together with some remarks on advantages, disadvantages, and proposed future improvements is found in the concluding chapter. TABLE OF CONTENTS page P R E F A C E .......................................................................................................................... iv A B S T R A C T .....................................................................................................................v iii CHAPTERS 1 A POTENTIAL THEORETIC FRAMEWORK FOR FIXED-INCOME MARKETS ....1 1.1. The Essentials of M them atical Finance ................................................................ 1 1.1.1 The Fundamentals of Fixed-Income Finance.......................................... 2 1.1.2 A Review of Arbitrage Pricing Theory................................................... 6 1.2 Potential Approach I: Riskless Bonds and the Martingale Measure.................... 8 1.3 Potential Approach II: Risky Bonds and the Forward Martingale Measure......... 12 1.3 .1 R isk y B o n d s ................................ ................................................................ 13 1.3.2 The Forward Martingale Measure......................................................... 21 1.4 A Simple Example of the Potential Theoretic Approach................................ 26 1.5 T he R isk-Spread O ption........................................ ......................... .............. 28 2 A CAUCHY PROBLEM FOR THE RISK-SPREAD OPTION.................31 2.1 D erivation of the C auchy Problem ........................................................................ 31 2.2 The Potential Theoretic Parametrix Method................................................... 37 2.2.1 The G aussian Sem group ........................................................ .............. 37 2.2.2 The Fundam ental Solution ....................... ......................................... 39 2.3 Prelim inary Technical R results ................ .......................................... .............. 43 2.3.1 Differentiability of the Gaussian Semigroup........................................ 43 2.3.2 B asic P potential T heory ........................................................... .............. 48 2.4 The Derivatives of the Gaussian Potential ............... ................................... 54 2.5 A Series Representation of the Fundamental Solution ................................... 61 2.5.1 Convergence and Continuity .................................................. .............. 62 2 .5.2 H older C ontinuity ........................................ ........................ .............. 65 2.6 The Solution to the Cauchy Problem.............................................................. 71 3 NUMERICAL RESULTS AND APPLICATIONS ..................................................73 3.1 The Fourier and Laplace Transform s ................................................ .............. 73 3.2 Delta Hedging with the Risk-Spread Option .................................................. 79 3.3 Numerical Properties of the Yield Curve........................................................ 82 4 SUM M ARY AND CONCLU SION S ........................................................ .............. 91 4.1 Sum m ary of R results ........... .. .......... ........ .............. .............. 91 4.2 Future Projects and M odel Extensions ............................................. .............. 92 REFERENCES ................................................... 94 BIOGRAPH ICAL SKETCH ................... ............................................................... 96 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE RISK-SPREAD OPTION IN A POTENTIAL THEORETIC FRAMEWORK By Michael C. Swearingen August 2000 Chairman: Joseph Glover Major Department: Mathematics A fixed-income economy, which includes defaultable securities, is developed through a potential theoretic approach to modeling the spot rate of interest. Under the assumption of an arbitrage free market, the riskless and risky state-price densities are used as inputs to generate the respective spot rates in a Markovian setting. The riskless state-price density is simply the discounted conditional expectation of the derivative of the martingale measure Q with respect to the reference probability P associated with the underlying Markov process X,. The risky state-price density is an original modification of its riskless counterpart. If the time to default is modeled as the first jump in a generalized Poisson process with intensity kt = k(Xt), then the risky state-price density is defined as the discounted conditional expectation of the derivative of the forward martingale measure F with respect to P. However, the discounting is done with respect to the default intensity X rather than the riskless spot rate. Furthermore, it is revealed through the resulting expression for the risky bond price, that the default intensity k is the risk spread between the riskless and risky spot rates. The main example used to illustrate this procedure is the well-known Ornstein- Uhlenbeck process from which a Cox-Ingersoll-Ross model of both spot rates is derived. In addition to computing bond prices with this example, a Cauchy problem for an originally designed option on the risk spread is derived through the Feynmann-Kac Theorem. A series solution is then developed using a modem potential theoretic version of the classical parametrix method for parabolic partial differential equations. CHAPTER 1 A POTENTIAL THEORETIC FRAMEWORK FOR FIXED-INCOME MARKETS 1.1 The Essentials of Mathematical Finance The two main prerequisites of mathematical finance that are imperative to an understanding of this dissertation are fixed-income finance and arbitrage pricing theory. This section begins by establishing the probabilistic setting in which these concepts will be reviewed. According to Musiela and Rutkowski (1998), an economy is a family of filtered probability space {(Q,.f ,) :te 7 }, where the filtration .T= {t}tI[0,T] satisfies the usual conditions, and P is a collection of mutually equivalent probability measures on the measurable space (, .7T). We model the subjective market uncertainty of each investor by associating to each investor a probability measure from P. Investors with more risk tolerance will be represented by probability measures that weight unfavorable events relatively lower, whereas conservative investors are characterized by probability measures that weight unfavorable events relatively higher. Moreover, it is assumed that investment information is revealed to each investor simultaneously as events in the filtration Y. Since the measures in P are mutually equivalent, the investors agree on the events that have and have not occurred. It is convenient to further assume that investors initially have no other information, i.e. .o0 is trivial with respect to each probability measure in P. This assumption asserts that the initial information available to investors is objective. 2 1.1.1 The Fundamentals of Fixed-Income Finance The foundation of a working knowledge of fixed-income finance rests on an understanding of the inherent relationship between the various interest rates and bonds. Consider the economy {(Q,f ,) : te j7} on the interval [0,T] and a Markov process Xt with .t7 o(Xs :0 < s < t). Implicit in this statement is the assumption that the state- variable probability P Px associated with Xt belongs to P for some fixed element x of the state space of Xt. A zero-coupon bond, or discount bond, of maturity T is a security that pays the holder one unit of currency at time T. The prices of government and corporate discount bonds at time t < T are denoted by B(t,T) and B(t,T), respectively. The local expectations h)ypothei, (L-EH) relates the discount bond to the instantaneous interest rate, or spot rate, for borrowing and lending over the time interval [t,t + dt]. Denote the riskless spot rate by rt = r(Xt) and assume that it is a nonnegative, adapted process with almost all sample paths integrable on [0,T] with respect to Lebesgue measure. The L-EH asserts that B(t,T) = Ep (exp (-T r(Xs) ds) (1.1) According to Musiela and Rutkowski (1998), the economic interpretation of this hypothesis is that "...the current bond price equals the expected value ... of the bond price in the next (infinitesimal) period, discounted at the current short-term rate" (p. 283). This statement is better understood in a discrete-time setting. In fact, using a left sum approximation to the integral in (1.1) with the partition {tiij 0 of [0,T] yields B(0,T) = Ep exp- r(Xt, 1Ati (1.2) =EP exp(-r(X)At )exp r (Xt, )Ati = exp(-r(x)tl)EP EP exp -2r(Xt )Ati1 = exp(-r(x) t)E (B(t, T)). Under the assumption of no arbitrage, it will be shown that (1.1) holds under the risk- neutral measure in Section 2 Naturally, a similar relationship holds between the risky bond B and the risky spot rate it, which will be derived in Section 3. The savings account Bt is a process that represents the price of a riskless security that continuously compounds at the spot rate. More precisely, it is the amount of cash at time t that accumulates by investing one dollar initially, and continually rolling over a bond with an infinitesimal time to maturity. Hence, we have Bt- exp( r(Xs)ds). (1.3) When a security St is divided by the savings account, the resultant process is the price process of the security discounted at the riskless rate. Another bond of importance is known as the coupon bond, which pays the holder fixed coupon payments c1,...,cn at fixed times T',...,Tn with Tn = T. The price of the coupon bond is simply the present value of the sum of these cash flows. Denoting the price of a riskless coupon bond at time t by B,(t,T), we have Be(t,T)= cB(t,T). (1.4) T,>t A similar relationship holds for the risky coupon bond Bc. In practice, the coupons are typically structured by setting c, = c for i = 1,...,n -1, and c = N + c, where N is the principal, or face value, and c is a fixed amount that is generally quoted as a percentage of N called the coupon rate. A problem that arises in comparing coupon bonds is that the uncertainty of the rate at which the coupons will be reinvested causes uncertainty in the total return of the coupon bond. Hence, coupon bonds of different coupon rates and payment dates are not directly comparable. The continuously compounded riskless yield-to-maturity (YTM) Y(t,T) is the unique solution to the equation Be (t,T) = c, exp (-Y(t, T)(T t)), (1.5) T, >t and represents the total return on the coupon bond under the assumption that each of the coupon payments occurring after t are reinvested at the rate Y(t,T). The risky YTM Y(t,T) is defined in a similar fashion. The interested reader should verify that there exists a unique, adapted, nonnegative process {Y(t,T)}0 tT given the adapted coupon bond process, coupon payments, and payment dates. In fact, this follows by noting that the LHS of (1.5) is a decreasing function of Y, and that the price of a coupon bond will never exceed the sum of the coupon payments. The yield-to-maturity expectations hypothesis (YTM-EH) relates the riskless YTM and the riskless spot rate. Musiela and Rutkowski (1998) state this hypothesis as the assertion that "...the [continuously compounded] yield from holding any [discount] bond is equal to the [continuously compounded] yield expected from rolling over a series of single-period [discount] bonds" (p.284). To gain a better understanding of this statement, we first observe that the YTM of a discount bond is simply the continuously compounded interest rate. Hence, in a discrete-time setting with the partition {ti}7 0 of [t,T], we have that the yield of a discount bond B(t,_1 1,t,) is given by Y(ti_ 1, ) = r(Xt ), (1.6) from which we deduce that the bond price is given by B(ti_ ,ti)= exp (-r(Xt )Ati). (1.7) Since the YTM-EH asserts that the yield of B(t,T) is equal to the yield expected from rolling over a series of discount bonds B (t1,, t,), it follows that Y(t,T) -- ln(B(t,T))=- 1 Ep In B(ti, ti) (1.8) T -t T -t Ep r Xt, )Ati Taking the limit as the mesh of the partition tends to zero, we obtain the continuous- time discount bond price and YTM under the YTM-EH: B(t,T)= exp I-Ep ( r(Xs)dst (1.9) and Y(t,T) = Ept r(Xs)ds (1.10) The last interest rate that we will consider is the instantaneous forward interest rate, or forward rate for borrowing or lending over the time interval [s,s + ds] as seen from time t _< s. This will be denoted by f(t, s) in the riskless case and f(t,s) in the risky case. If the dynamics of the process {f(t,s)}tas, are specified, then the price of the discount bond is defined by B(t,T) exp f(t,s)ds (1.11) Alternatively, if the dynamics of the discount bond are known, then we have f(t,T) --lnB(t,T), (1.12) OT provided that this derivative exists. By combining (1.9) and (1.12) we obtain f(t,T)= Ep (r(XT)| .). (1.13) Therefore, the YTM-EH asserts that the forward rate is an unbiased estimate of the spot rate under the state-variable probability measure P. Under the assumption of no arbitrage, it is shown in Section 3 that this holds under the forward martingale measure. 1.1.2 A Review of Arbitrage Pricing Theory The terminology presented in this review may be found in Musiela and Rutkowski (1998). Consider the economy {(Q,F,t) :.te T} on the interval [0,T]. A trading strategy, or portfolio, 4 is a vector of locally bounded, adapted processes of tradable asset holdings. Moreover, it is assumed that every sample path is right continuous with left limits. A trading strategy 4( is called self-financing, if the ei',,ihh process Vt (4b) of the trading strategy neither receives nor pays out cash flows external to the assets that comprise the strategy. More precisely, let 4' denote the holding of asset S'. Then, a n self-financing trading strategy 4 (=(4,..., 4n) is defined by asserting that Vt (4) -4tiS i=1 satisfies (1.14) dVt tl d St' + + tn d t * Sn A self-financing strategy 4) is called an arbitrage portfolio, if its associated wealth process satisfies all of the following conditions for some (thus for all) P 7': Vo (4) = 0 (Zero Investment) P(VT () > 0) = 1 (Zero Risk) P(VT ()) > 0) > 0 (Possible Gain). Hence, an investor taking advantage of an arbitrage opportunity may become infinitely wealthy without risk. Under the assumption that arbitrage portfolios do not exist, it has been shown that there exists a risk-neutral, or martingale measure, Q in our economy under which the discounted asset process Zt -B St follows a martingale. This result, known as the Fundamental Theorem of Asset Pricing, is proven in a quite general setting by Delbaen and Schachermayer (1994, 1998). The next topic for review is the arbitrage pricing of financial derivatives. A self- financing trading strategy 4) is Q-admissible if the discounted wealth process Vt (4) B Vt () is a Q-martingale and uniformly bounded below with respect to t G [0,T]. The uniform boundedness condition is included to disallow trading strategies in which the investor's debt may become arbitrarily large. A contingent claim, or T - measurable random variable, C is Q-attainable if there exists a Q-admissible trading strategy 4 that replicates the value of C at time T (i.e. VT ()) =C ). The market is defined by M(Q) (S,0), where 0D consists of the Q-admissible trading strategies. A market is said to be complete if every contingent claim is attainable. Under the assumption of no arbitrage, an attainable claim C is uniquely replicated for each martingale measure Q. In fact, we define the arbitrage price process 7it (C Q) of C to be the wealth process of the uniquely replicating trading strategy. Since this strategy is Q-admissible, it follows that ,t (CQ)= BtEQ(BT1C ). (1.15) It is shown in Musiela and Rutkowski (1998) that 7t (CIQ1)= =7t(C|Q2) for distinct martingale measures Q, and Q2, if C is attainable with respect to both measures. Hence, the definition of arbitrage price is independent of the choice of martingale measure and will be denoted by 7't (C). Therefore, if we assume that the market is complete, then the pricing of contingent claims does not depend on the choice of martingale measure. Alternatively, we may assume that the martingale measure is unique from which it follows that the market is complete in the restricted sense that every contingent claim C with BT1C e LC (Q,F, Q) is attainable (Bjork, 1996). Either assumption will suffice for the contingent claims considered in this dissertation. Furthermore, it will be shown in the next section that the expression for the arbitrage price (1.15) can be rewritten with respect to the state-variable probability P associated with the Markov process X, used to model market uncertainty. 1.2 Potential Approach I: Riskless Bonds and the Martingale Measure Equipped with the notions from our review of mathematical finance, we will now present the potential approach to developing models of the riskless spot rate. The fundamentals of this approach will carry over to the next section where a framework in which credit derivatives such as risky bonds and the risk-spread option can be priced. Consider the economy {(eQ,g,[t) : te P} on the interval [0,T] and a Markov process X, with !t o(Xs :0< s< t). Combining the concepts of fixed-income finance with those of arbitrage pricing theory, we see that a discount bond is simply a contingent claim with the constant value one. Under the assumption of no arbitrage, it follows from (1.15) that there exists a risk-neutral measure Q such that B(t,T)= t (1)= BtEQ (B1 )= EQ exp(- Tr(X,)ds).t (2.1) This proves the previously stated assertion that the L-EH given in (1.1) is satisfied under the risk-neutral measure Q. Rogers (1997) has shown that the expectation in (2.1) can be rewritten with respect to the state-variable probability P using the state-price density t -exp r(Xs)ds Ep t = BNt, (2.2) where Nt = EpdQ Before proving this result, we recall the following abstract S dP ) version of Bayes Rule: Lemma 1 Let Q and P be probabilities on the measurable space (0,J), W-L be a sub- G-algebra of J, f e L, (Q, J, Q), and N dQ Then dP Ep(f N ") E f 7-) = Ep.(-) (2.3) EP(N19 ) proof: See pg. 458 of Musiela and Rutkowski (1998). U Theorem 2 For any contingent claim C, we have 7t (C) = E[(2.4) proof: From (1.15) and Lemma 1, it follows that Ep (CBT1NT t) EPCCT t] t (C)= BtEQ(BT1 Ct)= Bt E(N- Ct (2.5) Ep(NTt) tt An immediate consequence of Theorem 2 is the desired expression: B(t,T)=- (2.6) This is the fundamental result of the potential approach. Since r is nonnegative, it is easily verified that Ct is a positive supermartingale with respect to P. In fact, we have EP( gt)=Ep Xp exp r(Xs)dsdQ gt (2.7) =exp(- r (Xs) ds)E dQ t t Furthermore, if we assume that Ep (C ) 0 as t -> 0, then t is a potential. From (2.6) it follows that this assumption translates into the reasonable financial assumption that the price of the riskless bond B(0,t) tends to zero as the time until maturity increases to infinity. The general potential approach to fixed-income finance outlined in Rogers (1997) is to generate models of the spot rate through a judicious choice of the state-price density. The only mathematical restriction is that Ct must be a positive supermartingale with respect to P. A specific procedure is to choose a positive function g defined on the state space of X, and use this to define Ct by at U'g(Xt) t e (2.8) Uag(Xo)' where (Uag)a is the ac-potential of g defined by Uag(x)= Ex Je g (Xs)ds (2.9) Since this Ct is clearly nonnegative, we must only verify that it is a supermartingale with respect to P. In fact, consider the martingale Mt= Ep e -asg(Xs)ds gt (2.10) Applying the Markov property of Xt, we deduce Mt = At +eatUag(Xt), (2.11) where At t eg(Xs)ds is an increasing process. It follows from (2.8) that 1 t = Ug(X0)(Mt At) (2.12) from which we deduce that Ct is a supermartingale. Given the above model for the state-price density, (2.6) may be employed to derive the price of a riskless bond. We will now derive an expression for the riskless spot rate by comparing (2.2) with (2.12). It follows from (2.2) that d t = Bt dNt r(Xt)Ctdt, (2.13) On the other hand, we deduce from (2.12) that d^t = 1 (dMt -e-atg(Xt)dt). (2.14) Uag(X0) Since Nt is a (local) martingale, a comparison of (2.13) and (2.14) reveals the desired expression for the riskless spot rate r(Xt) e- tg(Xt) g(Xt) (2.15) Uvg(XoK)t Ug(Xt)' where the model of the state-price density (2.8) was used in the final equality. Rather than specify g directly, it is convenient to model its a-potential by a nonnegative function f, defined on the state space of Xt, which lies in the domain of the infinitesimal generator G of Xt. With f(Xt)= Uag(Xt) we rewrite (2.8) as t = eat (Xt) (2.16) f(Xo) Since UI is the inverse of a G, we have g = (a G)f. Hence, the expression for the spot rate (2.15) may be rewritten as (a -G) f(Xt) (2.17) f (Xt) 1.3 Potential Approach II: Risky Bonds and the Forward Martingale Measure The main goal of this section is to extend the results of the previous section to the risky setting by developing a model in which the risky bond price is expressed in a similar fashion to the riskless bond in (2.6). The main difference is that the riskless state- price density t is replaced with the risky state-price density (pt, which is expressed in terms of the forward martingale measure instead of the risk-neutral measure. Consider the economy {(Q,.F,): t e P on the interval [0,T] with F defined below. Let the Markov process Xt denote the state-variable process, and v be a random time denoting the time of default. Following Lando (1998), we define Y so as to make v a stopping time and adapted to X, as follows: t G (Xs: 0 < s< t) -o([v>s]:0 In the first of the following two sub-sections, an expression for the risky bond price is derived under the assumption that v may be represented by the first jump of a generalized Poisson process. The forward martingale measure is introduced in the second sub- section to derive the risky bond analog of (2.6). 1.3.1 Risky Bonds A risky bond is a contingent claim that pays the holder one unit of currency at maturity in the event that there is no default. Hence, under the assumption of no arbitrage, it follows from (1.15) that there exists a risk-neutral measure Q such that B(t,T) t I[v>T]) = BtEQ (BT11[v>T] .)= EQexp(- Tr(Xs)ds)l[v>T] .t. (3.1) In applying the potential approach to risky bonds, it will be convenient to rewrite the conditional expectation in (3.1) with respect to !t. Before this can be achieved, we will recall the Monotone Class Theorem (MCT) and use it to prove some preliminary results. Theorem 1 (The Monotone Class Theorem) Let A and D be collections of subsets of a set C. Then o(A) c D if the following conditions are satisfied: (i) A B e A for every A and B in A (ii) A cD (iii) C D (iv) B \ A D for every A and B in D with A cD (v) JAi e D for every increasing sequence {Ai}11 of sets from D . i=1 A proof of this result may be found in Blumenthal and Getoor (1968) among other places. The first result that we will prove using the MCT is Lemma 2 For every t < T and A e W, we have that either Q([v > t]n A g)= Q([v > t]T) (3.2) or Q([v >t]n A |)= 0. (3.3) proof: It suffices to show that for every t < T, A e Wt, and C e g we have that either Q([v >t]n AnC)= Q([v >t]n A) (3.4) or Q([v> t]nAnc)= 0. (3.5) Let t < T. We will apply the MCT with C 0, A- {[v > t]:0 < s< t}, (3.6) and D {A e W,: Q([v > t] nA C)= Q([v> t]n C) or 0 for every C eT}. (3.7) Let 0s < s, < s < t. Since [v > s] [v > s2] = [v > s2], it follows that A is closed under intersections and A c "D. Hence, A satisfies the hypotheses of the MCT. We proceed to verify the hypotheses on D . Since it is clear that Q e 1D, we begin by showing that D is closed under proper differences. Let A e D, B e D, and C e!g with A c B. We have Q(B\An([v >t] C))= Q((Bn[v >t]nC)\(An[v>t]n C)) (3.8) = Q(Bn[v> t] c)-Q(A [v > t]n c). If Q(B [v >t]nc)= 0, then the RHS of (3.8) is also equal to zero since AcB. So, assume that Q(B l [v > t] n C) > 0. Since B e D we have that Q(B [v >t]n C)= Q([v >t]lnC). (3.9) Hence, if Q(An [v> t]n C)= 0, then the RHS of (3.8) is equal to Q([v >t]n C). On the other hand, if Q (Af [v > t] n C) > 0, then Q(An[v >t]nc)= Q([v> t]fnc) (3.10) since A e D. From (3.9) and (3.10) we deduce that the RHS of (3.8) is zero. It follows that B \ A e D. The final hypothesis of the MCT that must be verified is that D is closed under increasing sequences. Let {Ai}i1l be an increasing sequence of sets from D and define A JAi We will show that Ae D. We begin by defining the pairwise disjoint i=1 sequence of sets {Bi}1 by B,1 A, and B,1 -= A,1 \A,. Since D is closed under proper differences, we have that B1 e D for every i > 1. Furthermore, it is clear that A = Bi It follows that i=1 1>Q(An[v>t]nC)=Q QUBin[v>t]inc = Q(Bin[v>t]nC). (3.11) Since this series is finite, we have that Q(Bi n[v >t]n C) =0 (3.12) for all but finitely many i. Define I {i>1:Q(Bi n[v>t]nc)>0}. (3.13) Since B, E D for every i > 1, it follows that Q(An[v>t]nlC)= Q(Bin [v>t]n c) =Q UBi[v>t]c), (3.14) iel iel :! Q (Aim- n IV > ti n c):! Q (IV > ti n c), max{i e I} and the last inequality holds since Ai e u. since B1 E D for every i e I, we also have that Q(An[v >t]n C) ZQ(Bi n[v > t]n C) iel nQ([v > t]l C), where n denotes the cardinality of I. Comparing (3.14) with (3.15), we see that n must be zero or one from which we deduce that A e D. Therefore, we conclude from the MCT that Wt = cy(A) c- D either (3.2) or (3.3) holds for every t < T and A e ',. Theorem 3 For every t < T we have Q(v >T I 9Tv7=t) Q(v >T !9T) This implies that (3.16) proof: Define Yt -Q(v>t gT) for every t < T and rewrite (3.16) as I[v>t]YT . (3.17) In the following proof, be aware that Yt is T -measurable for each t < T and is not necessarily !t -measurable. Since the RHS of (3.17) is measurable with respect to QT v Wt, it suffices to show that for every t < T and De T v Wt we have EQ ([v>t]nDYT) . Let t < T. We will prove (3.18) using the MCT with C Q 0, A {A B: A e T,B t}, -DV tD e T v7-t:EQ (1[ >T~nD Y)EQ(V hDT (3.15) (3.18) (3.19) (3.20) where i.,,, Furthermore, E Q ('[v>T]yt gT V Wt ) E Q (I [v>T]nDyt) E Q (lv>tnDYT)j - We begin by checking that A satisfies the hypotheses of the MCT. Clearly, A is closed under intersections. Let Afl B e A for some A e T and Be W, We will show that A c_ D by considering the following two cases: Case 1 Q([v > T] nB Q) > 0 Since Be W c 7-T it follows from Lemma 2 that (3.21) Q([v > TinB )= Q(v>T ). Furthermore, since [v > T] c [v > t] we have that Q([v > t]nBs T)> Q([v> T]nB 9T)> 0. Thus, another application of Lemma 2 yields (3.22) Q(v> t gT). (3.23) From (3.21) and (3.23) we deduce that (3.24) EQ ('A MYt Y) EQ(l[v>t]n(AnB)YT) - Hence, Afl B e D which implies that A c_ D in this case. Case2- Q([v > T]nB ) =0 From the assumption of this case, we have that EQ ([v>T]n(AnB)Yt) (3.25) So, it suffices to show that EQ (l[v>t]n(AAB) YT) -0. EQ ('[v> T]n(AnB)yt) EQ (A YTQ (V > tj 9T)) E Q (A YTQ (IV > ti nB19T)) (3.26) Q([v >tin B19T) EQ (IAYQ (IV > Ti nB19T )) = EQ (A YIQ (IV > T11 9T)) EQ (IAYtQ([v > Ti nB19T)) Hence, we may assume without loss of generality that YT > 0. This implies that Q([v >t]nBc g )> Q([v > T]nBc g) (3.27) =Q([v>T]nBC Ig)+Q([v>T]pn3BT) YT>0. It follows from Lemma 2 that Q([v>tlB cg)=Q(v>t t ) (3.28) since Bce t. On the other hand, we have that Q(v >t jt)= Q)+([v >>tBct]lB~ 1). (3.29) Combining (3.28) and (3.29) we obtain Q([v> t]nBgT)= 0 (3.30) from which we deduce that EQ ([vt]n(AB)YT) = EQ(1AYTQ([v > t] B )) = 0. (3.31) Therefore, we deduce from (3.25) and (3.31) that A c_ D in this case as well. We proceed to verify that D satisfies the hypotheses of the MCT. Clearly, Q e D. In fact, we have that EQ (l[v>T]Y) = EQ(YQ(v > T T)) EQ (YYT) (3.32) = EQ (YTQ(v >t gT))= EQ (I[v>t]YT). The next step is to show that D is closed under proper differences. Let A and B be sets in D with A c B We deduce that B \ A e D from the following result: EQ([v>T]n(B\A) Yt) = EQl T]rI | >T]nA) Yt )EQ(I TB)Y t -EQ(([v>TPA)Yt) (3.33) = EQ( I t]B)YT)-EQ (1 >]fnA) T) = E lQ ( t]r I t]nA)YT )= EQ ([vt]n(B\A)YT) The final hypothesis of the MCT that must be verified is that D is closed under increasing sequences. Let {Ai}i 1 be an increasing sequence of sets from D and define A JAi We will show that Ae D. As in Lemma 2, we begin by defining the i 1 pairwise disjoint sequence of sets { Bi by B, -A, and B, 1 A,1 \A. Since D is closed under proper differences, we have that B, e D for every i > 1. Furthermore, it is clear that A = JBi Thus, it follows that A e D from the following equality: i=1 EQ (Y> El T]A)= EQ(Y[vT] B) = EQ(YT1[v t]B,) EQ(YT1[vt]n1A) (3.34) i=1 i=1 Hence, we conclude from the MCT that g! V t = (A) C- D Thus, (3.18) holds for every t < T and A eT v Wt from which we obtain the desired result (3.16). U We now return to the problem of rewriting the conditional expectation in (3.1) with respect to !9,. Conditioning with respect to 9T v Wt first yields B(t,T) = EQ EQexp(- r(Xs)ds)l[v>T] T v i t (3.35) =EQ exp T r(Xs)ds)Q(v > T |v )t) t T Q(v>T a) = 1[v>t]EQ exp(-Jt r(Xs)ds Q(V >) ,t where the last inequality follows from Theorem 3. This expression for the risky bond is more attractive than (3.1) since the argument of the conditional expectation is now OT- measurable. However, we are still unable to replace the conditioning on 't with conditioning on !t. This will require an independence assumption and the following lemma: Lemma 4 Let .Y, .2, and 3 be sub c-algebras of the c-algebra .Fsuch that F1 v . is independent of .3. Then for every integrable, YJ -measurable function Y we have E(Y v.) =E(Y| ). proof: See Section 9.2 of Chung (1974). U Assuming that gT is independent of ,t for each t < T, we may apply Lemma 4 with - G T, T2 -t G, and c --Lt to (3.35) to obtain B(t,T)= 1[v t]E, exp (-Jr(Xs)ds) Q(v>tg t (3.36) The financial interpretation of his independence assumption is that the riskless spot rate up until the time horizon T is independent of the default status of the bond prior to maturity. We will now present a model of the default time in which this independence assumption is somewhat relaxed. As in Lando (1998), we model the default time by v inf t> 0: X(X,)ds> S , where S is distributed under Q as a unit exponential random variable that is independent of !T, and k is a nonnegative, continuous function on the state space. The default time may be regarded as the first jump in a generalized Poisson process with intensity At f(Xs)ds, and we deduce that Q(v > t QT) = exp (-At). In fact, we clearly have that At is measurable with respect to t. So, it suffices to show that for every A e t we have Q([v > t]l A)= EQ(1Aexp(-At)). It follows that Q([v> t]lnA)= Q([At <$]S A)= EQ(lA e -udu)= EQ(A exp(-At)). (3.37) Hence, (3.35) becomes B(t,T) = [ t]EQ exp T(X,)ds) .t (3.38) where r+ k. Now, since is independent of !T, we may apply Lemma 4 with SQ 2- and 3 = o(9) to obtain EQ exp(- fi(Xs)ds)at9vG()] =EQexp(- TF(Xs)ds)gJ (3.39) Since !t c v c ((8)), we may condition with respect to 't on both sides of (3.39) to obtain EQ exp(- TF(Xs)ds). = EQ exp(-f T(Xs)ds) Qt. (3.40) Combining this with (3.38) we deduce the desired expression for the risky bond: B(t,T) = [ t]EQ exp (- T(X,)ds) gt (3.41) We conclude that the process kt represents the risk spread between the riskless spot rate r, and risky spot rate it. Intuitively, this makes sense since the probability of default increases with .t . 1.3.2 The Forward Martingale Measure The next step before applying the potential approach to risky bonds is to develop an understanding of the forward martingale measure (FMM) which is denoted by F. We define this probability measure through the derivative dF B, B(t,T) dQg BB(0,T) It should be obvious from (2.1) that (3.42) defines a Q-martingale, since it is shown there that the discounted bond price follows a martingale under Q. The FMM arises when pricing forward contracts in a market without arbitrage. Formally, a forward contract is an agreement established at time to < T to exchange an asset for a prearranged delivery price at time T. More precisely, we have Definition 3 A forward contract written at time to < T on an attainable contingent claim C for settlement at time T with delivery price K is an attainable contingent claim H -C -K, where the delivery price is a fixed amount of cash determined at time to. Since there is no initial exchange of money between the participants of a forward contract, the delivery price must be set such that 7t (H) = 0. In other words, the delivery price K is set equal to the arbitrage price of C. If this is not the case, then it can be shown that an arbitrage portfolio exists. Example 4 Let H be a forward contract as in the previous definition with to = 0 and suppose that 7ro (H) > 0. Let (c and e(H denote the replicating trading strategies of the attainable claims C and H, respectively. Then an arbitrage portfolio can be constructed by taking a long position in the forward contract, short selling (c, and purchasing 7C riskless bonds of maturity T. Denoting this portfolio by y, we see that B(0,T) t = B (0,T) 0...,0 +eH, (3.43) where the first coordinate denotes riskless bond holdings. It follows that Vo(Y)= 0 () B(OT)-Vo (c)+ V H) t(-V c) 0. (3.44) S B(0,T) This implies that the trading strategy \y requires no initial investment. Furthermore, 7V(0 C) (T,T)-VT C)+VTH) (C) -C+(C-K) (3.45) T B B(0,T) B (0,T) 7o0(C)-KB(0,T) _7o(H) >0 B(0,T) B(0,T) Hence, y also satisfies the zero risk and possible gain conditions of an arbitrage portfolio. In fact, y represents extreme arbitrage in the sense that a positive profit will almost surely be realized. On the other hand, if 7r0 (H) < 0 then an arbitrage portfolio can be constructed by taking a short position in the forward contract, short selling ) riskless bonds of B(0,T) maturity T, and purchasing 4( That is, we construct an arbitrage portfolio by negating the holdings of the previous case. Therefore, 70 (H) = 0 in an arbitrage free market. 0 Although the forward contract has an initial price of zero, its arbitrage price may fluctuate before it matures. We define the forward price of C at time t as the delivery price for which Tt (H) = 0 and form the adapted process {FF (t, T)}to (3.45) that the forward price is given by Fc (t, T) = t (3.46) B(t,T) Hence, the forward price C is simply the arbitrage price of C discounted by the riskless bond price. This is not surprising since the FMM is a special case of a change of numeraire with the riskless bond chosen as the new numeraire (Bj ork, 1996). The next result justifies the name "forward martingale measure" for F. Theorem 5 Let C be an attainable contingent claim that is integrable with respect to F. Then, the forward price process {Fc (t,T)}0 proof: Since C is integrable with respect to F and F, (T,T) for every t e [0, T] we have (3.47) We apply Bayes rule (Lemma 2.1) to obtain E K cl~~ E Q (C 0) F ( I EQ(YTlg,) EQ (Y 1YTC1 0, where the martingale Y, is defined in (3.42). From (3.42) and (3.46) we deduce EF BtEQ(BT1C gt) B(t,T) t(C) Fc (t,T). U B(t,T) Theorem 6 B(t,T) proof: An immediate consequence of (3.49) is 7t(C) = B(t,T)EF (C.). A comparison of (1.15) (with -t replaced by ) and (3.51) yields EQ exp (-T r(Xs)ds) C g Applying (3.52) to the attainable contingent claim C = exp ?L(X,) ds we obtain the desired result (3.50) from the risky bond equation (3.38). 0 It follows from this theorem that the YTM-EH (1.13) holds under the forward martingale measure. In fact, it follows from (3.52) with C = r, that EF(r1lgt) B(-T)EQ rTexp(-f T r(X,)ds)&'j 1 EQ (exp B(t,T) OBT I a B(t,T)= B(t,T)BT -_ Tr(Xs)ds)=t T -a lnB(t,T)= f(t,T). OT (3.48) (3.49) (3.50) (3.51) (3.52) (3.53) C, it suffices to show that Fc(tT) =EF(Cj91)- 7rt (C) = B (t, T) EF (C j!9t ) - I[V>t]B(tT)EF I exp (-ft T k(Xjdspt1. We proceed to extend the potential approach to the risky bond. A comparison of the expectation in (2.1) with that in (3.50) provides motivation for the following: dF Definition 7 The risky state-price density is given by (pt exp (-A )- . dP iR It follows from Theorem 6 that we may proceed as in Theorem 2.2 to obtain the risky analog of (2.6): EP [T(p 9t B(t,T) = [>t]B(t,T) PTt (3.54) In fact, it follows from Bayes Rule (Lemma 2.1) that EP e Xp TX (X s)ds dF T t EF [exp (-_f (Xs)ds) = t / P I t ]. (3.55) EP [fLt] [dP Hence, (3.54) follows from (3.50) and (3.55). Therefore, the risky bond price may be determined by specifying the risky state-price density in a similar fashion to the procedure outlined in Section 2. We finish this section by noting that the risky state-price density may be expressed in terms of its riskless counterpart and the risk spread. From the relation dF _dF dQ B(t,T) dQ B(t,T) (3.56) dP dQ g dP 9 BnB(0,T) dP g B(0,T) it follows that B(t,T) IPt= exp(-At) t (3.57) Inserting this into (3.54) yields Ep exp (-t T (X,)s)dsT t J B(t,T) = 1--t] (3.58)t This expression for the risky bonds resembles (2.5); however, it does not follow directly from Theorem 2.2 since 1[v>t] is not t -measurable. This result illustrates the importance of the independence assumption used to derive (3.41). 1.4 A Simple Example of the Potential Theoretic Approach Consider the economy {(QF,[t) : t e } on the interval [0,T] with F defined as in the beginning of the previous section. Let the state-variable process Xt denote the Gaussian diffusion with state space Rd satisfying dXt = dWt -0Xtdt (4.1) for some positive parameter 0, where Wt is a d-dimensional Brownian motion. Hence, Xt is the well-known stationary Ornstein-Uhlenbeck process given by Xt= eet(Xo + JtedWs). (4.2) It follows that Xt has distribution N(e-etX0, V), where Vt ( -e -21\). Also, the 20 generator of this process is given by Gf V2fV x Of (4.3) 2 1 ax1 for every f C2 (Rd) Define the function f : Rd -> R by f(x)= expd x l21 (4.4) for some positive parameter a. It follows from (2.17) that (a -G)f(Xt) 1 (4.5) f(Xt) 2 where ay 4&(0 &) and & -. Since the riskless spot rate must be positive, this d induces the condition 0 > &. We will show in Chapter 3 that this is the well-known mean-reverting Cox-Ingersoll-Ross (CIR) process. Next, the riskless state-price density given by (2.16) is Se-tf(Xt) f(Xo) exp(-(at + Kro))exp(Krt), (4.6) 2a where K -- . od Hence, the price of the riskless bond may be calculated from (2.6): B(t,T) = t Ct exp (- (aC+Krt))Ep(exp(KrT) |t) exp( (aT +c'c r)) E' (exp(Kr,)) d (1-2&V,) 7 exp (-(Qrt +a C)), V where V- T and -c 1-2&V, T t. We deduce that the riskless YTM is given by Y(t,T) -lnB(t,T) LTt 2cLTJ Finally, we use (1.12) to derive an expression for the riskless forward rate: a1nB (t, T) f(t,T) = nB(tT) OT exp(-20'c) r, (1-2&V,) 1-2&V, 2cL&V,2 (4.9) Similarly, in the risky setting we use a function h : Rd -> IR defined by h(x)= exp xI 2 d (4.10) to model the risky state-price density, where P3 is a positive parameter. We deduce that (1- G)h(Xt) = 1X 2 h(Xt) 2 (4.11) 1 ) and P1 -. Since the risk spread must be positive, this induces the d (4.7) (4.8) where & 4(0 condition 0 > 3 It follows that the risky state-price density is given by pt e-Pth(Xt) exp (Pt + flo))exp (flt), h(Xo) 2f where From (3.54), (4.11), and (4.12) we obtain the risky analog of (4.7): Bd B(t,T) = I v>t]B(t,T)EP[T I t I- -I ON V where V, We deduce that the risky YTM is given by 1- 2BV, Y(t,T)t--Iln(t,T)= 1[v>t] Y(t,T)+ I Vt +Pc+ n(1- 2V) . We conclude this example with an expression for the risky forward rate: We conclude this example with an expression for the risky forward rate: 1[ ~t](f(tT)+ ex(2c) 2f0f3VJ] Ol'Itj (4.12) (4.13) (4.14) (4.15) 1.5 The Risk-Spread Option Continuing with the economy of the previous section, we will conclude this chapter by presenting the payoff of the risk-spread option. Consider an investor who purchases a risky bond at t = 0 that matures at time T. We would like to construct an option that will guarantee that he would receive a minimum return of y above the riskless spot rate. We define the risk-spread option by its payoff of C= exp (J(y- )+ ds)-1, (5.1) and we refer to y as the risk-spread insurance level. d I[V> t] B (t, T) (I 2V, )_ 2 exp(-('"kt +P-C)), We begin with a simple deterministic, discrete-time example under the assumption that the risky bond does not default. Suppose that the riskless spot rate is given by rp, ift < to r, = (5.2) P2 if t > to and that the risk spread is given by kt ={q, ift where q, > y > q2. From (5.1) and (5.3) we have that C =exp((y- q2)(T- to))-1. (5.4) Since the risky rate i, = rt + kt is deterministic, it follows from (3.41), (5.2) and (5.3) that S(0,T)=exp (- Fds) exp(-to(p,+q ))exp(-(T-to)(p2 q2)). (5.5) Because we are assuming that the risky bond does not default, the investor receives one dollar at maturity. Hence, the total (continuously compounded) return the investor receives from the option and the risky bond at maturity is I 1+C R= ln (5.6) T 1B(0,T)+7o (C) Assuming that '7o (C) = 0, we have that It can be seen from this result that the investor is compensated when the risk spread drops below the insurance level y after time to. If to = 0 then the investor is guaranteed to receive a return of y above the riskless spot rate. On the other hand, if to = T, then the option expires out-of-the-money and the investor receives the risky spot rate of return. However, from (1.15) we see that n0 (C) = BT1C > 0. This implies that the return on the risk-spread option is the riskless spot rate. In fact, the rate of return of holding any deterministic derivative is the riskless spot rate. Of course, there is no need for the risk- spread option in a deterministic world. In the general stochastic, continuous-time case we deduce from (5.4) and (5.6) that 1 Iexp( 7- As)+ ds) R =I In -exp((y- ) ds) (5.7) T B (0,T) + 7o (C) = OT ( y- s) ds ln(t (0,T) + 7o(C))). We would like to find 9 > 0 such that R = T +f(9- s) )ds. (5.8) If there exists a constant y such that (5.7) and (5.8) are equal, we define the effective risk-spread insurance level 9 as the solution to Ep((_ T)+) =Ep ((y- T)+-rT) in( (0,T) + 7ro(C)). (5.9) In the next chapter, the general stochastic, continuous-time case for the risk-spread option is studied in detail. In particular, a representation for the arbitrage price of this option is derived as the solution to a Cauchy problem. CHAPTER 2 A CAUCHY PROBLEM FOR THE RISK-SPREAD OPTION 2.1 Derivation of the Cauchy Problem Consider the economy {(Q,YF,P) : P 7} on the interval [0,T] with F defined as in Section 1.3. Let C = exp( (y (Xt)) dt- 1 denote the payout at time T of a risk- spread option with fixed risk-spread insurance level y, where {Xtj0 variable process with natural filtration !t and probability P e P defined in Section 1.4. According to (1.2.6), the price of this option at time t is given by EP(C) [C gT Ep (exp(Jf(y- (Xs)) ds)-l)(T t (C) = (1.1) EP exp Tsds w 1t, -- B(t,T) , where s (Xs) (Y- (Xs)) (1.2) and B (t, T) denotes the riskless bond price. Using the model of the state-price density developed in the simple example of Section 1.4, it follows from (1.4.6) that Ep exp ( Tsds ~T = exp(-(u.T+Kro))Ep exp(KrT) exp (JT ds) g~t (1.3) = exp(-(cT + Kro))exp( Ytsds )Ex exp(Kr,)exp(j'fsds)1. Combining (1.1) and (1.3) yields 7rt(C)=exp(-(aT+K1crt))exp(tsds)u(Xt,T)-B(t,T). (1.4) where u(x,T)-E exp(Kcr,)exp Tsds) =Ex g(X,)exp TYds, (1.5) vd g(x)-exp (&x 2), c- ,and 'c T-t. The goal of this section is to prove that (1.5) is the unique solution of class u e C2'1Rd1 x [0,T]) to the following Cauchy problem: Lu(x, )= 0 (1.6) for every ( x, ) e Rd x (0,T], and u(x,0) = g(x) (1.7) for every x e Rd, where L G+V-- (1.8) and G is the generator of the state-variable process given by (1.4.3). This result is known as the Feynman-Kac Theorem and is proven in (Karatzas & Shreve, 1991) under the assumption that there are constants M > 0 and [ > 1 such that for every x e Rd we have max u(x,T)_ We will extend this result by replacing (1.9) with a less restrictive bound using the following lemma, which appears as Problem 3.4.12 in (Karatzas & Shreve, 1991). Lemma 1 Let {Mt, Jt }o, be a continuous, real-valued martingale such that Mo = 0 almost surely. Also, let Ct be a continuous, real-valued process of bounded variation such that Ct + (M) < p almost surely for some p > 0 and every t > 0. Then, for every n > 2p, the semimartingale Yt Ct + M satisfies P(max Y, >n :<3(27rp) 2exp (1.10) O proof: We will first construct a time-changed Brownian Motion as in (Karatzas & Shreve, 1991). For every t >0 define the stopping time ft inf s 0: (M) > tj if 0:< t <(M)( 00 if t <(M) Without loss of generality1, assume that our probability space is rich enough to contain a standard one-dimensional Brownian motion {Bt, {Ytt< and define Bt B Bt + M (1.12) for every t >0. It follows from Problem 4.7 of (Karatzas & Shreve, 1991) that {Bt,' }10t is a standard one-dimensional Brownian motion such that the filtration gt }0t satisfies the usual conditions and for every t e [0,T] we have Mt = B() a.s. (1.13) For every positive integer n, we define the stopping time Rn -inft >0:Bt >I2. (1.14) Let n > 2p and note that Mt > Yt- Ct 4>Y Yt p> Y n (1.15) 2 for every t e [0, T]. Hence, [maxY n]c max Mt> crM > R. [p > Rn]. (1.16) W 0 Using integration by parts, it can be verified that for every x > 0 we have exp du<-exp (1.17) 1 See Remark 3.4.1 in (Karatzas & Shreve, 1991) for a technical justification. From (1.17), the symmetry of Brownian motion, and the reflection principle (Revuz & Yor, 1991), it follows that P (max Y >n)
o the first integral on the RHS of (4.3), we apply Theorem 3.2, Lemma 3.1, and Lemma 2.2 to obtain the estimate JZI [fs](x) < x -s (x |