Spectral Approximation of Infinite-Dimensional Black-Scholes Equations with Memory
This paper considers the pricing of a European option using a -market in which the stock price and the asset in the riskless bank account both have hereditary price structures described by the authors of this paper (1999). Under the smoothness assumption of the payoff function, it is shown that the infinite dimensional Black-Scholes equation possesses a unique classical solution. A spectral approximation scheme is developed using the Fourier series expansion in the space for the Black-Scholes equation. It is also shown that the th approximant resembles the classical Black-Scholes equation in finite dimensions.
The pricing of contingent claims in the continuous-time financial market that consists of a bank account and a stock account has been a subject of extensive research for the last decades. In the literature (e.g., [1–5]), the equations that describe the bank account and the price of the stock are typically written, respectively, as where is a one-dimensional standard Brownian motion defined on a complete filtered probability space and , , and are positive constants that represent, respectively, the interest rate of the bank account, the stock appreciation rate, and the stock volatility rate. The financial market that consists of one bank account and one stock account will be referred to as a -market, where stands for the bank account and stands for the stock.
A European option contract is a contract giving the buyer of the contract the right to buy (sell) a share of a particular stock at a predetermined price at a predetermined time in the future. The European option problem is, briefly, to determine the fee (called the rational price) that the writer of the contract should receive from the buyer for the rights of the contract and also to determine the trading strategy the writer should use to invest this fee in the -market in such a way as to ensure that the writer will be able to cover the option if it is exercised. The fee should be large enough that the writer can, with riskless investing, cover the option, but be small enough that the writer does not make an unfair (i.e., riskless) profit.
In , we noted reasons to include hereditary price structures to a -market model and then introduced such a model using a functional differential equation to describe the dynamics of the bank account and a stochastic functional differential equation to describe those of the stock account. The paper then obtained a solution to the option pricing problem in terms of conditional expectation with respect to a martingale measure. The importance of including hereditary price structure in the stock price dynamics was also recognized by other researchers in recent years (see, e.g., [7–14]).
In particular,  was one of the firsts that took into consideration hereditary structure in studying the pricing problem of European option. There the authors obtained a solution to the option pricing problem in terms of conditional expectation with respect to a martingale measure. The two papers [7, 9] developed an explicit formula for pricing European options when the underlying stock price follows a nonlinear stochastic delay equation with fixed delays (resp., variable delays) in the drift and diffusion terms. The paper  computed the logarithmic utility of an insider when the financial market is modelled by a stochastic delay equation. There the author showed that, although the market does not allow free lunches and is complete, the insider can draw more from his wealth than the regular trader. The paper also offered an alternative to the anticipating delayed Black-Scholes formula, by proving stability of European call option proces when the delay coefficients approach the nondelayed ones. The paper  derived the infinite-dimensional Black-Scholes equation for the -market, where the bank account evolves according to a linear (deterministic) functional differential equation and the stock dynamics is described by a very general nonlinear stochastic functional differential equation. A power series solution is also developed for the equation. Following the same model studied in , the work in  shows that under very mild conditions the pricing function is the unique viscosity solution of the infinite-dimensional Black-Scholes equation. A finite difference approximation scheme for the solution of the equation is developed and convergence result is also obtained. We mention here that option pricing problems were also considered by [12–14] for a financial market that is more restricted than those of [10, 11].
This paper considers the pricing of a European option using a -market, such as those in , in which the stock price and the asset in the riskless bank account both have hereditary price structures. Under the smoothness assumption of the payoff function, it is shown that the pricing function is the unique classical solution of the infinite-dimensional Black-Scholes equation. A spectral approximation scheme is developed using the Fourier series expansion in the space for the Black-Scholes equation. It is also shown that the th approximant resembles the celebrated classical Black-Scholes equation in finite dimensions (see, e.g., [4, 5]).
This paper is organized as follows. Section 2 summarizes the definitions and key results of  that will be used throughout this paper. The concepts of Fréchet derivative and extended Fréchet derivative are introduced in Section 3, along with results needed to make use of these derivatives. In Section 4, the results regarding the infinite-dimensional Black-Scholes equation and its corollary are restated from [6, 10]. Section 5 details the spectral approximate solution scheme for this equation. Section 6 is the paper's conclusion, followed by an appendix with the proof of Proposition 3.2.
2. The European Option Problem with Hereditary Price Structures
To describe the financial model with hereditary price structures, we start by defining our probability space. Let be a fixed constant. This constant will be the length of the time window in which the hereditary information is contained. If with , denote the space of continuous functions by . Define Note that is a real separable Banach space equipped with the uniform topology defined by the sup-norm and is a closed subset of . Throughout the end of this paper, we let and for simplicity. If and , let be defined by , .
Let , the space of continuous functions , and let , the Borel -algebra of subsets of under the topology defined by the metric , where
Let be the Wiener measure defined on with Note that the probability space is the canonical Wiener space under which the coordinate maps , , defined by for all and is a standard Brownian motion and . Let the filtration be the -augmentation of the natural filtration of the Brownian motion , defined by for all and
Equivalently, is the smallest sub--algebra of subsets of with respect to which the mappings are measurable for all . It is clear that the filtration defined above is right continuous in the sense of .
Consider the -valued process , where and , for all . That is, for each , and . In , it is shown that for and for , where
The new model for the -market introduced in  has a hereditary price structure in the sense that the rate of change of the unit price of the investor's assets in the bank account and that of the stock account depend not only on the current unit price but also on their historical prices. Specifically, we assume that and evolve according to the following two linear functional differential equations: with initial price functions and , where and are given functions in . In the model, , , and are bounded linear functionals on the real Banach space . The bounded linear functionals can be represented as (see ) where the above integrals are to be interpreted as Lebesgue-Stieltjes integrals and , , and are functions that are assumed to satisfy the following conditions.
Assumption 2.1. The functions , are nondecreasing functions on such that and , and the function is a function of bounded variation on such that for every .
We will, throughout the end, extend the domain of the above three functions to by defining for and for , and so forth.
Proposition in  provides an existence and uniqueness result under mild conditions, so the model makes sense mathematically to use. Note that the equations described by (2.7)-(2.8) include (1.1) as a special case. Therefore, the model considered in this paper is a generalization of that considered in most of the existing literature (see, e.g., ).
For the purpose of analyzing the discount rate for the bank account, let us assume that the solution process of (2.7) with the initial function takes the following form: and . Then the constant satisfies the following equation: The existence and uniqueness of a positive number that satisfies the above equation is shown in .
Throughout the end, we will fix the initial unit price functions , and , and the functional for the stock price described in (2.8) and (2.10). For the purpose of making the distinction when we interchange the usage of and in (2.8), we write the stock price process as for simplicity. And, when the functional , is used in place of in (2.8), its solution process will be written as .
A trading strategy in the -market is a progressively measurable vector process defined on such that for each , where and represent, respectively, the number of units of the bank account and the number of shares of the stock owned by the writer at time , and is the expectation with respect to .
The writer's total asset is described by the wealth process defined by where again and are, respectively, the unit price of the bank account and the stock described in (2.7) and (2.8). This wealth process can clearly take both positive and negative values, since it is permissible that .
We will make the following basic assumption throughout this paper.
Assumption 2.2 (self-financing condition). In the -market, it is assumed that all trading strategies satisfy the following self-financing condition: or equivalently,
Define the process by The following results are proven in .
Lemma 2.3. The process defined by (2.19) is a martingale defined on .
Lemma 2.4. There exists a unique probability measure defined on the canonical measurable space such that where is the indicator function of .
Lemma 2.5. The process defined by (2.17) is a standard Brownian motion defined on the filtered probability space .
From the above, it has been shown (see [6, equation (14)]) that with . It is also clear that the probabilistic behavior of under the probability measure is the same as that of under the probability measure ; that is, they have the same distribution.
Define the process , called the discounted wealth process, by We say that a trading strategy from belongs to a subclass if a.s. where is the expectation with respect to , is a nonnegative -measurable random variable such that . We say that belongs to if .
Throughout, we assume the reward function is an -measurable nonnegative random variable satisfying the following condition: for some . Here, is the expiration time. (Note that the above condition on implies that .)
Let be a nonnegative -measurable random variable satisfying (2.24). A trading strategy is a -hedge of European type if and a.s. We say that a -hedge trading strategy is minimal if for any -hedge strategy .
Let be the set of -hedge strategies from . Define The value defined above is called the rational price of the contingent claim of European type. If the infimum in (2.28) is achieved, then is the minimal possible initial capital for which there exists a trading strategy possessing the property that a.s. .
Let be defined by where . In , it is shown that the process is a martingale defined on and can be represented by where that is -adapted and ( a.s.).
Theorem 2.7. Let be an -measurable random variable defined on the filtered probability space that satisfies (2.24). Then the rational price defined in (2.28) is given by
where is the positive constant that satisfies (2.12). Furthermore, there exists a minimal hedge , where
and the process is given by (2.30).
If in addition, the reward is intrinsic, that is, for some measurable function , then the rational price does not depend on the mean growth rate of the stock and
3. Fréchet and Extended Fréchet Derivatives
In this section, results are proven that allow the use of a Dynkins formula for stochastic functional differential equation as found in [16, 17]. We assume contingent claims of European type in which the -measurable reward function has the explicit expression , where again , and is the unit price of the stock described by the following equation: where . Throughout this section, we assume that , and therefore , are uniformly bounded almost surely. This assumption is realistic for the price of a stock during time interval in a financial system with finite total wealth.
The remaining sections make extensive use of Fréchet derivatives. Let be the space of bounded linear functionals . is a real separable Banach space under the supremum operator norm For , we denote the Fréchet derivative of at by . The second Fréchet derivative at is denoted as .
Let be the vector space of all simple functions of the form , where and is defined by Form the direct sum and equip it with the complete norm Then has a unique continuous linear extension from to which we will denote by , and similarly for ; see  or  for more details.
Finally, we define for all and , where is defined by Let . We say that if has a continuous Fréchet derivative. Similarly, if has a continuous th Fréchet derivative. For , we say that if is infinitely differentiable in its first variable and has a continuous th partial derivative in its second variable.
Proposition 3.1. Let and with . Define by Then .
Proof. That is is clear, so we have only to show that , where given that .
We have that with . Under Assumption 2.1 on and the properties of , it can be shown that there exists such that . Therefore, By Theorem , Chapter 2 of , as a function of . By a second application of the same theorem (since ), we have that as a function of . Define by . Since and in its third variable, . Hence, for , where is a function mapping continuous functions into the reals such that The last integral is clearly and is bounded and linear in , so this integral is the first Fréchet derivative with respect to . Since , the process can be repeated, giving a second Fréchet derivative with respect to and so .
Proposition 3.2. Let and . Further assume and let be defined by Then if and are globally Lipschitz, then so is .
Recall from Proposition 3.1 that is where with .
Proposition 3.3. Let and . Further assume and let be defined by Then if and are globally bounded, then so is .
Proof. We have that where we used the assumption that and hence are globally bounded to move the limit inside the integral and .
Remark 3.4. Note that since is bounded for all , exits. Also, if is bounded, exits (see ).
4. The Infinite-Dimensional Black-Scholes Equation
It is known (e.g., [4, 5]) that the classical Black-Scholes equation is a deterministic parabolic partial differential equation (with a suitable auxiliary condition) the solution of which gives the value of the European option contract at a given time. Propositions 3.1 through 3.3 allow us to use the Dynkin formula in . With it, a generalized version of the classical Black-Scholes equation can be derived for when the -market model is (2.7) and (2.8). The following theorem is a restatement of Theorem in .
Theorem 4.1. Let , where and . Let be a function with and globally Lipschitz and let and . Finally, let and be globally bounded. Then if is the wealth process for the minimal -hedge, one has where and the trading strategy is defined by Furthermore, if (4.1) and (4.2) hold, then is the wealth process for the -hedge with and
Proof. The theorem is a restatement of Theorem in  and is therefore omitted.
5. Approximation of Solutions
In this section, we will solve the generalized Black-Scholes equation (4.1)-(4.2) by considering a sequence of approximations of its solution. By a (classical) solution to (4.1)-(4.2), we mean satisfying the following conditions:(i),(ii) for all ,(iii) satisfies (4.1).
The sequence of approximate solutions is constructed by looking at finite-dimensional subspaces of , solving (4.1)-(4.2) on these subspaces, and then showing that as the dimension of the subspaces goes to infinity, the finite-dimensional solutions converge to a solution of the original problem. Theorem 5.2, Remark 5.3, and Corollary 5.4 show that the generalized Black-Scholes equation can be solved by solving two simpler equations. The first of these, a first-order partial differential equation, can be handled by traditional techniques once the second equation is solved. Theorem 5.5 provides a solution to the second. Proposition 5.7, which uses Lemma 5.6, gives a generalized Black-Scholes formula for the standard European call option when used in conjunction with Theorem 5.2.
We start by noting that where is the space of all square-integrable functions on the interval . Furthermore, is dense in . It is well known (e.g., ) that even extensions of a function in may be represented by a cosine Fourier series where as where Here, for . If is Hölder-continuous, then the convergence is also point wise (see, e.g., ).
Throughout this section, we let be the subspace of consisting of functions that can be represented as a finite Fourier series, that is, if for all .
We will see that it is convenient having a spanning set for where for such that for all and for all . Here, . Let be any function such that . For example, let where and . To this end, we define the following functions. Let and for , Recall that is defined by and let Here again is any function such that . For example, can be chosen as in (5.5). In this case, the constant is nonzero but otherwise arbitrary, and so on with for .
Lemma 5.1. The set defined in (5.6) and (5.7) forms a spanning set for in the sense that as , where the are defined by and continuing using until This set of functions has the properties that and for all
Proof. For any , we can construct an even extension where for all and for all . The function may be represented by a Fourier series of cosine functions
where the “’’ is used to indicate that
as . In what mentioned before,
for all For simplicity, we will replace the “’’ with an equality sign knowing that mean-square convergence is implied.
For the Fourier series, the basis is so the first term of this basis and are the same, namely, the constant “.’’ Clearly and . The first part of this proof is to show that for all , and .
For , we have that which implies that Also, . Since we do not want , we require that There are no restrictions on other than .
For , requires that Since we want , then
The rest of the , that is, where , are handled alike. In order that , we require that . To ensure that , We have now shown that the sequence of functions is such that and for all Now it must be shown that this sequence is a spanning set for . To do this, we will compare this sequence of functions with the cosine Fourier sequence of functions.
Consider where We would like for some set of real numbers. By the Fourier expansion, We want where Equating the last coefficients gives Continuing, and finally Hence, with the above choice of , and so as .
To find an approximate solution to the generalized Black-Scholes equation we start by letting (from ) and approximating by We define the space as the set of all continuous functions that can be represented by this summation for some . Note that . Also define by so that . Define by provided that the is formed by the coefficients of in the spanning set . In general, is formed by the coefficients of in the spanning set . Also, define by Last, let be defined by where the are the coefficients of using the spanning set . Finally, define the operator by where the right-hand side is the first terms of the -expansion of .
We are now ready for a theorem which enables us to approximate the solution of the infinite-dimensional Black-Scholes equation by solving a first-order real-valued partial differential equation and an equation similar to the generalized Black-Scholes equation but without the term. The lack of this term allows approximate solutions to be found using traditional techniques.
Theorem 5.2. Let and . Let be a function satisfying the conditions of Theorem 4.1 and let . Then where has a solution of the form Here, is a solution to for and for some and , and is a solution of where and is a uniformly bounded function satisfying the conditions of Theorem 4.1.
Proof. We assume a solution of the form , then
If is the solution to (5.44), then
Define by , that is, is a shift operator. Now let for a fixed . Then
where the superscript denotes a right-hand derivative with respect to . Then
A slightly more restrictive, but more familiar form is where and for some . There is the additional requirement that so that (5.44) holds.
Remark 5.3. It can be easily shown that is -Hölder continuous a.s. for provided that is -Hölder continuous for the same . Therefore, for each as where is a solution to (5.44) and is an approximate solution, since is in its second variable and
Corollary 5.4. If is Hölder continuous, then where has a solution of the form . Here, is a solution to for and for some . is the solution to (5.44) where one lets . In addition, .
Now we must solve (5.44), which is done in the following theorem. With this solution, the first-order partial differential equation can be solved by traditional means.
Proof. Since , the definition of the Fréchet derivatives and the properties of the set give The consists of the first coefficients of in the set of functions . By the Feynman-Kac theorem (see [15, Theorem 5.7.6]), where is the solution to for . Noting that