Selected Problems in Financial Mathematics

UPPSALA DISSERTATIONS IN MATHEMATICS 38

Selected Problems in Financial Mathematics

Erik Ekström

Department of Mathematics Uppsala University

UPPSALA 2004

Dissertation at Uppsala University to be publicly examined in Room 2146, Building 2, Polacks- backen, Uppsala, Friday, October 29, 2004 at 13:15 for the Degree of Doctor of Philosophy.

The examination will be conducted in English Abstract

Ekström, E. 2004. Selected Problems in Financial Mathematics. Uppsala Dissertations in Mathematics 38. 17 pp. Uppsala. ISBN 91-506-1774-5

This thesis, consisting of six papers and a summary, studies the area of continuous time financial mathematics. A unifying theme for many of the problems studied is the implications of possible mis-specifications of models. Intimately connected with this question is, perhaps surprisingly, convexity properties of option prices. We also study qualitative behavior of different optimal stopping boundaries appearing in option pricing.

In Paper I a new condition on the contract function of an American option is provided under which the option price increases monotonically in the volatility. It is also shown that American option prices are continuous in the volatitlity.

In Paper II an explicit pricing formula for the perpetual American put option in the Constant Elasticity of Variance model is derived. Moreover, different properties of this price are studied.

Paper III deals with the Russian option with a finite time horizon. It is shown that the value of the Russian option solves a certain free boundary problem. This information is used to analyze the optimal stopping boundary.

A study of perpetual game options is performed in Paper IV. One of the main results provides a condition under which the value of the option is increasing in the volatility.

In Paper V options written on several underlying assets are considered. It is shown that, within a large class of models, the only model for the stock prices that assigns convex option prices to all convex contract functions is geometric Brownian motion.

Finally, in Paper VI it is shown that the optimal stopping boundary for the American put option is convex in the standard Black-Scholes model.

Keywords: American options, convexity, monotonicity in the volatility, robustness, optimal stopping, parabolic equations, free boundary problems, volatility, Russian options, game options, excessive functions, superreplication, smooth fit

Erik Ekström, Department of Mathematics. Uppsala University. PO-Box 480, SE-752 32 Uppsala, Sweden

Erik Ekström 2004 c ISBN 91-506-1774-5 ISSN 1401-2049

urn:nbn:se:uu:diva-3344 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3344)

List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Ekström, E. (2002) Properties of American option prices, to ap- pear in Stochastic Processes Appl.

II Ekström, E. (2003) The perpetual American put option in a level- dependent volatility model, J. Appl. Probab., 40(3), 783-789.

III Ekström, E. (2004) Russian options with a finite time horizon, J.

Appl. Probab. 41(2), 313-326.

IV Ekström, E. (2004) Properties of game options, submitted for publication.

V Ekström, E., Janson, S., Tysk, J. (2003) Superreplication of op- tions on several underlying assets, to appear in J. Appl. Probab.

42(1).

VI Ekström, E. (2004) Convexity of the optimal stopping boundary for the American put option, J. Math. Anal. Appl., 299(1), 147- 156.

Reprints were made with permission from the publishers.

iii

Acknowledgments

I am deeply indebted to my adviser Johan Tysk for sharing with me his deep knowledge of mathematics. His guiding and his constant support and opti- mism have been invaluable for me, and his suggestions of improvements of manuscripts have significantly contributed to the contents of this thesis.

My thanks also go to all my friends and colleagues at the Department of Mathematics in Uppsala for providing a nice and friendly atmosphere. In particular, I would like to thank Gustaf Strandell for great companionship, and the group of financial mathematics for many interesting and stimulating seminars and discussions.

I thank my friends outside the department for filling my spare time with activities not even remotely connected to mathematics. In particular, I would like to mention Helena and Kristina.

Finally, I thank Kajsa for everything.

v

Sammanfattning på svenska (Summary in Swedish)

Denna avhandling består av en introduktion och sex stycken artiklar som alla berör finansiell matematik i kontinuerlig tid. Ett typiskt problem inom detta område är prissättningen av finansiella instrument definierade i termer av nå- gon underliggande tillgång. Eftersom den underliggande tillgångens framtida värde i allmänhet inte är känt så modelleras den med en stokastisk process.

I denna avhandling används nästan uteslutande stokastiska differentialekva- tioner för att modellera den underliggande tillgångens prisprocess. Värdet av det finansiella instrumentet kan då uttryckas som ett väntevärde av den framtida avkastningen.

Det är välkänt att väntevärden av funktioner av lösningar till stokastiska dif- ferentialekvationer även är lösningar till paraboliska partiella differentialek- vationer. Således kan optionspriser studeras både med hjälp av stokastiska metoder och med metoder från teorin för partiella differentialekvationer. I artiklarna i denna avhandling används en kombination av båda dessa metoder.

En svårighet vid modellering av den underliggande tillgången är att be- stämma volatiliteten. Volatiliteten hos till exempel en aktie är ett mått på fluktuationernas storlek hos aktiepriset. Denna storhet kan uppskattas med historiska data, men några säkra förutsägelser om det framtida beteendet kan givetvis ej göras. Således är det viktigt att veta vad en eventuell fel-speci- fikation av volatiliteten har för effekt vid prissättning av det finansiella instru- mentet. Något förvånande är kanske att denna fråga är nära besläktad med studiet av konvexitetsegenskaper hos lösningar till paraboliska ekvationer.

Studier av en eventuell fel-specifikation av modellen och av konvexitets-

egenskaper hos lösningar till paraboliska ekvationer återkommer i fyra av

artiklarna i denna avhandling, se artikel I, II, IV och V. I artikel I studeras

amerikanska optioner. Vi ger bland annat ett nytt villkor under vilket det

amerikanska optionspriset är en växande funktion av volatiliteten. Vi visar

även att amerikanska optionspriser är kontinuerliga i volatiliteten, det vill säga

att en liten förändring av volatiliteten endast medför en liten förändring av op-

tionspriset. I artikel II ger vi en explicit formel för priset av en amerikansk

säljoption i den så kallade CEV-modellen. Med hjälp av denna formel un-

dersöker vi optionsprisets beroende av volatiliteten. I artikel IV studerar vi

så kallade speloptioner. Vi ger en karakterisering av priset av en speloption

vii

med hjälp av konkava funktioner, samt ett villkor under vilket värdet av en speloption är en växande funktion av volatiliteten. I artikel V studerar vi europeiska optioner med flera underliggande tillgångar. Vi visar det kanske förvånande resultatet att, i en stor mängd av möjliga modeller, den enda mod- ellen i flera variabler som ger konvexa optionspriser för alla konvexa kontrakt är geometrisk Brownsk rörelse.

I de två återstående artiklarna studeras andra kvalitativa egenskaper hos op- tionspriser. I artikel III studeras så kallade ryska optioner med ändlig tids- horisont. Vi visar att det ryska optionspriset tillsammans med randen till fortsättningsområdet löser ett fritt randvärdesproblem. Denna information an- vänds sedan för att approximativt bestämma utbredningen av fortsättnings- området nära inlösensdatumet.

Slutligen, i artikel VI visar vi med hjälp av klassiska metoder för paraboliska partiella differentialekvationer att fortsättningsområdet för den amerikanska säljoptionen är konvext.

viii

Contents

1 Introduction . . . . 1

1.1 European Options . . . . 2

1.2 Arbitrage Pricing . . . . 2

1.3 The Black-Scholes Equation . . . . 3

1.4 Mis-Specification of Models . . . . 5

1.5 American Options and Optimal Stopping Problems . . . . 5

1.6 American Options and Free Boundary Problems . . . . 7

2 Included Papers . . . . 9

2.1 Paper I . . . . 9

2.2 Paper II . . . . 9

2.3 Paper III . . . . 10

2.4 Paper IV . . . . 11

2.5 Paper V . . . . 12

2.6 Paper VI . . . . 12

References . . . . 15

ix

1 Introduction

This thesis is devoted to the study of continuous time financial mathematics.

In this area typical problems are the pricing and hedging of various finan- cial instruments defined in terms of some underlying asset. The value of the instrument is determined by the expected value of some future prices of the underlying asset. These future values are of course not known and cannot be determined by some law of nature. The price processes are therefore modeled with stochastic processes. Surprisingly enough, there are strict theoretical limitations for what models should be used, compare Section 1.2. Roughly speaking, the discounted asset price should be modeled using a martingale when pricing financial instruments. Brownian motion, and more generally solutions to stochastic differential equations, are therefore used for modeling purposes. Thus the models appearing in mathematical finance have a lot sim- ilarities with models used in for example physics where Brownian motion is used to model particle movements and heat conduction.

Expected values of solutions to stochastic differential equations solve para- bolic partial differential equations. Thus both option pricing and heat conduc- tion can be studied using both stochastic methods and parabolic differential equations. Indeed, in this thesis methods from both of these areas are used.

However, the problems motivated by financial applications are in nature often different from the problems motivated by physics. The diffusion coef- ficient of financial models always represents more or less educated guesses and can never be measured precisely. The volatility of a stock can be approxi- mated with historical volatility, but when pricing financial instruments written on this stock an estimate of the future volatility is needed. Due to this uncer- tainty of coefficients it is of great importance to study the issues of robustness of models and the implications of a possible mis-specification of models. In- timately connected with the issue of mis-specification of models is, perhaps surprisingly, convexity properties of solutions to parabolic equations.

Mis-specification of models and convexity of option prices together consti- tute one of the main themes in this theses, compare Papers I, II, IV and V.

The remaining two papers, III and VI, deal with qualitative properties of free boundaries appearing in option pricing. Almost as a coincidence, the notion of convexity again occurs but in a different context.

1

1.1 European Options

A substantial part of the literature in financial mathematics deals with option pricing. Examples of options are the so-called call options and the put options.

The holder of a call option written on a certain stock has the right, but not the obligation, to buy this particular stock at some pre-determined date T at some pre-determined price K. If the stock at time T is worth more than K, say S (T), then the holder typically exercises the option and makes a profit S (T)−K. On the other hand, if the stock value S (T) is less than K, then he does not use his option so the option is worthless. Thus the value at time T of a call option is (S(T) − K) ⁺ : = max{S(T) − K,0}. Similarly, the holder of a put option has the right to sell the option at a fixed price K, so the value at T of a put option is (K − S(T)) ⁺ . The functions (s − K) ⁺ and (K − s) ⁺ are called the contract functions of the call option and the put option, respectively.

More generally, given a (preferably non-negative) function g (s) one can of course consider the option that gives the option holder the amount g (S(T)) at time T . These kind of options that are exercised at T and with a pay-off only depending on S (T) are usually referred to as European options. Later on we will also discuss other types of options, compare for example Section 1.5 in which American options are described in some detail. A central question in option theory is how to price options at times before T .

1.2 Arbitrage Pricing

An arbitrage is a risk-free profit, i.e. a strategy of buying and selling finan- cial instruments in such a way that the initial endowment is 0 and the wealth at time T is non-negative almost surely and strictly positive with a positive probability. In the seminal papers [7] and [28] it was shown how to price op- tions in order not to introduce arbitrage in the market. The surprising result of these papers can be formulated as follows: Consider a market consisting of two traded assets. One of them, the bank account, grows deterministically according to

dB = rBdt,

and the other one, the stock, is a risky asset with price process described by the stochastic differential equation

dS = µSdt + σSdW t (1.1)

Here the interest rate r ≥ 0, the appreciation rate µ and the volatility σ are

constants and W is a standard Brownian motion. Then the only price at time t

of an option that at time T > t pays g(S T ) that does not introduce arbitrage in

2

the market is P (s,t) given by the expected value

P(s,t) = E s ^Q ,t e ^−rT g(S T ) (1.2) where the indices indicate that the diffusion S is started in s at time t. What is surprising is that this expected value should not be calculated under the physical measure P which is used to describe S in (1.1), but rather under a so- called risk-neutral measure Q defined as the unique measure equivalent to P under which the discounted price process e ^−ru S(u) is a martingale. The stock price process can then be described in terms of a standard Q-Brownian motion W ^Q as

dS = rSdt + σSdW ^Q . (1.3)

The solutions to this stochastic differential equation and the one in (1.1) are so- called geometric Brownian motions, and they can be written down explicitly.

Indeed, the solution to (1.3) is given by

S (u) = S(t)exp{(r − σ ² /2)(u −t) + σ(W u ^Q −W t ^Q )}.

Since the increments of the Brownian motion W ^Q are Gaussian one can in the case of for example a call option calculate the expected value in (1.2) explicitly in terms of the cumulative distribution function for the normal distribution.

This explicit formula for the value of a call option is usually referred to as the Black-Scholes formula.

Note that the dynamics of S under the risk-neutral measure Q are the same as the dynamics under the physical measure P except that the drift of S under Q equals the interest rate r instead of µ. This is so regardless of how the appreciation rate µ is specified. Consequently, µ does not come into play when pricing options, so it is not necessary to specify µ in the model. In other words, if two agents disagree about the appreciation rate of the stock, but they agree about all other components in the model, then they will price all options in the same way.

1.3 The Black-Scholes Equation

There is a well-known connection between the expected value of a function of a diffusion process and parabolic equations through the Feynman-Kac repre- sentation theorems. Using this connection one finds that the value P = P(s,t) of the option given in (1.2) as an expected value also can be determined by solving the Black-Scholes equation

P t + σ ² s ²

2 P ss + rsP s − rP = 0

3

with terminal value P (s,T) = g(s) where g is the contract function (here and in the sequel indices on functions denote differentiation with respect to the indicated variable).

In the Black-Scholes model as presented above the stock price is assumed to follow a geometric Brownian motion. There is, however, a lot of statistical evidence that real stock prices are not given by such a process. For example, the volatility tends to increase if the stock value decreases, so the assumption of constant volatility is a little simplistic. To deal with this one might introduce the class of models

dS = rSdt + α(S,t)dW (1.4)

for different diffusion functions α = α(s,t). Here we assume that the model is specified directly under the risk-neutral measure Q, so there will be no need to change measures when pricing options written on an asset with price process (1.4). The Black-Scholes equation for these kind of models now becomes

P t + α ² (s,t)

2 P ss + rsP s − rP = 0,

again with terminal condition P (s,T) = g(s). Note that the Black-Scholes equation is of parabolic type with the time parameter going backwards but with a terminal condition instead of the usual initial condition. Using the time variable t ^ = T − t instead of t the equation is transformed into a parabolic equation with initial condition instead of a terminal condition.

Parabolic equations similar to the Black-Scholes equation have of course been studied extensively in both physics and mathematics. For example, the classical heat equation

u t = cu xx

which describes the propagation of heat is of the same type. The problems under consideration in finance, however, are often somewhat different in na- ture from the typical problems studied in physics. For example, in option pricing the unknown ingredient is the diffusion coefficient, i.e. the function α, whereas the terminal condition, i.e. the contract function, in general is completely known. Many natural questions in option pricing thus deal with a whole class of possible prices corresponding to different models for the stock price process, but for a fixed contract function. The natural emphasis in physics would perhaps rather be to assume that the diffusion coefficient is known and investigate problems for different initial temperatures. Another difference is that the coefficients in the Black-Scholes equation are in general degenerate in the sense that lim _{s →0} α(s,t) = 0 for every fixed time t. Thus the Black-Scholes equation is not always necessarily covered by the general theory of parabolic equations.

4

1.4 Mis-Specification of Models

As presented above, a possible starting point when pricing options is to specify a model for the underlying stock price. Then the true model for the stock is assumed to be known to everybody. Within such a framework one can apply standard arbitrage theory to determine prices of options, compare above.

One natural question that arises is what happens if the true stock price is not given by the specified model. In particular, will an agent who uses this model overestimate or underestimate the “true” option price?

Recalling the results of Black and Scholes it suffices to specify the stock price S under the risk-neutral measure Q as we did in (1.4). The interest rate r can be read off directly from the market (for example using bond prices), so the only remaining unknown ingredient when pricing options is the volatility.

Hence, when investigating robustness of option prices it is natural to investi- gate the implications for the option prices of a mis-specification of the volatil- ity. In particular, consider a scenario in which one hedger believes that the diffusion function appearing in (1.4) is α 1 , and another hedger believes that it is given by α 2 which is dominating α 1 in the sense that |α 1 (s,t)| ≤ |α 2 (s,t)|

for all s and t. Is it then necessarily true that the corresponding option prices P 1 and P ₂ satisfy the inequality P ₁ (s,t) ≤ P 2 (s,t)? If this is true we say that the option price is monotone increasing in the volatility (or in the diffusion coefficient).

It turns out that a question which is closely related to the issue of mono- tonicity in the volatility is the question of convexity of the option price in the underlying stock value, i.e. if the function s → P(s,t) is convex for every fixed t ≤ T. If the option price is convex, then an increase in the volatility implies an increase in the option price, compare [6], [14], [18] and [21]. This fact can be seen heuristically for example by looking at the Black-Scholes equation.

The issues of monotonicity in the volatility and convexity preserving models together constitute one of the main themes in this thesis, compare Papers I, II, IV and V.

1.5 American Options and Optimal Stopping Problems

American options may, in contrast to European options, be exercised at any time before the final time T . At each instant the holder of an American option needs to decide whether to exercise immediately or to wait. If the contract function is g, and if the holder of the American option decides to exercise it at time γ, then he receives the amount g(S(γ)) at γ. It is shown in [4] and [22]

that the unique arbitrage free price V at time t of an American option is given

5

by

V (s,t) = sup

t ≤γ≤T E s ^Q ,t e ^−r(γ−t) g(S(γ)). (1.5) In this expression the supremum is taken over all random times γ that are stopping times with respect to the filtration generated by the Brownian motion used to specify the dynamics of the stock price process S. This has the intu- itively clear interpretation that when determining the optimal time to exercise the option one may only take into account past information, not future infor- mation, about the stock price. Note that the value satisfies V (s,t) ≥ g(s) since the possibility γ = t is included in the supremum. Also note that choosing γ = T gives the inequality V(s,t) ≥ P(s,t) where P(s,t) is the corresponding European option price.

Unlike the European case, closed expressions for American option prices are very rare. However, it can be shown that the supremum in (1.5) is attained for the stopping time γ = T if g(s) = (s−K) ⁺ is the contract function of a call option. Thus the price of an American call reduces to the price of a European call which is given explicitly by the Black-Scholes formula.

The price V in (1.5) of an American option is given as the solution to a so-called optimal stopping problem. There are two main issues when dealing with optimal stopping problems. The first is obviously to determine the value V , i.e. to price the option. The second is to characterize a good strategy for the option holder, i.e. to find a stopping time that realizes the supremum, or if the supremum is not attained to find a sequence of stopping times along which the expected value in (1.5) converges to V . Clearly, if the value V (S(t),t) at some time t is strictly larger than the amount g (S(t)) which corresponds to immediate exercise, then it is not optimal to exercise the option. A key result in the theory of optimal stopping states that if g is continuous and the random variable

sup

t ≤u≤T g(S(u))

is integrable, then the supremum in (1.5) is attained for the stopping time γ ^∗ : = inf{u ≥ t : V(S(u),u) = g(S(u))},

compare for example [13] or Appendix D in [23]. This means that it is optimal to exercise the option at the first time t that the value V (S(t),t) of the option is equal to g (S(t)). Alternatively, the optimal stopping time γ ^∗ can be described as the first exit time from the continuation region

C := {(s,t) : V(s,t) > g(s)},

6

i.e. as

γ ^∗ : = inf{u ≥ t : (S(u),u) /∈ C }.

In many problems the continuation region is not possible to determine ex- plicitly. One of the main themes in the current thesis is to derive qualitative properties for the shape of the continuation region for some different options, compare Papers II, III and VI.

Above, the optimal stopping time γ ^∗ is described in terms of the value func- tion V , but it is not clear how to determine V itself. The most well-known characterization of V is in terms of so-called excessive functions, compare [11] and [15]. This characterization is not very explicit, however, except for perpetual options. For these options T = ∞, i.e. the option holders can choose to wait arbitrarily long to exercise them. The excessive functions are here the same as the concave (in a generalized sense) functions, compare [9] and [12].

1.6 American Options and Free Boundary Problems

Another useful characterization of the value of an American option (or more generally, an optimal stopping problem) is in terms of variational inequalities and free boundary problems, compare [3], [5], [24], [27] and [34]. It can be shown that the Black-Scholes equation holds at all points in the continuation region and that boundary conditions at the optimal stopping boundary, the boundary of the continuation region, are given by the smooth fit principle.

The smooth fit principle states that the function s → V(s,t) is continuously differentiable, not only in C , but also over the boundary ∂C (at least at points of ∂C where the contract function g is continuously differentiable).

To illustrate, let us consider the American put option in the standard Black- Scholes model, i.e. when the underlying stock has constant volatility. Then it is possible to show that the continuation region is described as all points (s,t) satisfying s > b(t) for some function b(·) describing the optimal stopping boundary {(s,t) : s = b(t)}. This means that for s > b(t) the value V satisfies V (s,t) > (K − s) ⁺ , and for s ≤ b(t) the value satisfies V(s,t) = (K − s) ⁺ . It is known, compare for example [19], that b (t) is monotone increasing, that it approaches K as t approaches T , and the asymptotic behavior of b (t) close to maturity is also known. In Paper VI we show that b (t) is a convex function.

The value V together with the function b (t) can be described as the solution

7

to the free boundary problem

 

 

 

 

 



V t + ^σ ₂

s ² V ss + rsV s − rV = 0 if s > b(t)

V = K − s if s = b(t)

V s = −1 if s = b(t)

V = (K − s) ⁺ if t = T,

(1.6)

compare [23]. Note here that the condition at the free boundary states that V s (b(t)+,t) = V s (b(t)−,t) = −1.

Explicit solutions to parabolic free boundary problems like the one above are not very often known. The situation is quite different, however, if one in- stead considers perpetual American options, compare [27], [32] and Paper II.

For these options the dependence on time, or rather on time left to maturity, is removed, so the parabolic equation in (1.6) reduces to an ordinary differential equation. Indeed, the value of the perpetual American put option can be found

by solving 

 

 

σ

2 s ² V ss + rsV s − rV = 0 if s > b

V = K − s if s ≤ b

V s = −1 if s = b,

compare [27] or [23]. This problem has the explicit bounded solution V (s) = (K − b)( b

s )

where b = _2r ^2rK _+σ

. It should not come as a surprise that the solution V (s,t) and b (t) to (1.6) converges to V(s) and b as t → −∞, i.e. as the time left to maturity tends to ∞.

The put option plays a central role in the theory of American option pricing.

It is probably the most studied American option, compare [2], [19], [29], [30], and the references therein. One of the most studied free boundary problems in physics is the so-called Stefan problem which describes the melting of ice, compare Chapter 8 in [16]. In this problem the free boundary describes the interface between water and ice. The condition at the free boundary is not given by the smooth fit condition, but it is given in the one-dimensional case by the condition V _s = −˙b(t) = − _dt ^d b(t) which is a condition ensuring the con- servation of energy in the system. There is a link between the put option and the Stefan problem since the time derivative V t of the American put option can be shown to satisfy a Stefan problem.

8

2 Included Papers

In this chapter we give short summaries of each of the papers included in this thesis.

2.1 Paper I

It is well-known, see [6], [14], [18] and [21], that the price of a European option written on a single stock with price given by (1.4) and with a con- vex contract function is convex also at times before maturity. This convexity property has certain implications in the case of a possible mis-specification of models, compare above. For example, the price of a European option with a convex contract function is increasing in the diffusion function α(s,t). More- over, it is easy to see that the convexity of the contract function is not only sufficient but also necessary for this monotonicity property.

This paper deals with American options with a finite time horizon. These contracts are analyzed using the notion of volatility time introduced in [21], see also [18]. It is shown that for American options, convexity of the contract function is a sufficient condition (this is also shown in [14] and [18] under slightly different assumptions) but no longer a necessary condition for the option price to be increasing in the volatility. In fact, it is shown that the price is increasing in the volatility if the contract function g satisfies the condition

g(s)

s is decreasing for all s > 0. (2.1) One immediately notices that this class includes for example all decreasing functions. It is also shown, by means of an example, that not all American op- tions are increasing in the volatility. It seems like a challenging open problem to determine the precise class of contract functions that guarantee monotonic- ity in the volatility.

We also study time-decay of option prices and continuity in the volatility.

2.2 Paper II

In this paper the value of an American perpetual put option is determined

explicitly when the underlying stock price is assumed to be given by the Con-

9

stant Elasticity of Variance (CEV) model, compare [8]. Thus the diffusion coefficient is given by α(s) = σs ^γ for some constants σ > 0 and 0 ≤ γ < 1.

Candidates for the value function and the optimal exercise strategy are found via a smooth fit guess, i.e. by solving the free boundary point problem

 



 

σ

2 s ^{2 γ} V ss (s) + rsV s (s) − rV(s) = 0 if s > b

V (s) = K − s if s ≤ b

V s (b) = −1.

Solving this problem here refers to finding an exercise level b ∈ R ⁺ and a bounded function V ∈ C ¹ [0,∞) ∩C ² (b,∞) that satisfies the equations. Once this is done we use martingale techniques to prove that the obtained candi- date for the value function indeed equals the value of the option, and that the optimal exercise policy is given as the first hitting time of the level b.

In the second part of the paper we analyze the value function V and the optimal exercise level b. More precisely, we show by direct calculations that the value of the American put option in the CEV-model is convex in the un- derlying stock price s and increasing in the volatility parameter σ. These results can also be seen as consequences of the results in Paper I, [14] or [18].

Moreover, we show that as γ 1, i.e. as the model converges to the standard Black-Scholes model, the price and the optimal exercise level approaches the

“correct” limits.

2.3 Paper III

The holder of a Russian option receives the supremum of the stock price pro- cess up till the time when he chooses to exercise it. The value of the perpetual Russian option has been found explicitly, compare [32] and [33]. In this pa- per we investigate the Russian option with a finite time horizon, i.e. the case when the option has to be exercised before some pre-determined time T , see also [10] and [31]. The differences between the finite time horizon option and the perpetual option are, naturally, quite similar to the differences between the finite horizon American put option and the perpetual put option.

We show that the optimal exercise policy is to exercise as soon as the ratio

between the current stock price and the running maximum falls below a cer-

tain level. As time goes by, i.e. as the time to maturity decreases, this level

increases to 1. Not surprisingly, it is possible to show that the value together

with the optimal stopping boundary is a solution to a parabolic free boundary

problem. We use this fact to show that the optimal stopping boundary is con-

tinuous as a function of time left to maturity if the interest rate is less than the

appreciation rate of the stock. Moreover, in this case the asymptotic behavior

10

of the optimal stopping boundary for times close to maturity is determined.

The method we use to determine the asymptotic behavior of the optimal stopping boundary is inspired by the method developed in [25] to examine the optimal stopping boundary for the American put option, see also [2]. Co- incidentally, it turns out that the optimal stopping boundaries of the Russian option and of the American put option exhibit the same type of asymptotic behavior.

2.4 Paper IV

This paper deals with a natural generalization of optimal stopping problems and American options, namely with so-called game options. A game option is like an American option but with the added feature that not only the option holder but also the option writer has the possibility to terminate the contract at any time. However, if the option writer terminates the contract, then he will have to pay a certain extra amount to the option holder. Given the two contract functions 0 ≤ g 1 ≤ g 2 and a constant discounting factor r ≥ 0 the value of such a contract is defined by

V (x) = sup

τ inf

γ Ee ^{−r(τ∧γ)} (g 1 (X τ )1 _{τ≤γ} + g 2 (X γ )1 _{γ<τ} ).

Here the supremum and infimum are taken over random times that are stop- ping times with respect to the Brownian motion driving the diffusion X. It is an interesting and non-trivial result of [26] that the order of the supremum and the infimum does not matter (at least not if g 2 is bounded).

Recall that the value of an ordinary optimal stopping problem can be char- acterized in terms of excessive functions, compare [11] and [15]. In this paper we manage to characterize also the value of perpetual stochastic games in terms of excessive functions. It is shown that

V (x) = inf

f ∈F f (x)

where the set F is the set of continuous functions f satisfying g 1 ≤ f ≤ g 2

and f is excessive in the regions where the second inequality is strict. This characterization allows us to use the equivalence between excessive functions and concave functions to give a method to explicitly calculate the value of a game option. This method is illustrated in two examples where the values of the game versions of the American put option and the American capped call option are determined explicitly.

In the last section we investigate convexity and monotonicity properties for

game options. Not all perpetual game options are convex in the stock value

in the continuation region and increasing in the volatility. This is in contrast

11

to the case of perpetual American options, compare [1]. We show that if the contract functions satisfy the condition

s 1 g 1 (s 2 ) ≤ s 2 g 2 (s 1 ) for all s 1 < s 2 , (2.2) then the value is indeed convex in the continuation region and thus also in- creasing in the volatility. It might be of interest to point out the similarity between this condition and the condition (2.1) appearing in Paper I: If g 1 sat- isfies (2.1), then the condition (2.2) is also satisfied.

It remains an interesting open problem to determine conditions under which game options with a finite time horizon are increasing in the volatility.

2.5 Paper V

It is well-known that for European options written on one underlying asset the price is convex in the stock price provided the pay-off function is convex, com- pare above. For options written on several underlying assets the situation is quite different. In this paper we show that, within a large class of models, the only model for the stock prices that preserves convexity in higher dimensions is the geometric Brownian motion (with a possibly time-dependent volatility matrix). The class of models considered contains essentially all models in which the volatility of an asset is a deterministic function of the value of that asset and time and for which the volatility is not an increasing function (unless constant). The proof of the theorem relies upon the methods developed in [20]

to determine which parabolic operators are convexity preserving.

We also give an application of our result to the problem of superreplication in several dimensions.

It should be noted that our result states that geometric Brownian motion is the only model (within the class) that gives a convex price for any convex contract function. It is a challenging open problem to determine the class of models that preserve convexity for specific contract functions.

2.6 Paper VI

In this paper we consider the following free boundary problem: Find functions f = f (τ,x) and x = x(τ) such that

 

 

 

 

 



f _τ = f xx + (C − 1) f x −C f if x > x(τ)

f = 1 − e ^x if x = x(τ)

f x = −e ^x if x = x(τ)

f = (1 − e ^x ) ⁺ if τ = 0.

12

It is well-known that the value of the American put option in the standard Black-Scholes model is a solution to this problem (after a logarithmic change of coordinates). The main result in this paper states that the free boundary x(τ) is a convex function of τ. The idea of the proof, inspired by [17], is to study the level curves of the function

v := f _τ f x + e ^x .

This function satisfies at the free boundary v (τ,x(τ)) = − ˙x(τ), so to show that x (·) is convex it suffices to show that v(τ,x(τ)) is decreasing. This is done by showing that all level curves of v that start at the free boundary leave the continuation region at the origin (the origin here corresponds to the point (s,t) = (K,T) in the coordinates used in Section 1.6), and by showing that these level curves have to be ordered at the origin.

13

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