• TABLE OF CONTENTS
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 Title Page
 Dedication
 Preface
 Table of Contents
 Abstract
 A potential theoretic framework...
 A Cauchy problem for the risk-spread...
 Numerical results and applicat...
 Summary and conclusions
 References
 Biographical sketch














Group Title: risk-spread option in a potential theoretic framework
Title: The risk-spread option in a potential theoretic framework
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Title: The risk-spread option in a potential theoretic framework
Physical Description: ix, 97 leaves : ill. ; 29 cm.
Language: English
Creator: Swearingen, Michael C
Copyright Date: 2000
 Subjects
Subject: Mathematics thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Mathematics -- UF   ( lcsh )
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theses   ( marcgt )
non-fiction   ( marcgt )
 Notes
Summary: 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.
Summary: KEYWORDS: interest rate derivative, potential theory, state-price density, fixed-income finance, stochastic processes
Statement of Responsibility: by Michael C. Swearingen.
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.
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Holding Location: University of Florida
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Resource Identifier: oclc - 45825348
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Table of Contents
    Title Page
        Page i
        Page ii
    Dedication
        Page iii
    Preface
        Page iv
        Page v
    Table of Contents
        Page vi
        Page vii
    Abstract
        Page viii
        Page ix
    A potential theoretic framework for fixed-income markets
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        Page 2
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        Page 4
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        Page 25
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        Page 29
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    A Cauchy problem for the risk-spread option
        Page 31
        Page 32
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    Numerical results and applications
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
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    Summary and conclusions
        Page 91
        Page 92
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    References
        Page 94
        Page 95
    Biographical sketch
        Page 96
        Page 97
Full Text










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 q2 ift > to

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

(1.18)


4P B> 4
JiT


exp du
J2 l 2


6 n2 1 n


We will use this lemma to prove a version of the Feynman-Kac Theorem for the

state-variable process Xt defined in Section 1.4. Recall that


Xt e-X0j e- "dW~,


(1.19)


where Wt is a d-dimensional Brownian motion and 0 is a positive parameter. Define


Yt eetXt= X 0+Mt,


(1.20)


where the process {Mt, o }J is the continuous, real-valued martingale defined by


Mt f JeodW,


(1.21)


for every t > 0 Furthermore,


(M) =Jte2sds= 2--(e2et
/ *' 20


(1.22)


Hence, for every t > 0 we have

IN 11 + ~t : II 1 + oI (e 2
Xo +(M)t
Therefore, it follows from Lemma 1 that


P(max Yt >n) <3(27p) 2 exp .
O

(1.23)


(1.24)


1):! I (e 2ff
20









Theorem 2 (Feynman-Kac) Let u C2'1 (Rdx [0,T]) satisfy the Cauchy problem

given by

Lu(x, c)= 0 (1.25)
for every ( x, ) e Rd x (0,T], and
u(x,0) = g(x) (1.26)
for every x e Rd, where
L -G + (1.27)
Ox
and G is the generator of the state-variable process defined by

Gf -V2f- x d (1.28)
2 1=1 ax

for every f e C2 (Rd). Furthermore, assume that there exists positive constants K and h,
1
with h < -- and p defined by (1.23), such that for every x e Rd we have
8pd
max u(x, c) < Kexp(h I|x 2). (1.29)

Then, u is uniquely represented by

u(x,T) =E g(X,)exp(j sds)1. (1.30)


proof: Let ( x, C) e Rd x (0,T] with X0 = x almost surely. We will apply Ito's formula

to the process {Vs }1s defined by

Vs, u(X,T -s)Es. (1.31)
where
Ss exp(J sqdq) (1.32)

It follows that for every s e [0, c] we have


Vs-V0= q U(X, -q)dX)+ q V2-+ u(Xq,t-q)dq. (1.33)
Recalling that X=1 satisfies dt, we see that the first term becomes

Recalling that Xt satisfies dXt = dWt XN dt, we see that the first term becomes









X -0 q)dWi) -0 X) u(Xq,'-q)dq. (1.34)

Combining (1.25), (1.27), (1.28), (1.33), and (1.34) we deduce that

Vs-Vo= f -- U(Xq,T -q)dW) (1.35)
i=1
for every e [0, -c]. Moreover, the expectation of the RHS of (1.35) is zero, hence

u(x,c)= Vo = E(Vs)= E(u(Xs,T -s),s). (1.36)

Let Sn inf t > 0 : Y > n } for every n > 1, and fix me (0, c). Then,
u(x,Tc) = E (U(Xs mT -Sn Am)sm) (1.37)

= E (u(X,',T- m) El[s>m])+Eu( XSn Sn)'Sn l[Sn m] ).

As n -> oo and m 1T c, we see that the first term on the RHS of (1.37) approaches

E (u(X ,)) = Ep ig(X,)exp( Tsds). (1.38)

From (1.2), (1.24), and (1.29), we see that the second term is dominated by
E (u(XsT-Sn)S l[sm])
d
n
1l 1

< yKexp (hdn2) 3(27Tp) 2exp -_)
1 18phd p
<3yKd(27rp) exp 8P n2 )

which approaches zero as n -> o since l-8hpd> 0. Therefore, by combining (1.38)

and (1.39) with (1.37) we deduce the desired result (1.30). U









The main goal of this chapter is to construct a unique solution u e C2'1 (Rd x [0,T]) to

this Cauchy problem that satisfies (1.29). The next section introduces the basic

terminology inherent in the potential theoretic parametrix method. In Sections 3 and 4,

the relevant potentials are studied and technical differentiability results are provided. The

fifth section provides a series representation for the fundamental solution of the Cauchy

problem and establishes some continuity results. Finally, the chapter concludes by

constructing the solution to the Cauchy problem in Section 6.


2.2 The Potential Theoretic Parametrix Method


2.2.1 The Gaussian Semigroup


The solution presented in this section updates the parametrix method (Friedman,

1964) by using the modem theories of potentials and semigroups. The canonical family

of semigroups associated with the differential operator L is the Gaussian family

{Z6: 0 < 0} defined by

Z~ (x) exp x (2.1)
(2 2 -c

for every (x,'c) e Rd x (0,T] and 6 > 0 .2 We will omit the subscript -c when we wish to

refer to the Gaussian semigroup as a function on Rd x (0,T].

For each Tc e (0,T], we define the operator Z [.] by

Z[f ](x) (Z* f )(x) Z(x- y)f (y)dy (2.2)

for every f e C(R d) for which this convolution exists. Because of the bound (1.29) it is

important that the convolution is finite for the following subset of C (Rd ) :


2 The superscript 6 will be dropped for the standard Gaussian semigroup (i.e. 6 = 1).








Definition 1 For every positive 6, let

A =- f e C(IRd): f < Aexp (h for some positive h < and A > 0 (2.3)
32pd

denote the set of &admissible functions and denote A1 by A, where

p-= max{ -T,I|X0 + (e2T -1)}. Also, the bound parameter of a 6-admissible function
*4d 20
f is defined by
hf inf 00 (2.4)
32pd f

It follows that the initial function g appearing in our Cauchy problem (1.26) is 6-

admissible for some 6 depending on &. The next result gives an example of a class of

admissible functions for which the convolution in (2.2) is finite.

Lemma 2 Let c e (0,T], 0 < s < 26, and f e A. Then Z [f] e A, More precisely,

for every x e Rd we have that

Z [f](x) h exph 2) Z (x -)d (2.5)

d
6 2- exp 6I xI- 1 < Cexp IXII 2),
6-2Thf 6-2Thf
d
where C and hf hf < 2hf.
6 2Thf 6-2Thf

proof: We first note that since Z6 and f are continuous functions on Rd it follows

from (2.5) that Z [f]eC(Rd). Therefore, to prove that Zj[f] eA,,, it suffices to

verify that (2.5) holds. After completing the square, the integrand becomes

2---6 I Sh 1 (6- 2Tah) X 6.
exp hfx exp -- x (2.6)
S2 7rc 6- 2Thf 2 6 2 ) 6-2-hdf

Since hf e A25 we have 6 2chf > 0, thus we see that (2.5) holds for every x e Rd.








The final property of the family of Gaussian semigroups that we will review is

continuity in time. Clearly, for every x e Rd and f e A, with 0
Tl- Z->Z~(x) and Tcl->Z[f](x) are continuous. Furthermore, we may continuously

extend the action of the Gaussian semigroup by defining Z0 [f](x) limZ [f](x) = f(x).

In fact, this follows from the weak convergence of the measure defined by the Gaussian

semigroup to the point mass measure as c -> 0.


2.2.2 The Fundamental Solution

We begin this section by noting that for a given function F e C (Rd x (0,T]), we will

refer to the family {F} (0,T] as a continuous family of functions from C(Rd), where

F (x) F(x,c) for every (x,T) Rd x(0,T].

Definition 3 A fundamental solution for L is a continuous family {F}1 (0,T] of

functions from C(IRd) such that for every c e (0,T] and f e A we have:

(i) L(F [f]) L(F f) = 0

(ii) 0 [f] lim(F *f) =f.

It will be shown that the solution to the Cauchy problem given by (1.25) and (1.26)

may be represented by
u(x,c)= F [g](x), (2.7)

where the fundamental solution F, shall be constructed using the following parametrix

method. Consider the following Cauchy problem for a given f A :
Lpu(x,T)= 0 (2.8)

u(x,0)= f(x), (2.9)
where the principal part Lp of the differential operator L is defined by

1V2 a
L V2 C (2.10)
2 0-








It follows from the Feynman-Kac Theorem that the solution to this simple heat equation

may be represented by
u(x,c)= Z [f]. (2.11)

A fundamental solution for L will be known as a parametrix associated with L. In

our case, it follows that the Gaussian semigroup Z, is a parametrix for L. This result will

be used to motivate an expression for a fundamental solution for L that satisfies (2.7).

Definition 4 The Gaussian potential is a family of operators {U1}0~ T defined by

U, [f ] Z,-s [f]ds (2.12)

for every f e C(IRd) for which this integral is defined. For the continuous family

f -{f}0<,T of functions from C(Rd), we define

U [f]- 'Z s[fs]ds. (2.13)

As with the Gaussian semigroup, we will omit the subscript -c when we wish to refer to
the Gaussian potential as a function on Rd x (0,T].

Recall that our state-variable process satisfies

dXt = dWt -OX dt. (2.14)

Hence, if 0 and are identically zero, then it follows from (1.5) and (2.14) that

L = Lp. This inspires us to look for a fundamental solution for L that satisfies (2.7) in

terms of the Gaussian potential of some continuous family p- { p}0<, T of functions

from C(IRd), as follows:
F= Z +U [4]. (2.15)

The function (p will be determined by condition (i) of Definition 3. This condition

implies that for every Tc e (0,T] and f e A we have

0= L(F[f])=L(Z,[f])+L(U,[p][f])=L(Z *f)+L(U, [p]*f). (2.16)








Consequently, we are led to an investigation of the differential properties of the

Gaussian semigroup and the Gaussian potential. In particular, we will show in Section 3

that the differential operator L may be passed through the convolution

Z, [f]. More explicitly, for every (x,'c) e Rd x (0,T] and f e A we will show that

L(Z [f])(x)= JdLZ (x- )f ()d. (2.17)

In Section 4, we will derive sufficient conditions on (p so that the differential operator
L may be passed through the convolution LU1 [p][f]. More precisely, it will be shown

that U, [p] is well-defined provided that there exists positive constants C and 6

with 6 _< 1 such that for every (x, ') e Rd x (0,T] we have

I (x) < CZ (x). (2.18)

In this case, we shall say that (p is Z3 -bounded. Furthermore, if (p is locally, uniformly

Holder continuous in space, then we will prove that

L(U [[f])= J'L(Zs [s][f])ds- p[f] (2.19)

for every c e (0,T] and f e A, where

(p[f] (p *f (2.20)
and
L(Z,-s s][f])= L(Z,- s[ps]*f). (2.21)

In Section 5, we will construct a series representation for a function (p satisfying

(2.16) and prove that it satisfies the conditions above, which have been asserted to be

sufficient for (p to satisfy (2.19). We will define (p in terms of the following potential:









Definition 5 The L-potential is a family of operators {VM }10T defined by

V [f] L(Z, [f])ds (2.22)

for every f e C (Rd) for which this integral is defined. For the continuous family

f- {f}0
V, [f ] L (Z [f ]) ds. (2.23)

As with the Gaussian Potential, we will omit the subscript -c when we wish to refer to

the L-potential as a function on Rd x (0,T]. It follows that (2.19) may be rewritten as

L(U, [p][f])= V,[p][f]- [f], (2.24)

where
V1 [1][f]- V []* f. (2.25)

Combining (2.24) with (2.16), we deduce that p must satisfy the following Volterra

integral equation for every Tc e (0,T] and f e A:

T, [f] = L (Z, [f])+ V, [][f]. (2.26)

It is shown in Section 5 that the solution of (2.26) may be represented by the series

S= Vm [LZ], (2.27)
m=0

where Vm is an operator defined by

Vo I (2.28)
and
VM + V1 o Vm (2.29)


for every T e (0,T] and m > 0.









2.3 Preliminary Technical Results


2.3.1 Differentiability of the Gaussian Semigroup


In this sub-section we begin presenting the technical results that are required to

complete the construction of a fundamental solution as outlined in Section 2. The main

result is that for every (x, c) e Rd x (0,T] and f e A we have

L(Z [f])(x)= dLZ (X- ((3.1)

This will require a few preliminary results.

Lemma 1 Let (x, ') e Rd x (0,T], and let n, m, s, and 6 be positive constants with

2m- n+j
s < 86. For every constant [t with 0 < m _< there exists a positive constant A
2
such that for j = 0,1, and 2 we have


Ixrn InZ A
S-' (x) < (X) (3.2)
SM O II |x||2m-n+j-2t

2m- n +2
Furthermore, for every constant [t with 0 <- K-< there exists a positive
2
constant B such that

mXn (x)< Zl B n+22Z(x) (3.3)
-m a T x2 t mn+2-2T


proof: Let n, m, 6, and a be positive with a < 6 and define W(x, c) - Then for
2-c
2m n
every [t with 0 < t < -- there is a positive constant A such that
2








d (
mZ(x) = mT-m ( x -exp X
T "1 T 2x 2-


L(2W(x, ))m- exp(- (6 8)W(x,) J ep (x,C))
HI 2m-n 272C)


A
< 2m-n-224 Z (x).
T X12m- 1,


This establishes (3.2) for j


0. For j = 1 we deduce from (3.4) that


ii{a(X) xD i Z (x) < -+1 lZ(x)
-CM 2x, T n T '

< |||2m-n+1-2 T (x)
2m -n +1
holds for every 0 < K -< Similarly,
2


1 xxn =iii -
aTm CmX2 () z T


+x.az


(x) < 1 Z(x)
'C T1 T T


(3.6)




2m- n+2
holds for every 0 < t--< which implies (3.2) for j = 2. Finally, we obtain
2
(3.3) from the following inequality:

x --!(x) = d z [x) m+2 dZ (x) (3.7)
m 2m 2 22- T 2Tm+2 2T T
T6x||XIn2 dIxIIz> A
+ lii+ Z (x) < A Z- (_x))
S2tm+2 2-,+l )JT ( |x d+2m-n+2-2Z T(x)

We will now apply this lemma to obtain a result on the differentiability of the


Gaussian semigroup.


(3.4)


(3.5)







Theorem 2 For every 6 > 0 and f e A, we have Z6[f] G C2'1 (IRdx (0,T]). In fact,


Oz [f](x)= fd -Z x )f ()d (3.8)
and
Z 6 )= Z6 (x ) f()d (3.9)
O- T I[f] (x)N d OTC

for every (x,T) Rdx(0,T] and je{1,2}. Furthermore, Z [f] and -Z [f] are
functions in A28 for every T e (0,T].

proof: Let 6 > 0 and f e A,. Clearly, the integrands above are continuous functions
of (x,c) on Rd x (0,T]. Hence, to prove assertions (3.8) and (3.9) as well as show that

Z [f C2,1 (Rd x (0,T]), it suffices to show that the integrals in these assertions are
locally, uniformly bounded with respect to (x, ) e Rd x (0,T].

Let f e A, je {1,2}, and B { Rd : <1} From (3.8) we obtain


d CJZ (x- )f ) d:< C Z(x- )f() d + Z C (x-()fx )d (3.10)

Clearly, the first integral on the RHS of (3.10) is bounded uniformly in
(x, -) e Rd x (0,T]. For the second integral, consider a compact subset M c- Rd and let
(x, ) e M x (0,T]. We deduce from Lemma 1 with [t = 0 that for every positive g < 6
there exists a positive constant A such that

Sa-- (x )f (Q)d
-
Since we may choose g arbitrarily close to 6, we may assume that f e A,. It follows
from Lemma 2.2 that there exists positive constants C and hf < 2hf such that








Sexp(h x2)Z (x- )d
where N = diam(M) .
By combining (3.10), (3.11), and (3.12), we deduce that the integral in (3.8) is locally,

uniformly bounded with respect to (x,'c) e Rd x (0,T]. Moreover, this result may also be

established for the integral in (3.9) by repeating the proof using (3.3) to obtain the
analogue of (3.11). Consequently, we have shown that assertions (3.8) and (3.9) hold for

every (x,_C) e Rd x(0,T] and have proven that Z[f ] C21(Rd x(0,T]). Finally, it

follows from (3.12) that Zs[f] and -Z [f] are in A26 for every c e (0,T].

We proceed to prove the main result of this sub-section.

Theorem 3 For every f e A we have LZ[f] e C(Rd x (0,T]) In fact,

L(z [f])(x)= d LZ,(x- ) f()d (3.13)

for every (x,'c) e Rd x (0,T]. Furthermore, LZ, [f] e A, for each c e (0,T].

Remark: The differential operator L acts in the variables (x, _c) e Rd x (0,T].

proof: Clearly, the integrand in (3.13) is continuous in (x,'c) e Rd x (0,T]. Hence, to

prove (3.13) as well as show that Z6[f]e C(Rd x(0,T]), it suffices to show that the

integral in (3.13) is locally, uniformly bounded in (x, c) e Rd x (0,T].

We begin by proving that for every compact M cR d there is a C > 0 such that

LZ (x )
for every (x,-c) e M x (0,T] and ffI'1. Let M d_'1 be compact with diameter N,

(x,c) e Mx(0,T], and eI Rd. It follows from Lemma 1 with =t- that there is a
2
C > 0 such that for every positive 6 < 1 we have









LZ,(x- 0j <9 |xj Z (x -) + (x)Z,(x-() (3.15)


il
< ONC Z (x y) + -Z C (x 2 (X
1 1


Now, let (x,'c) e M x[q,T] for some q > 0 and 6 < 1 be a positive constant such that

hf < -- It follows from (3.15) and Lemma 2.2 that there exists positive constants C1,
32pd
C2, and hf < 2hf such that

L LZ,(x )f ()d< _C

We deduce that the integral in (3.13) is locally, uniformly bounded with respect to

(x,'C) Rd x(0,T] from which we conclude that L(Z[f])e C (Rd x(0,T]) and (3.13)

holds for every (x,'c) e Rd x (0,T]. Also, the bound (3.16) implies that L(Z, [f]) e A2

for each Tc (0,T]. 0

This result is the first step towards the construction of the fundamental solution of our

Cauchy problem. Recall that we wish to express the fundamental solution in the form

F, =Z, +U[p] (3.17)

for some continuous family (p {( 'p T of functions from C(Rd) that satisfies

0= L(F, [f])= L(Z, [f]) + L(U, [(p][f]) (3.18)

for every Tc e (0,T] and f e A. Theorem 3 allows us to pass the differential operator L

through the convolution in the first term on the RHS of (3.18).








2.3.2 Basic Potential Theory


Before turning our attention to the second term on the RHS of (3.18), we will

determine two classes of functions for which the Gaussian and L-potentials are defined.

Naturally, the class of admissible functions is a good place to start.

Definition 4 For every 6 > 0, the bound parameter of a family f {fj-0- T of

functions from A, is defined by hf sup hf < Furthermore, the family f is said
O to be uniformly bounded if hf < --
32pd

Theorem 5 Let gs < 2 and f {f } 0
functions from A,. Then, U[f] and V[f] are functions in C(Rd x (0,T]). Also, for

each Tc e (0,T] we have that U, [f] and V, [f] are elements of A2,.

proof: Recall from Definitions 2.4 and 2.5 that for every (x, c) e Rd x (0,T] we have

U,[f](x) Zs[f ](x)ds (3.19)
and
V [f](x) L(Z- s[f])(x)ds. (3.20)

From Lemma 2.2 and Theorem 3 we see that the integrands in (3.19) and (3.20) are

continuous for every (x,', s) e Rd x (0,T] x (0, ') Therefore, the desired assertions

follow from Lemma 2.2 and (3.16). In fact, we have that

U~[f](x) < Z~ [fs](x) ds :< Cexp(hf x)ds CTexp(hf x2) (3.21)
and
V[f](x) _< L(Z [fs])(x) ds_
<2CfTexp(hf x 2)

for every (x,c) e Rd x (0,T], where C > 0, C > 0, and h <2hf. .







We will use this result in conjunction with the following class of functions:

Definition 6 Let 6> 0. A family x {,}O T of functions from C(IRd) is called
Z3 -bounded, if there exists a positive constant C such that

p,(x) for every (x,'c) e Rd x (0,T].
Consider a continuous family i ({0x) } TT of Z- bounded functions and an
admissible function f e A,. It follows from Lemma 2.2 that

I[f ] < ] ld (x- )f ()d (3.24)

where C > 0 and hf < 2hf. Hence, {kV [f]}0) family of functions from A2,,. In particular, it follows from Theorem 5 that

U[Y[f ]]C(Rd x (0,T]) and UT[ y[f]] e A2 for each c e(0,T] and 6 <1.
The next goal is to show that \ is in the domain of the Gaussian potential and
combine our results to obtain
UT[x [f ]]= U [][f] (3.25)

for every -ce(0,T]. The following lemma is the first step towards this goal and
essentially follows from the semigroup property of the Gaussian kernel.
Lemma 7 For every a and b with -oc < a,b < 1 and positive 6 we have


L (T s) -aS bZ s(x-_ )Z'()dds= B(1- a,l-b)T'c1- bZ(x). (3.26)
JO' IdO'


where B denotes the beta function.








proof: We begin by substituting w in place of y where

S/ 6s
wi -i + sxi. (3.27)
2(T s)s 2(T- s)T'

After some algebraic manipulation, it is easy to verify that

6 12 x t- 1 I2 6 I w 2. I2
+ + 1w|U (3.28)
2(c- s) 2s 21c

Denoting the integral in (3.26) by I, we obtain the desired result
d
I= -exp 2 exp- w2) dw (C-2s ls s) 2 s 2 ds (3.29)
27r 2T d 6CT

=Z(x) (T-s)-as-bds= T-a- bZ(x) f(1-p) -p bdp

=B(1-a,l-b)T'-a-bZa(x),

where the substitution p = was used in the second equality. U

Theorem 8 Let 0 <8 < 1 and 1{}0< ~< be a continuous family of Z6- bounded

functions. Then, U, [x] is well-defined for every -c e (0,T]. In fact, {U, [x]}0T is a

continuous family of Z6- bounded functions.

proof: From Lemma 7, it follows that there exists a positive constant C such that


U[x|] < fZo s[Z, s]ds< J] d Z-,(x ) Is ()d(ds (3.30)



for every Tc (0,T]. 0

This result will now be used to establish (3.25).







Corollary 9 Let 0<6<1, f eA, and y { }0 Z'- bounded functions. Then, for every Tc e (0,T] we have
UT [f]] = U, [y][f]. (3.31)
proof: It follows from (3.24) that {, [f]}0 family of functions from A28 c .A2 Consequently, we deduce from Theorem 5 that the
LHS of (3.31) is well-defined. In fact, U [[f]] C(Rd x(0,T]) and UT[y[f]] e A2
for each c e (0,T]. Fix (x,'c) e (0,T] From Theorem 8 we have that U, [x] is well-
defined. Furthermore, we apply Fubini's theorem to obtain

U,[v][f](x) =U,[v]*f(x)=(JZ s[,- s]ds)*f(x) (3.32)
= d Ra -s [s](x )f ()dsd

= Rd JfRd ZI-s(x -- y)ys (y)f ()dydsd

= o Rd dZ -(x -- y)ys (y)f(Q)dydyds

= oRd Rd Z- s(x -w)ys (w &)f (&)dwd~ds
= Z Zs(X w) (d S(w )f ()d&)dwds

= Rd Z s(x- w) s [f]dwds
= J'Z-s[ys[f]](x)ds= U,[[f]](x).

The final objective of this section is to prove that continuous, Z- bounded
functions are in the domain of the L-potential for every 6 < 1. In particular, the analogue
of (3.25) for the L-potential will be established. We will first prove a result for
continuous, Z6- bounded functions that is similar to Theorem 3.








Lemma 10 Let 0<6<1 and ={ }0<~T be a continuous family of Z6-bounded

functions. Then, L(Z[[y]) eC (Rd x (0,T]). In fact, for every (x, ') e Rd x (0,T] we have

L(Z [y])(x)= d LZ (x- ) ()d (3.33)

proof: We proceed as in the proof of Theorem 3. Clearly, the integrand in (3.33) is a

continuous function of (x,'c) on Rd x (0,T]. Hence, to prove (3.33) as well as show that

L(Z[y]) C(Rd x(0,T]), it suffices to show that the integral in (3.33) is locally,

uniformly bounded with respect to (x,'c) e Rd x (0,T].

Let (x,c) e M x[q,T] for some q > 0 and compact M c Rd. It follows from (3.29)

and (3.15) that there exists a positive constant C such that
1 1 1
Rd LZ,(x-) (() d
where C -sup Zt(x): (x,'c) eMx[q,T]}. We deduce that the integral in (3.33) is

locally, uniformly bounded with respect to (x, ') e Rd x (0,T] from which we obtain the
desired conclusions. U

This lemma allows us to prove the following L-potential analogue of Theorem 8.

Theorem 11 Let 0 <6< 1 and y- {Y}0<,T be a continuous family of Z6- bounded

functions. Then, V, [x] is well-defined for every c e (0,T]. In fact, {V, [x]}0< T is a

continuous family of Z6- bounded functions.

proof: From (3.15) and Lemmas 10 and 7, it follows that there exists a C > 0 such that

V [y]

=CB l1,) Zj(x)< CZj(x)

for every c e (0,T]. 0







The final result in this section is the L-potential analogue of Corollary 9.
Corollary 12 Let 0 <6 < 1, f e As, and x {xY}0 Z6- bounded functions. Then, for every Tc e (0,T] we have

VI[Yf]] = M y][f]. (3.36)
proof: We proceed as in the proof of Corollary 9. It follows from (3.24) that
f{, [f]}0 Consequently, we deduce from Theorem 5 that the LHS of (3.31) is well-defined. In fact,
V [f]] C (Rd x (0,T]) and V, [y[f]] e A2 for each c e(0,T]. Fix (x,'c)e(0,T].
From Theorem 11 we have that V, [x] is well-defined. Furthermore, we apply Fubini's
theorem to obtain

VI [][f](x) = V [U]*f(x)= (fTL(ZT s[ys])ds)*f(x) (3.37)

= fRdL(Zs [s ])(x )f () dsd

= Rd f Rd LZ-s (x - y) ys (y)f( )dydsd

= JoRd RdLZ-s (x -- y) ys (y)f ()dyd~ds
= o Rd RdLZ-s (x w)ys (w -- )f () dwdyds

= o Rd LZT-s (x w)(d dS (w--) f ()d)dwds

= JJ d LZ-s (x- w)ys [f]dwds
=J L(Z -s[ys[f]])(x)ds= V, [[f]](x).








2.4 The Derivatives of the Gaussian Potential


In this section we turn our attention to the second term on the RHS of (3.19). In

particular, we will derive the following relationship between the Gaussian and L-

potentials expressed in (2.24):

L(U[L [f]])= V,[[f]] I[f] (4.1)

for every fe A,, 6 < 1, and continuous family of Z6-bounded functions Y {j0
for which y [f] is locally, uniformly Holder continuous in space with exponent 3 < 1.

Theorem 1 Let 6 < 1 and f = {f}T(0,T] be a uniformly bounded, continuous family of
functions from A,, that is locally, uniformly TIlder continuous in space with exponent

3 <1. Then, forj e {1,2} we have that U[f] e C(Rd x (0,T]). In fact,





and --U [f] e A28 for every T e (0,T].

Q1
proof. From Theorem 3.2 we have that (x,'c,s)l->---Z sfs](x) is continuous on

E {(x,'c, s): x Rd,0
prove (4.2) as well as show that .U[f] eC(IRd x(0,T]), it suffices to show that the

integral in (4.2) is locally, uniformly bounded in (x,c) e Rd x (0,T]. We break up the

integration in (4.2) as follows:
Z-[fs]ds= [fs]ds+ Z- [fs]ds (4.3)



where q e (0,T] is fixed.








{(x,'c,s) eE:x eM,s

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




0)fS (0 d


(4.4)


) d


< C2 exp(hf x2)(f s) ,

where hf < 2hf This implies that

--Z.-[fs](x) ds < C2exp(f x 2) (T s)-ds < C3exp(f x 2).
IO ix I o


(4.5)


Now, let (X, -, s) Gq


{(x,jI, s) E:x e M,0 < q< s < c}. For the second integral


on the RHS of (4.3), we define B {e G Rd : ||| < 1} and rewrite (4.4) as


-s [fs ](x) < Z -s(x
Ox Ox1


)fS (0 d + fS (x) 0i a Z,-s (x )d


(4.6)


+ Z -s(x-4 ) fs()-fs(x) d .

For the first integral on the RHS of (4.6), it follows from (3.11) and (3.12) that


(4.7)


We use the divergence theorem to obtain the following bound for the second integral:


JB a1ZT-S(X


-)d


< IB c O-x j 's(


(4.8)


Let (x, -c, s) e Eq


aB Ox ij-ai Z T-S (x )cos(yj)dS


Oi (fIIX112).
-Z'-S (x )fs (0 d:! C4exp
Oxi








where y, is the angle between ,1 and the outwardly directed normal to the boundary OB,
and dS, is the (d -1) -dimensional volume element on OB. Since x | = 1 on aB, we

use Lemma 3.1 with t to deduce that
2
J Z,-s (x ) d < Z (x- )dS (4.9)


d+j-1 X 2

d+j-1


where we used the fact that the expression in square brackets is uniformly bounded. It

follows that the second term on the RHS of (4.6) is bounded by


fs(x) Z,- (x -)d
Since f is locally, uniformly Holder continuous in space with exponent P3 < 1, we

have that there exists a positive constant C9 such that

fs(x)-fs()|< C9 x-( (4.11)

for every x and in M and q < s < T Without loss of generality, we may assume that

B c- M. It follows from Lemma 3.1 with 1 -p < 1 that the final integral on the RHS
2

exp -6x 2- 2
__ Z-s (x -) fs() -fs(x) d-< C9 2-- s) -2 d (4.12)
xi C (-s) x-
_C< (T-S) B|X-^|








Combining (4.7), (4.10), and (4.12) with (4.6), we obtain

Zxi [fs] < C4 exp x 2)C8 exp(hf x 2) Cl10 s) (4.13)

< C (- s) exp(2hf I|x||2).
Since [t < 1, it follows that

_f aZ'[fjds
Therefore, we deduce from (4.3), (4.5), and (4.14) that for every (x,c) eMx(0,T]
we have

S-- Z-s [fs(x) ds
We conclude that U[f] C(Rd x (0,T]), U [f]e A25, and (4.2) holds for every
Tce(0,T]. U
We have a similar result for the time derivative of the Gaussian potential.

Theorem 2 Let 6 < 1 and f = {f}T(0,T] be a uniformly bounded, continuous family of
functions from A, that is locally, uniformly -Holder continuous in space with exponent
S< 1. Then, we have that -aU[f] e C(Rd x (0,T]). In fact, we have

a [f]= J-Zs [fs]ds+f, (4.16)


from which we conclude that -U, [f] e A25 for every T e (0,T].

proof: Let M c Rd be compact, (x, ) M x (0,T], and AT > 0. We apply the mean


value theorem to the following difference quotient to obtain








U U [f](x) = J ZA -s [fs](x)ds Z- [fs](x)ds (4.17)


=- +A ZA A-s[fs](x)ds+ Z+ ZT-S [fs](x)ds
AT Sc JTKo Atc
1 +A IT
= ZA+-s [fs](x)ds + -z -s[fs](x)ds

for some t e (r,T + At). In case the notation in the last line is unclear, we note that

-Zs [fJ(x) = Zas s ](x) (4.18)

We will show that the RHS of (4.17) approaches the RHS of (4.16) as AT decreases
to zero. A symmetric argument can then be made to show that this result also holds as
AT increases to zero. This will establish (4.16) since the limit of the LHS of (4.17) as
A -> 0 is the LHS of (4.16).

Since s --> Z1+A, s[f ](x) is continuous on [t,T + At], it follows from the mean value

theorem for integrals that the first integral on the RHS of (4.17) may be rewritten as

1 +
SZ~ -s[fs](x)ds= ZA [f ]j(x) (4.19)

for some e (t,T + At). Letting AT -> 0 yields

ir- f1 + Z T+Ats [fs](x)ds= oirnZT [f,](x)= f (x). (4.20)

For the second integral on the RHS of (4.17), it remains to be shown that

lim zt-s[fs](x)ds= Zs[fs](x)ds. (4.21)
AT -- >0 J /O I L t









Let s>0, 1 --< < 1, 61
2


1
- and Tc e (C 1,,c), where the positive
2 3C )


constant C will be determined below. Consider

I Z-[fs](x)ds- Z,- [fs](x)ds


<" Z- o
: Cra -


-s I [fs](x) ds+ z s[fs](x) ds+ Z,- [fs](x) ds.


For the first integral on the RHS of (4.22), fix


se (0, T ] and note that


i-s > Tc- T > 0. We deduce from Theorem 3.2 that q -> Zqs [fs ](x) is continuous on

['c, T]. It follows that there exists a E2 <: E such that for AT < sE we have


Zi-s Zs [fs](x) <-3T
s-B O--J 3T


from which we deduce


f -C ti-zS


aZ


s [fs](x) ds< .
3


(4.23)


(4.24)


To evaluate the second and third integrals on the RHS of (4.22), recall that Z. is a
parametrix for L. This implies that


aZc 1
a- 2


(4.25)


Let 1 < < 1 and N denote the diameter of M. Combining (4.25) with (4.13) yields
2


a :[f](x)_ at IsI x


S) - < exp(2hfN 2)( T


s)

for every s e ['e T], where C is some positive constant and C Cexp(2hfN 2).


(4.22)


(4.26)







We use this estimate to see that for AT < 6 and q e {c, } we have

J Zq-s[j(x)ds C 1- --
C ( q -)1c < C [ ( q -c) + ( -c C,) ]
1-t 1-t
< [ATc +(Tc-c)]1 < (26)1 = -
1- 1- l 3

Combining the estimates (4.24) and (4.27) with (4.22) yields I < s when AT < 6 from
which we deduce (4.21). It follows from (4.20) and (4.21) that
1 T+A- I Ta
mli Z ~tZ I[fs](x)ds + J Zi- [fs](x)ds (4.28)

= -Z,-s [fs](x)ds +f,(x).

A symmetric argument can be made to establish the analogous result for the left-handed
limit. Combining this with (4.17) yields the desired result (4.16).
The remaining assertions follow from Theorem 3.2 and (4.16). In fact, it follows that

|U[f](x) < Z0 [fs](x) ds + (x) (4.29)


for every (x,t) e Rd x (0,T] from which we deduce that U[f] eC(R x (0,T]) and

U,[f]z A26 *
ax T








We combine Theorems 1 and 2 to obtain the desired relation (4.1) between the

Gaussian and -potentials. Let -j {- } 0
functions and f e A, for some positive 6 < 1. It follows from (3.25) that {X [f]}0)
a continuous, uniformly bounded family of functions from A25. Assuming that y[f] is
locally, uniformly Holder continuous in space with exponent 3 < 1, we deduce from
Theorems 1 and 2 that for every (x, c) e (0,T] we have

L(U [y [f]])= L(Z s[f]])ds y, [f]= V, [v[f]] -y, [f]. (4.30)


2.5 A Series Representation of the Fundamental Solution


We will now apply relation (4.30) to the problem of representing the fundamental

solution for L by
F, = Z, +U,[] (5.1)

for some continuous family (p (} 0<< of functions from C(Rd) that satisfies

0= L(F [f])= L(Z [f]) + L(U, [(p][f]) (5.2)

for every -c e (0,T] and f eA A. Let f e A and assume that (p is Z6- bounded for some
6<1. If (p[f] is locally, uniformly Holder continuous in space (with exponent 3 <1),

then we deduce from Corollary 3.9 and (4.30) that

L(U, [(p][f]) = L(U, [(p[f]]) = V, [(p[f]] (p [f]. (5.3)

Combining this with (5.2) implies that p must satisfy the following Volterra integral

equation for every Tce (0,T] and f e A:

(p,[f] = L(Z, [f])+ V [p[f]]. (5.4)









2.5.1 Convergence and Continuity


The goal of this sub-section is to show that the solution to (5.4) is given by


( = Vm[LZ], (5.5)
m=0

where Vm is an operator defined by

V I (5.6)
and
VmC -V oV Vm (5.7)

for every c e (0,T] and m > 0. We will see in the proof of Theorem 2 below that

Vm" [LZ] is well-defined for every Tc e (0,T] and m > 0. Furthermore, it is shown that

the series representation (5.5) is a continuous family of Z6 -bounded functions for every

6 < 1. Finally, our objective is obtained in Corollary 3, where we deduce that (5.5) is a

solution to (5.4). We begin with a preliminary result:

Lemma 1 LZ is Z6 -bounded for every positive 6 < 1.

proof: Let0 < 6 < 1. From the definition of L we have

LZ,(x)=-e x, Z +(x)+Y(x)Z(x) = x + Y(x) z,(x) (5.8)

and

Y(x) = Y- 1x12 (5.9)


From Lemma 3.1 with t = 0, it follows that there is an A > 0 such that for every

(x,'C) e Rd x (0,T] we have

0 < LZ (x) <9 OAZ (x) + yZ, (x) < CZ (x), (5.10)


where C A + y. m








Throughout the remainder of this section, let (p be given by the series (5.5).

Theorem 2 The series (p converges absolutely. Moreover, (p is a continuous, Z6 -

bounded function for every 6 < 1.

proof: Let 0 < 6 < 1. The first step is to prove by induction on m that {V" [LZ]}

is a continuous family of Z6- bounded functions such that for some C > 0 we have
Cm m-1
V, [LZ] C< (x) 2- Z(x). (5.11)
m +1
( 2

From Lemma 1 we see that {Vo [LZ]}0<
functions that satisfies (5.11) for some C > 0. To use induction, assume that this holds

for some m > 0. Since {Vm [LZ]}0<
it follows from (3.36) and (5.11) that


vm+i[LZ](x) = VLVm[LZ]](x) < d LZ,-s(x )Vm [LZ](Q) d0ds (5.12)

Cm+1 1 m-1
< m(T-s) 2 S Zs (x )Zs(o)dds
m+1 0 d
2
Cm+l 1 m+1 m



^m +1

2 )

Therefore, {Vm [LZ]}0
satisfies (5.11) for every m > 1. To obtain the desired bound on p, we will break the

series apart as follows:









(p(x, T) < Vm[LZ](x) m=l m=l m=2
modd meven
For the odd series we have

Vl ct2n+1 (^nT (5.14)
Vm[LZ](x) v~n+I[LZ](x) <;Z (x) nl (5.14)
m=l n=0 n=0
modd

n= 0

where C1 Cexp(7tC2T). For the even series we have

GO GO GO c r 2n \n-
SVm [LZ](x) V2n [LZ](x) < Z (x) /C 2 (5.15)
m=l1 n=l n=1 F n +-
meven ) 2

< 2Zn 7i ^n- -
< 2Z6 x) __ n 2) 2Z6 Cn
n=1 n=0(

< 2C2 exp(7C2T)Z (x) < C2Z (x),

where C2 2C2 ff-exp(7TC2T). By inserting (5.14) and (5.15) into (5.13) we conclude

that p is Z6-bounded. Furthermore, since Vm [LZ]e C (Rd x (0,T]) for every m >l 1 and

the series representation for (p is uniformly bounded on compact subsets of Rd x (0,T],

we also have that p e C(Rd x (0,T]). *

Corollary 3 (p is a solution to the following Volterra integral equation for every

c e (0,T] and f e A:


(p [f]= L(Z [f])+V [(p[f].


(5.16)








proof: It follows from Theorem 2 that for every Tc e (0,T] we have

V1[(] = JL(Z,-s[Ps])ds = L L(Z,-[Vm[LZ]])ds= V,[Vm[LZ]] (5.17)
0 m=O0 m=0

= Vm+ [LZ] = Z Vm [LZ] = V [LZ] = LZ .
m=0 m=l

Therefore, we deduce from Corollary 3.12 that for every Tc e (0,T] and f e A we have

V [ f]] = V[][f] =( -LZ)[f]. (5.18)

Hence, we conclude that (5.16) holds for the series representation (5.5) for p. U

We would like to be able to claim that

L(F,[f]) 0, (5.19)
where
F, = Z, +U, [p] (5.20)

for every Tc e (0,T] and f e A. However, it remains to be shown that (p[f] is locally,

uniformly Holder continuous in space with exponent P3 < 1. As stated in the beginning of

this section, this condition combined with Theorem 2 and Corollary 3 is sufficient to
conclude that (5.19) holds for every c e (0,T] and f e A.


2.5.2 Holder Continuity


The first objective of this sub-section is to show that p is uniformly Holder

continuous in space with exponent P < 1. We deduce from (5.17) that

(p,(x) p(y) _< LZ,(x) -LZ,(y) + V,[qp](x) -V [p](y) (5.21)

for every x and y in Rd and -c e (0,T]. Our goal will be obtained by proving the desired

result for each of the terms on the RHS of (5.21). We begin with a preliminary lemma:








Lemma 4 Assume that |x -y2 < -. Suppose that C lies along the line segment
connecting x and y. Then, for any positive constant 6 there is a C > 0 such that

Z ) < CZ(y). (5.22)

proof: Since C lies on the line segment connecting x and y, we have C = kx + (1 k)y

for some X ez (0,1). Hence,

02 2=|x +(1- )y2 2= |(x -y)+y12 (5.23)


It follows that

Z ) 2 exp 2 exp -- 2_- =CZ6(y), (5.24)


where C exp .

Theorem 5 Let x and y be elements of the compact set M c Rd, d I e (0,T],

0 < P < 1, and 0 < 6 < 1. Then, there exists a positive constant C such that
1+p
LxZ P(x-)-L 2 (y-)
where the subscripts x and y denote the spatial variable of the action of the operator L.
proof: We divide this proof into two cases:

Case 1 x -y| > T'

From (3.15) it follows that for some C > 0 we have
P 1+p 1+P
LxZ (x -) < CT2C 2 Z(x-)_
A similar inequality holds for LyZT(y ) thus we obtain (5.25) in this case.









Case 2 x y2 < T

Throughout this case we will use the following result for every a e [0,1 +13]:
a 1+p-a 1 1+p 1+p
T 2 x-y= T 2 x-ylX T2 lx-yl
Consider the inequality


d
LxZ (x -) LyZ, (y ) < O xi a-- Z (x
i=1 i

+ (x)Z,(x -)-

We first break up the summation and find that


T()y Z, (y


d
xi Z,(x
i=1 i


) yi Z (y
CIi


d a
) < xi yi Z,(x -
i=1 axi


d
+ Z(x
i= 1 ^i


a Z(
___jyi


For the first term on the RHS of (5.29), we see from Lemma 3.1 with[L

(5.27) that there exists a constant C, > 0 such that


1 1+p
Yi Z(x-

) (5.30)


Applying the mean value theorem to the second term on the RHS of (5.29), it follows

that there exists a point C lying on the line segment between x and y such that for

some C2 > 0 we have


d a
yax
ii


a Z,(y
a,-yi


d a2
yi1 2Z() xi -yhi Yi
i=1 ^i


(5.31)


d a,2
i=1 1


(5.27)


(5.28)


(5.29)


d
xi=
i=1


1
- and
2


o) yi -








It follows from Lemma 3.1 with [L = 1 that for some C3 > 0 we have
d a2
2 () <( C3 1Z( ) T


Since |x y 2 < -, we deduce from Lemma 4 that


d C2
-Z (() 1=1 ai


1+p 1- p
()=C4T 2Z Z(y


1+13
< 'C4 2 ||x-y|| Z (y -).

Combining this with (5.31) yields


d
- x


a)- Z(y


Syi

0). (5.34)


Finally, by combining (5.30) and (5.34) with (5.29) we obtain


d Zx


Q) yi aZ(y
By.


We now consider the second term on the RHS of (5.28). Recalling the definition of

Y given in (5.9), we have


Y(x) Z (x


Z (x -).


(5.36)


Applying the mean value theorem, Lemma 3.1, Lemma 4, and (5.27) to the RHS of


(5.36), it follows that there exists a point C lying on the line segment between x


and


y such that


Y Z'(x- )- Z(y


)|=Yxi-Yid Z (() 7

(5.37)


< C8' 2 x jX yjj'Z (y


Combining (5.35), (5.36), and (5.37) with (5.28), we see that (5.25) holds for case 2. 0


(5.32)


(5.33)


+ Z6 (y -c)) (5.3 5)
T


0 T(y)Z' (y Ok YIz' (x -)


1+P
'C 2 JJX yll P (Z' (X
6 T








Applying this result with = 0, we deduce that for every q > 0 and compact
M c IRd, there exists positive constants C and C such that

LZ,(x)-LZ,(y)l
for every x and y in M and -c e [q,T]. Hence, LZ is locally, uniformly Holder continuous

in space with exponent P3 < 1. We proceed to establish this result for the second term on
the RHS of (5.21).

Corollary 6 V[2p] is locally, uniformly Holder continuous in space with exponent
3 < 1. More precisely, for every compact M c Rd and positive 6 < 1, we have

V [p](x) V [p](y) < Cl x- yll Z (x) + Z' (y)) (5.39)

for every x and y in M and c e (0, T].

proof : Let x and y be elements of the compact set M, 0 < 3 < 1, and 0 < 6 < 1. Since (p

is Z6-bounded, it follows from Theorem 5, Theorem 2, and Lemma 3.7 that for some
C > 0 we have

Vj[2](x)-V[2p](y) < f (L(Z, s[sp])(x) -L(Z [_s])(y)) ds (5.40)


< d (LxZ, (x- )-LYZT s(y -)S (p ) dds


d

+Z4-s(y- )]d~ds

= CB 2,1 x-y 2(Z(x)+ Z(y))
SCB 2 1 1x l


:!qx-yllP(z6(x)+z6(y)). 0








We conclude from (5.21), (5.38), and Corollary 6 that for every compact M c Rd
and positive 6 < 1, there is a positive constant C such that
1+p
p1(x)- (y)
for every x and y in M and c e (0,T]. Hence, (p is locally, uniformly Holder continuous

in space with exponent P < 1. This result allows us to prove the following theorem from

which we will deduce that LF, [f] = 0 for every f e A and c e (0,T].

Theorem 7 p [f] is locally, uniformly Holder continuous in space with exponent P < 1
for every f e A. More precisely, for every compact M Rd there is a positive constant

C such that we have

j [f](x)- j[f](y)
for every x and y in M and c e (0, T], where 0 < hf < 2hf.

proof: Let x and y be elements of the compact set M, 0 < P < 1, and f e A. It follows

from (5.41) that for some positive constants C and 6 with < 6 < 1 we have
2

r[f](x)- [f](y) < id I ( x- (X ) --(y -))f ()Jd. (5.43)



Since f e A c As2, it follows from Lemma 2.2 that there exists positive constants C

and hf with hf < 2hf such that

~p,[f](x)- p,[f](y)







2.6 The Solution to the Cauchy Problem


Before proceeding to prove the final results necessary to provide the solution to the

Cauchy problem, let us summarize the work we have completed in the past few sections.

We began by looking for a fundamental solution for L of the form

F, =Z,+ U,[] (6.1)

for some function p eC (Rd x (0,T]). It was shown in Section 4 that

L(U, [p][f])= V,[q[f]] [f], (6.2)

provided that (p is Z6-bounded for some 6 _<1, and that qp[f] is locally, uniformly

Holder continuous in space (with exponent P <1) for every f e A. Combining

(6.1) and (6.2) with the condition that L(F [f]) = 0 for every ce (0,T] and fe A, it

was deduced in Section 5 that (p must satisfy the following Volterra integral equation:

p [f] =LZ, +Vqp[f]. (6.3)

Finally, it was shown that the series pD = Vm [LZ] satisfies (6.3) and the conditions
m=0
necessary for (6.2) to hold from which we deduced that L(F, [f])= 0 for every T e (0,T]

and f e A. Thus, to conclude that F, is a fundamental solution for L, it suffices to show

that F0[f] lim(F1[f]) =f for every f eA. Furthermore, from (6.1) and Corollary 3.9
T->0 O
we have that
limF, [f] = lim(ZT + UT [])[f] = f + limU [ [f]] (6.4)

for every f e A. Therefore, it suffices to prove the following theorem:

Theorem 1 For every fe A, we have limUT [(p[f]] = 0.
T-proof L A. It follows from Theorem 5.2 and (3.24) that [f0 a
proof Let f eA. It follows from Theorem 5.2 and (3.24) that fpT [f]) O







continuous, uniformly bounded family of functions from A2. Hence, we deduce from

Lemma 2.2 that for some positive constants C and h,[f] < 2h [f] we have


U[qp[f]](x) < f (pjs[f]]ds0asTc->0. 0 (6.5)

Therefore, F, is a fundamental solution for L. It follows that the solution to our

Cauchy problem may be represented by

u(x, )= F,[g](x)= Z,[g](x)+U, [p[g]](x), (6.6)

where g(x) exp & x ) is the initial condition, provided that & <-
32pd
Finally, we must verify that u satisfies the hypotheses of the Feynman-Kac Theorem

from which we will deduce that

u(x,c)= Ex [g(X )exp (Tds) (6.7)

for every (x,'c) e Rd x [0,T]. From (6.6), Theorem 3.2, Theorem 4.1, and Theorem 4.2

we have that u is of class C21( Rd x [0,T]). Furthermore, by assuming that & <32
32pd
we see that g e A. Hence, we deduce from Theorem 5.2 and (3.24) that {(, [g]}0)
a continuous, uniformly bounded family of functions from A2. Consequently, it follows

from Lemma 2.2 and Theorem 3.5 that u e A4, thus there exists positive constants A and
1
h < such that
8pd
max u(x,'c) < Aexp(h||x||2) (6.8)

for every x e Rd We conclude from the Feynman-Kac Theorem that (6.7) holds for

every (x, ') e Rd x [0,T]. Therefore, we deduce from (1.5) that the arbitrage price of the

risk-spread option is given by

7t (C) =exp(-(cac + Kr))exp (Jt ds)F,[g](Xt)-B(t,T). (6.9)














CHAPTER 3
NUMERICAL RESULTS AND APPLICATIONS



3.1 The Fourier and Laplace Transforms


The series solution of convoluted potentials presented in the previous chapter is an

elegant exhibition of the ability of the modem mathematical prose to convey a relatively

simple method in a clear, concise manner. In fact, for those pure mathematicians who are

familiar with the antiquated notation used to present the parametrix method in Friedman

(1964), the refined potential theoretic approach is a significant contribution to the

literature. However, it remains to be shown that this solution is of practical value.

Although the iterated convolutions prevent us from directly integrating the series,

they are ideal for calculating the Fourier and Laplace transforms of our solution. Recall

that these transforms are defined by

d
F [f]() =(27r) d exp(-i- x)f(x)dx (1.1)
and
[h](s) -f exp(-st)h(t)dt (1.2)

for every f :Rd -> R and h : R, -> R for which these integrals are defined. A useful

result is that these transforms change convolutions into products. In fact, we have

d
S[f, f,2]= (27r)1 2[fl ]F"[f,] (1.3)
and
[h,*h2] =[hj][h2]. (1.4)








Recall from (2.6.6) that the solution to our Cauchy problem may be represented by

u(x, ) = F, [g](x) = Z, [g](x) + U, [[g]](x), (1.5)

where g(x)- exp(& ||x|2) and ,= Vm [LZ] for every (x,C) G Rd x[0,T]. It follows
mO0
from (2.2.5) that the first term on the RHS of (1.5) is given by

Z, [g](x) = (1- 26cT)) 2 exp &-- IIxl 1. (1.6)

The remainder of this section is devoted to the Gaussian potential in (1.5). Moreover, we

will consider the case where d = 1 henceforth.

Since the Fourier transform of g does not exist, we will consider the truncated

approximation

N n Nfg(x) if x [0 ifx > n

Hence, we approximate the Gaussian potential in (1.5) by the sequence {U, [p[g]]
We proceed to take the Fourier and Laplace transforms of the approximated Gaussian

potential in the spatial and temporal variables, respectively. We recall that

U [([gn] J= fZ1, [[g9 ]ds. (1.8)

It follows from (1.3) that for every (, c) e R x [0,T] we have

.[U,[ p[g]] = 27(j'F[z ](~) q]()dq)[gn](). (1.9)

By defining Z, 0 for every < 0, we see that

wFhe e[Zdcs ]() co[nl ] (t) ds = (we [Z] f [])(, c) (1.10)

where denotes convolution in time. Thus, we deduce from (1.4) that








C[.F[u[q[gj]]](,s) =2C[YF[Z]](,,s)C[[[]](,s)YF[g.](() (1.11)


2s+-

From the series representation for p, we see that

CF [] = FVm [LZ]]]. (1.12)
m=0
It follows from the definition of the L-potential that

Vm + [LZ](x)= V,[ Vm [LZ]](x) = J L(Z [vm[LZ]])ds (1.13)

= LxZ-s (x y)Vm [LZ](y) dyds

= LxZs (x y)Vm [LZ](y)dsdy

= (LxZ(x y)*Vm [LZ](y))(Tc)dy

for every (x, c) e R x [0, T] and m > 0. Applying (1.4) to (1.13), we deduce that

SV[vm+ [LZ]](x,s)= f [LxZ](x-y,s)L [vm[LZ]](y,s)dy. (1.14)

Let (x, c) e R x [0,T] and y e R. From the definition of L, we have that

LxZ (x-y) y) = xx )+WY(x) ZT(x-y) (1.15)

4= x(x-y)+2(R2 -x2) Z (x-y),

where R- Ideally, we would like to express LxZ, (x y) as a function of x y
and y so that we may rewrite (1.14) as a convolution in x. However, the plus operator in
the second term on the RHS of (1.15) makes this impossible. For the first term, we will
use the following identity:


x(x- y) (x -y)2 +y(x -y).


(1.16)








Inserting this identity into (1.15) yields

LxZ (x-y) = (x-y)2+y(x-y)) +Y(x) Z (x-y). (1.17)

We will write the Laplace transform of LxZ(x y) in terms of the following functions:

F(x,s) -x2 C[Z(x,s)=_ x exp( -x ) (1.18)
and
G(x,s)- OxC[](x,s)= sgn(x)Oexp(- x| ), (1.19)
where
Z(x, C) -Z (x). (1.20)
It follows from (1.17) that

[LxZ,(x y)]= F(x- y,s)+ yG(x- y,s)+ Y(x). (1.21)

Let m > 0. Combining (1.21) and (1.14) yields

C [Vm+l [LZ] (x,s)= f F(x-y,s)C [Vm[LZ]] (y,s)dy (1.22)


+ f G(x- y,s) (y[Vm[LZ]](y,s))dy


+Y(x) C[Z](x- y,s)CVm[LZ]](y,s)dy.

The first two integrals may be expressed as convolutions. In fact, we have

oF(x- y,s)C[Vm[LZ]](y,s)dy= (F* [Vm[LZ]])(x,s) (1.23)
and
f G (x y,s)(y[Vm [LZ]](y,s))dy = (G (x[Vm [LZ]]))(x, s). (1.24)

For the third integral on the RHS of (1.22), we have

Y(x) f [Z](x-y,s)C[Vm[LZ]](y,s)dy= (x)([Z]* C[Vm[LZ]])(x,s). (1.25)







We now compute the Fourier transform. Applying (1.3) to (1.23) and (1.24) yields

.F[F*,[m [LZ]]](,s)= .F[F]F [, [m [LZ]]]( ,s) (1.26)
and
F [G(x [Vm [LZ]]](, s) = 2F[G]F[x [Vm[LZ]]]((,s) (1.27)

= mF[G]i (F[C[Vm[LZ]]])(,s),

respectively. The last equality follows from the fact that

F [xf ( -in a [fn ( (1.28)

for every function f for which these Fourier transforms are defined.
To compute the Fourier transform of (1.25), we will also need the following property:
d
."[ff2]= (27r) 2 (.[f J"[f2]) (1.29)

Applying (1.28), (1.29), and (1.3) to (1.25) yields

".F[(c[Z] C m[LZ]])](, s) F (R2 -X2) ([Z]* Vm[LZ]])](,s) (1.30)


SR2 2 [ R,R] (c[Z] Vm [LZ]])](',s)

22
= (R2 + (Y2 [ -R,R] *] ,C[Z]]F ,C[Vm [LZ]]])(', s).

Combining (1.26), (1.27), and (1.30) with (1.22) yields the following iterative relation for
every m > 0 :

.F[[vm+ 1[LZ]]](,s) = 2h7. F[F]+i. [G] .F[L C[Vm[LZ]]](,s) (1.31)

+ 2 -( "1[ R,R] FEl[Z]F]LY vm [LZ]]])(,s) .








Computing the various Fourier transforms in (1.31), we have

.F [F](, s) = 0 2 (1.32)
S(2s+ (1.32)

iF [G]( s)= 2s (1.33)

[ R,R]]() sin(R) (1.34)

and
1Z. (1.35)

Inserting these results into (1.31) yields

F[C Vml [LZ]]] (, s) (1.36)

(20 )2 2s-2 + (2s + F2) F.['L [vm [LZ]]] (,s)
(2s + )

2 __2 sin((( y)R)
+ 9-R)(2s 2)FL CVm[LZ]]](y,s)dy.

Combining (1.11), (1.12), and (1.6), we see that (1.5) becomes
d2
u(x,c) =(1- 2&) 2exp x2 (1.37)
I 2&-c


+ 1imn L L 2V Ls2 T m
n 2s + 0 m=0

where the terms of the series are given by (1.36). Although this is not as aesthetically

pleasing as (1.5), we have removed the temporal convolution in the Gaussian potential

term. Unfortunately, the spatial convolution remains in the iterative relation (1.36).

Hence, we will set aside the numerical analysis of the risk-spread option for a future

project.









3.2 Delta Hedging with the Risk-Spread Option


The introduction of a financial derivative invariably leaves the writer of the option

with the task of hedging against the risk he incurs by assuming a short position in the new

derivative. In the case of a call option on a stock, the hedging of the option writer against

changes in the underlying stock price is commonly known as delta hedging. The delta of

a portfolio is the first derivative of the portfolio with respect to the stock price.

Moreover, the portfolio is said to be delta neutral when it is insensitive to changes in the

stock price, that is, when it has a delta of zero. If we denote the delta of a call option by

A, then a portfolio consisting of a short position in the option and A shares of the

underlying stock is delta neutral. In fact, denoting the call option by C and the portfolio

value by H, we find H = -C + AS which implies -A + A = 0.
as
For more exotic derivatives, delta hedging refers to the act of protecting the writer of

the option against changes in various risk factors. Unlike the call option on a tradable

security such as a stock, the risk-spread option must be hedged against the untradable risk

factors that drive the interest rates. However, there are numerous traded bonds that are

affected by the interest rate risk factors. Assuming that the number of risk factors is d,

then a portfolio of d distinct bonds together with a short position in the risk-spread option

that is delta neutral can be constructed.

Let C denote the risk-spread option and Ai for each of the d risk factors.
xaB

Consider d distinct bonds B, with 6ij and a portfolio with value H. The hedging
axi
problem is to find the amount hi of bond B. holdings such that the portfolio is delta

neutral. The portfolio value is given by
d
H= -C + hjB. (2.1)
i 1









This implies that the bond holdings may be determined by setting

anlJ d
^ -A, + dh o
for each i. We represent this by Dh = A, where

for each i. We represent this by Dh = A, where


D (61 )d

h (hl,...,hd )T,

A (A1,...,Ad)T.


(2.2)




(2.3)

(2.4)

(2.5)


Hence, the vector of bond holdings is given by h = D 1A, provided that D is invertible.

Returning to the example of the previous sections, consider the case of two risk

factors. We will construct a hedging strategy using the risky and riskless bonds. Fix

t G [0,T] and recall the prices of the risky and riskless bond from Section 1.4:

d
B(t,T) =(1-2&V, ) 7 exp (- rt +cc)) (2.6)


(2.7)


where V -- V,
1-2V,



and


Hence,


and V -(- e 2T). Also, we recall that
1- 23PV 20

rt =C IXt 2
2


2

(t,T)= -cxiVB(t,T)
Bx,

-(t,T)= -oxi.V (t, T).
Oxi


(2.8)


(2.9)


(2.10)


(2.11)


where ||Xt = x almost surely.


d
B(tT) =l[V>t]B(tT)(l-2V, 2 exp(-(-'kt +P-C)).









From (2.10) and (2.11), we see that the matrix D in this simple example is singular.

Hence, under the assumption that that there is more than one risk factor, we see that it is

not possible to simultaneously hedge the risk-spread option against all of the risk factors

using a portfolio consisting of only riskless and risky bonds. Consequently, we must add

a different type of interest rate derivative to our portfolio to make it delta neutral.

On the other hand, if we assume that there is a single risk factor, then it follows from

(2.2), (2.10), and (2.11) that we may hedge the risk-spread option by holding hf riskless

bonds, or hf risky bonds, where


hf =B A (2.12)
ox, B(t,T)

and
h=- A (2.13)
oxVB (t,T)

Alternatively, we may be able to hedge the risk-spread option against the riskless spot

rate directly, if it can be shown that the risk-spread option only depends on the risk

factors through the riskless spot rate. In this case, we compute


ar
and

(t, T) = -V (t, T) (2.15)
Or a

from which we deduce that holdings of either
h B A

hB= (t,T) (2.16)

in the riskless bond, or
V 'G(t,T) (2.17)


in the risky bond will provide the desired hedge.








3.3 Numerical Properties of the Yield Curve


Recall that the state-variable process Xt from Section 1.4 satisfies

dXt = dWt -OXtdt (3.1)

for some positive parameter 0, where Wt is a d-dimensional Brownian motion on

(Q,F,P). Continuing with the example of Section 1.4, we also recall that the riskless

spot rate and risk spread are given by


rt = I y Xt 2
2


(3.2)


k=t = 1a Xt 2, (3.3)
2

respectively, where c 4&(0 &), C 43(0 3), & -, and PJ are positive
d d
constants. We will show that rt follows the Cox-Ingersoll-Ross (CIR) process given by

drt = a(b rt)dt + 2ctdMt (3.4)

for some positive constants a and b, where Mt is a one-dimensional Brownian

Motion on (Q, F,P) with respect to the natural filtration !t of Wt (Elliot & Kopp, 1999).

In fact, from Ito's lemma we deduce from (3.1) and (3.2) that

d d
drt = o7XdX +- Zd(X)t (3.5)
i=d 2 i d
= CZ X dWi -O jt j2 dt + dt
il 2
= d + C d XldWi.
= 20rt dt+ Xt Xt

= a(b rt) dt + d2tdMt,


where a 20, b -, and Mt s
40' i=1 Xs









It remains to be shown that Mt is a one-dimensional Brownian Motion on (fQ, ,P)

with respect to !9. We first assert that Mt is a continuous martingale. In fact, we have


EP X dW__ < T < (3.6)


for every i. Furthermore, it follows from Ito's lemma that

t t d t X 2
M2 = 2 MdMs + d(M) = 2 MdMs + ds=2 MsdM +t. (3.7)


We deduce that M2 t is a martingale from which we conclude that Mt is a standard

Brownian motion with respect to !,. Hence, we have shown that rt follows the CIR

process given by (3.4). Similarly, we have that

d\ =a(b -t)dt + 2 tdMt, (3.8)

where b -d
40

The CIR process has two properties that correspond with empirical spot rates and risk

spreads. First, it is shown in (Lamberton & Lapeyre, 1996) that the CIR processes (3.4)

and (3.8) are almost surely positive, provided that d > 2, ro >0, and k0 > 0 almost

surely. Unfortunately, in the one dimensional case we have that the probability that these

CIR processes vanish for infinitely many times is one.

The second ideal property of the CIR process is that of mean reversion. Consider the

following ordinary differential equation:

dx = a(b- x)dt. (3.9)

The solution of (3.9) is given by


x(t) = exp(-at) +b .


(3.10)








This function exponentially decays to the mean reversion level b at the mean reversion

rate a as t tends to infinity. Comparing this with (3.4) and (3.8), we deduce that the CIR

process has a mean-reverting drift term.

We conclude this chapter with a graphical comparison of the yield curves of the

riskless and risky bonds. Recall from Section 1.4 that the riskless and risky yield curves

are given by

Y(t,T) -lInB(t,T) r+auc+-ln(1-2V, ) (3.11)
and

Y(t,T) 1--In(t,T)=[v>t] Y(t,T)+- rV t +P+In (1-2V 2 (3.12)


respectively. For fixed t, we see from the definitions following (2.7) that


Y(t,T)= 2c) -rt + c+dln 1- & -exp(-2T)) (3.13)
4 2(0-&)+2aexp(-20T) 2 0 p

and

Y(t,T) = l[v>t]Y(t,T) (3.14)


S1 1 )exp(-2 t)) df3 l+ P exp(
+1[v>t] 2( exp( 20'c) rt + 1C+dIn 1- 1-exp(-20c)).
2( 2(9- )+2pexp(-20T) 2 0

It follows that
limY(t,T) = a (3.15)
and
limY(t,T) =ca+ (3.16)

In the graphs below, we present the initial riskless and risky yield curves for various

parameter values. In particular, we see that the rate of convergence in (3.15) and (3.16)


depends on the values of 0, 0 &, and 0 3 .



















N I' Mv -20
P-i--, RIi' F: ; T M 1 1",.


K-1i K I IP1
R.- Ri S Vi F: SS T PA


K 1 K I I P.1 I _,,t
R i-FdS i F TPA~h~


Fig. 1 The Risky and Riskless YTM with d = 5, 0 = .05, a = .09, and P = .04


N I'S r I I M t; X- I".,
P i .-, i/ i F ; .-, T M S,,..



















K lbl r I ',,I -'u '- ,:.
--- RISL FSS TM 1 1.


I- i b I, I ',,I 11:. '.:,
- R--- RISILESS TPi .c':'.,,


K-- Ibr I I I M 1 -' ,:.
---- RISL FSS TM 7'',


-- ISlr I I M d-',:,
RISLI FSS I, TM ',-,


Fig. 2 The Risky and Riskless YTM with d = 5, 0 = .05, a = .09, and P = .04



















I- 1b r i i i,,i 1i i 1'i'.:,
- RI-- RISI FSS i TP 11 :',,


l.FI r i i I.I 1 r i '. 1'.:,
R-- RISL FSS TM c':.,,


KI-- I Lb I M 11 i .:.
- RI- RIS-I F TM 7M


-. Ibr i i I .I '
RIS FLSS i Ti V',-,


Fig. 3 The Risky and Riskless YTM with d = 5, 0 = .05, a = .09, and P = .04

















RIS i, .TM (.1 .,:.
--RISKLESS i TM 11 V:.


RISK TPA 159 1.
RISKLESS TPA l':'.-


--RISF, TM 11 12'.,
--RISKLESS TM 7:,


RISKRISV TM 7'2"'--.
RISKLESS TM


Fig. 4 The Risky and Riskless YTM with d = 5, 0 = .05, a = .09, and P = .04



















KIbI I I M 1l l1' :.
---- RISL FLSS TM 1 1..


F< ibr, i ri ~iD
RIS-KlES F i-,- TPA i ''-.-


--- bIS I M 11 Y.':.
----- RISFL FSS TM M '',


I-- Ir I I M .',.
RISLI FSS I, TM ,,.


Fig. 5 The Risky and Riskless YTM with d = 5, 0 = .05, a = .09, and P = .04

















RIS ,i TM l. .1 ,:.
-RISKLESS TM 11V':.


RISK TPA 159 1.
RISKLESS TPA''


- RIS iTM 11 12'.,
- RISFLESS TM 7':,


RISKRISV TM 7'2"'--.
RISKLESS TM 5"r


Fig. 6 The Risky and Riskless YTM with d = 5, 0 = .05, a = .09, and P = .04















CHAPTER 4
SUMMARY AND CONCLUSIONS



4.1 Summary of Results


The potential theoretic framework developed in Chapter 1 extends the work of Rogers

(1997) to include the case of defaultable bonds. Among the numerous examples of both

the riskless and risky spot rates of interest that may be generated from this procedure, the

familiar, tractable example of the Omstein-Uhlenbeck process was used as an illustration.

This example is realistic in the sense that the resulting Cox-Ingersoll-Ross model of the

spot rates is strictly positive, provided the dimension of the driving Markov process is at

least two. Furthermore, this model exhibits the mean-reverting behavior that has been

observed empirically.

This example was carried forward to the Cauchy problem for the risk-spread option

treated in Chapter 2. The potential theoretic parametrix method used to develop a series

solution to the Cauchy problem represents a significant contribution to mathematical

literature. To gain an appreciation for the modem language, the interested reader should

compare this method with the parametrix method outlined in the first chapter of

(Friedman, 1964).

In Chapter 3, the Fourier and Laplace transforms were used to derive an expression

for the risk-spread option. This removed the temporal convolution in the Gaussian




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