Spectral Approximation of Infinite-Dimensional Black-Scholes Equations with Memory

Abstract

This paper considers the pricing of a European option using a (𝐵,𝑆)-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 𝐶[−ℎ,0] for the Black-Scholes equation. It is also shown that the 𝑛th approximant resembles the classical Black-Scholes equation in finite dimensions.

1. Introduction

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, as𝑑𝑆(𝑡)=𝑑𝛼𝐵𝑆((𝑡𝑡))=𝑑𝑡𝑟+𝐵𝜎(𝑡𝑆)𝑑(𝑡𝑡),𝑑𝑊𝐵((0𝑡)),=𝑥𝑆,(0)=𝑦,(1.1) where 𝑊={𝑊(𝑡),𝑡≥0} is a one-dimensional standard Brownian motion defined on a complete filtered probability space (Ω,𝐹,𝐏;{𝐹(𝑡),𝑡≥0}) and 𝑟, 𝛼, and 𝜎 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 (𝐵,𝑆)-market, where 𝐵 stands for the bank account and 𝑆 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 (𝐵,𝑆)-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 [6], we noted reasons to include hereditary price structures to a (𝐵,𝑆)-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, [6] 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 [8] 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 [10] derived the infinite-dimensional Black-Scholes equation for the (𝐵,𝑆)-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 [10], the work in [11] 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 (𝐵,𝑆)-market, such as those in [6], 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 𝐶[−ℎ,0] for the Black-Scholes equation. It is also shown that the 𝑛th 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 [6] 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 0<ℎ<∞ be a fixed constant. This constant will be the length of the time window in which the hereditary information is contained. If 𝑎,𝑏∈ℜ with 𝑎<𝑏, denote the space of continuous functions 𝜙∶[𝑎,𝑏]→ℜ by 𝐶[𝑎,𝑏]. Define 𝐶₊[𝑎,𝑏]={𝜙∈𝐶[𝑎,𝑏]∣𝜙(𝜃)≥0∀𝜃∈[𝑎,𝑏]}.(2.1) Note that 𝐶[𝑎,𝑏] is a real separable Banach space equipped with the uniform topology defined by the sup-norm ‖𝜙‖=sup_{𝑡∈[𝑎,𝑏]}|𝜙(𝑡)| and 𝐶₊[𝑎,𝑏] is a closed subset of 𝐶[𝑎,𝑏]. Throughout the end of this paper, we let 𝐂=𝐶[−ℎ,0] and 𝐂₊={𝜙∈𝐂∣𝜙(𝜃)≥0∀𝜃∈[−ℎ,0]}(2.2) for simplicity. If 𝜓∈𝐶[−ℎ,∞) and 𝑡∈[0,∞), let 𝜓_𝑡∈𝐂 be defined by 𝜓_𝑡(𝜃)=𝜓(𝑡+𝜃), 𝜃∈[−ℎ,0].

Let Ω=𝐶[−ℎ,∞), the space of continuous functions 𝜔∶[−ℎ,∞)→ℜ, and let 𝐹=𝐵(𝐶[−ℎ,∞)), the Borel 𝜎-algebra of subsets of 𝐶[−ℎ,∞) under the topology defined by the metric 𝑑∶Ω×Ω→ℜ, where𝑑𝜔,𝜔^=^∞_𝑛=112^𝑛sup_{−ℎ≤𝑡≤𝑛}||𝜔(𝑡)−𝜔^(𝑡)||1+sup_{−ℎ≤𝑡≤𝑛}||𝜔(𝑡)−𝜔^(𝑡)||.(2.3)

Let 𝐏 be the Wiener measure defined on (Ω,𝐹) with 𝐏{𝜔∈Ω∣𝜔(𝜃)=0∀𝜃∈[−ℎ,0]}=1.(2.4) Note that the probability space (Ω,𝐹,𝐏) is the canonical Wiener space under which the coordinate maps 𝑊={𝑊(𝑡),𝑡≥0}, 𝑊(𝑡)∶𝐶[−ℎ,∞)→ℜ, defined by 𝑊(𝑡)(𝜔)=𝜔(𝑡) for all 𝑡≥−ℎ and 𝜔∈Ω is a standard Brownian motion and 𝐏{𝑊₀=0}=1. Let the filtration 𝐹^𝑊={𝐹(𝑡),𝑡≥−ℎ} be the 𝐏-augmentation of the natural filtration of the Brownian motion 𝑊, defined by 𝐹(𝑡)={∅,Ω} for all 𝑡∈[−ℎ,0] and 𝐹(𝑡)=𝜎(𝑊(𝑠),0≤𝑠≤𝑡),𝑡≥0.(2.5)

Equivalently, 𝐹(𝑡) is the smallest sub-𝜎-algebra of subsets of Ω with respect to which the mappings 𝑊(𝑠)∶Ω→ℜ are measurable for all 0≤𝑠≤𝑡. It is clear that the filtration 𝐹^𝑊 defined above is right continuous in the sense of [15].

Consider the 𝐂-valued process {𝑊_𝑡,𝑡≥0}, where 𝑊₀=0 and 𝑊_𝑡(𝜃)=𝑊(𝑡+𝜃), 𝜃∈[−ℎ,0] for all 𝑡≥0. That is, for each 𝑡≥0, 𝑊_𝑡(𝜔)=𝜔_𝑡 and 𝑊₀=0. In [10], it is shown that 𝐹₀=𝐹(𝑡) for 𝑡∈[−ℎ,0] and 𝐹(𝑡)=𝐹_𝑡 for 𝑡≥0, where 𝐹_𝑡=𝜎𝑊_𝑠,0≤𝑠≤𝑡,𝑡≥0.(2.6)

The new model for the (𝐵,𝑆)-market introduced in [6] 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 𝐵 and that of the stock account 𝑆 depend not only on the current unit price but also on their historical prices. Specifically, we assume that 𝐵 and 𝑆 evolve according to the following two linear functional differential equations: 𝑑𝑆(𝑡)=𝑀𝑑𝐵(𝑆𝑡)_𝑡=𝐿𝐵_𝑡𝑑𝑡,𝑡≥0,(2.7)𝑑𝑡+𝑁𝑆_𝑡𝑑𝑊(𝑡),𝑡≥0,(2.8) with initial price functions 𝐵₀=𝜙 and 𝑆₀=𝜓, where 𝜙 and 𝜓 are given functions in 𝐂₊. In the model, 𝐿, 𝑀, and 𝑁 are bounded linear functionals on the real Banach space 𝐂. The bounded linear functionals 𝐿,𝑀,𝑁∶𝐂→ℜ can be represented as (see [6]) 𝑀𝐿((𝜙𝜙))==^0−ℎ_0−ℎ𝜙(𝜃)𝑑𝜂(𝜃),𝑁(𝜙)=_0−ℎ𝜙(𝜃)𝑑𝜉(𝜃),(2.9)𝜙(𝜃)𝑑𝜁(𝜃),𝜙∈𝐂,(2.10) where the above integrals are to be interpreted as Lebesgue-Stieltjes integrals and 𝜂, 𝜉, and 𝜁 are functions that are assumed to satisfy the following conditions.

Assumption 2.1. The functions 𝜂,𝜉∶[−ℎ,0]→ℜ, are nondecreasing functions on [−ℎ,0] such that 𝜂(0)−𝜂(−ℎ)>0 and 𝜉(0)−𝜉(−ℎ)>0, and the function 𝜁∶[−ℎ,0]→ℜ is a function of bounded variation on [−ℎ,0] such that ∫^0−ℎ𝜙(𝜃)𝑑𝜁(𝜃)≥𝜎>0 for every 𝜙∈𝐂₊.

We will, throughout the end, extend the domain of the above three functions to 𝑅 by defining 𝜂(𝜃)=𝜂(−ℎ) for 𝜃≤−ℎ and 𝜂(𝜃)=𝜂(0) for 𝜃≥0, and so forth.

Proposition 2.3 in [6] 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., [5]).

For the purpose of analyzing the discount rate for the bank account, let us assume that the solution process 𝐵(𝐿;𝜙)={𝐵(𝑡),−ℎ≤𝑡<∞} of (2.7) with the initial function 𝜙∈𝐂₊ takes the following form: 𝐵(𝑡)=𝜙(0)𝑒^𝑟𝑡,𝑡≥0,(2.11) and 𝐵₀=𝜙∈𝐂₊. Then the constant 𝑟 satisfies the following equation: 𝑟=^0−ℎ𝑒^𝑟𝜃𝑑𝜂(𝜃).(2.12) The existence and uniqueness of a positive number 𝑟 that satisfies the above equation is shown in [6].

Throughout the end, we will fix the initial unit price functions 𝜙, and 𝜓∈𝐂₊, and the functional 𝑁∶𝐂→ℜ for the stock price described in (2.8) and (2.10). For the purpose of making the distinction when we interchange the usage of 𝑀∶𝐂→ℜ and 𝐿∶𝐂→ℜ in (2.8), we write the stock price process 𝑆(𝑀,𝑁;𝜓) as 𝑆(𝑀)={𝑆(𝑡),𝑡≥−ℎ} for simplicity. And, when the functional 𝐾∶𝐂→ℜ, 𝐾(𝜙_𝑡)=𝑟𝜙(𝑡) is used in place of 𝑀∶𝐂→ℜ in (2.8), its solution process will be written as 𝑆(𝐾)={𝑆(𝑡),𝑡≥−ℎ}.

In [6], the basic theory of European option pricing using the (𝐵,𝑆)-market model described in (2.7)-(2.8) is developed. We summarize the key definitions and results below.

A trading strategy in the (𝐵,𝑆)-market is a progressively measurable vector process 𝜋={(𝜋₁(𝑡),𝜋₂(𝑡)),0≤𝑡<∞} defined on (Ω,𝐹,𝐏;𝐹^𝑊) such that for each 𝑎>0, ^𝑎0𝐄𝜋^2𝑖(𝑡)𝑑𝑡<∞,𝑖=1,2,(2.13) where 𝜋₁(𝑡) and 𝜋₂(𝑡) represent, respectively, the number of units of the bank account and the number of shares of the stock owned by the writer at time 𝑡≥0, and 𝐄 is the expectation with respect to 𝐏.

The writer's total asset is described by the wealth process 𝑋^𝜋(𝑀)={𝑋^𝜋(𝑡),0≤𝑡<∞} defined by 𝑋^𝜋(𝑡)=𝜋₁(𝑡)𝐵(𝑡)+𝜋₂(𝑡)𝑆(𝑡),0≤𝑡<∞,(2.14) where again 𝐵(𝐿;𝜙) and 𝑆(𝑀,𝑁;𝜓) 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 (𝜋₁(𝑡),𝜋₂(𝑡))∈ℜ².

We will make the following basic assumption throughout this paper.

Assumption 2.2 (self-financing condition). In the (𝐵,𝑆)-market, it is assumed that all trading strategies 𝜋 satisfy the following self-financing condition: 𝑋^𝜋(𝑡)=𝑋^𝜋(0)+^𝑡0𝜋₁(𝑠)𝑑𝐵(𝑠)+^𝑡0𝜋₂(𝑠)𝑑𝑆(𝑠),0≤𝑡<∞,a.s.(2.15) or equivalently, 𝑑𝑋^𝜋(𝑡)=𝜋₁(𝑡)𝑑𝐵(𝑡)+𝜋₂(𝑡)𝑑𝑆(𝑡),0≤𝑡<∞.(2.16)

Using the same notation as in [6] (see also [10]) the set of all self-financing trading strategies 𝜋 will be denoted by SF(𝐿,𝑀,𝑁;𝜙,𝜓) or simply SF if there is no danger of ambiguity.

For the unit price of the bank account 𝐵(𝐿;𝜙)={𝐵(𝑡),𝑡≥0} and the stock 𝑆(𝑀,𝑁;𝜓)={𝑆(𝑡),𝑡≥0} described in (2.7) and (2.8), define 𝑊(𝑡)=𝑊(𝑡)+^𝑡0𝛾𝐵_𝑠,𝑆_𝑠𝑑𝑠,𝑡≥0,(2.17) where 𝛾∶𝐂₊×𝐂₊→ℜ is defined by 𝛾(𝜙,𝜓)=𝜙(0)𝑀(𝜓)−𝜓(0)𝐿(𝜙)𝜙(0)𝑁(𝜓).(2.18)

Define the process 𝑍(𝐿,𝑀,𝑁;𝜙,𝜓)={𝑍(𝑡),𝑡≥0} by 𝑍(𝑡)=exp^𝑡0𝛾𝐵_𝑠,𝑆_𝑠𝑑𝑊(𝑠)−12^𝑡0||𝛾𝐵_𝑠,𝑆_𝑠||²𝑑𝑠,𝑡≥0.(2.19) The following results are proven in [6].

Lemma 2.3. The process 𝑍(𝐿,𝑀,𝑁;𝜙,𝜓)={𝑍(𝑡),𝑡≥0} defined by (2.19) is a martingale defined on (Ω,𝐹,𝐏;𝐹^𝑊).

Lemma 2.4. There exists a unique probability measure 𝐏 defined on the canonical measurable space (Ω,𝐹) such that 𝐏(𝐴)=𝐄𝟏_𝐴𝑍(𝑇)∀𝐴∈𝐹_𝑇,0<𝑇<∞,(2.20) where 𝟏_𝐴 is the indicator function of 𝐴∈𝐹_𝑇.

Lemma 2.5. The process 𝑊 defined by (2.17) is a standard Brownian motion defined on the filtered probability space (Ω,𝐹,𝐏;𝐹^𝑊).

From the above, it has been shown (see [6, equation (14)]) that 𝑑𝑆(𝑡)=𝑟𝑆(𝑡)𝑑𝑡+𝑁𝑆_𝑡𝑑𝑊(𝑡),(2.21) with 𝑆₀=𝜓∈𝐂₊. It is also clear that the probabilistic behavior of 𝑆(𝑀) under the probability measure 𝐏 is the same as that of 𝑆(𝐾) under the probability measure 𝐏; that is, they have the same distribution.

Define the process 𝑌^𝜋(𝐿,𝑀,𝑁;𝜙,𝜓)={𝑌^𝜋(𝑡),𝑡≥0}, called the discounted wealth process, by 𝑌^𝜋(𝑡)=𝑋^𝜋(𝑡)𝐵(𝑡),𝑡≥0.(2.22) We say that a trading strategy 𝜋 from SF(𝐿,𝑀,𝑁;𝜙,𝜓) belongs to a subclass SF^𝜍⊂SF if 𝐏 a.s. 𝑌^𝜋(𝑡)≥−𝐄𝜍∣𝐹_𝑡,𝑡≥0,(2.23) where 𝐄 is the expectation with respect to 𝐏, 𝜍 is a nonnegative 𝐹-measurable random variable such that 𝐄[𝜍]<∞. We say that 𝜋 belongs to SF⁺⊂SF if 𝜍≥0.

In [6, 10], it is shown that 𝑌^𝜋 is a local martingale; for 𝜋∈SF^𝜍, 𝑌^𝜋 is a supermartingale, and is a nonnegative supermartingale if 𝜋∈SF⁺.

Throughout, we assume the reward function Λ is an 𝐹_𝑇-measurable nonnegative random variable satisfying the following condition: 𝐄Λ^1+𝜖<∞,(2.24) for some 𝜖>0. Here, 𝑇>0 is the expiration time. (Note that the above condition on Λ implies that 𝐄[Λ]<∞.)

Let Λ be a nonnegative 𝐹_𝑇-measurable random variable satisfying (2.24). A trading strategy 𝜋∈SF is a (𝑀;Λ,𝑥)-hedge of European type if 𝑋^𝜋(0)=𝜋₁(0)𝜙(0)+𝜋₂(0)𝜓(0)=𝑥(2.25) and 𝐏 a.s. 𝑋^𝜋(𝑇)≥Λ.(2.26) We say that a (𝑀;Λ,𝑥)-hedge trading strategy 𝜋^∗∈SF(𝑀) is minimal if 𝑋^𝜋(𝑇)≥𝑋^𝜋∗(𝑇)(2.27) for any (𝑀;Λ,𝑥)-hedge strategy 𝜋∈SF(𝑀).

Let Π(𝑀;Λ,𝑥) be the set of (𝑀;Λ,𝑥)-hedge strategies from SF⁺(𝑀). Define 𝐶(𝑀;Λ)=inf{𝑥≥0∶Π(𝑀;Λ,𝑥)≠∅}.(2.28) The value 𝐶(𝑀;Λ) defined above is called the rational price of the contingent claim of European type. If the infimum in (2.28) is achieved, then 𝐶(𝑀;Λ) is the minimal possible initial capital for which there exists a trading strategy 𝜋∈SF⁺(𝑀) possessing the property that 𝐏 a.s. 𝑋^𝜋(𝑇)≥Λ.

Let 𝑌(𝑀)={𝑌(𝑡),0≤𝑡≤𝑇} be defined by 𝑌(𝑡)=𝐄Λ𝐵(𝑇)∣𝐹^𝑡,0≤𝑡≤𝑇,(2.29) where 𝐹_𝑡=𝜎(𝑊_𝑠,0≤𝑠≤𝑡). In [10], it is shown that the process 𝑌(𝑀) is a martingale defined on (Ω,𝐹,𝐏;𝐹𝑊) and can be represented by 𝑌(𝑡)=𝑌(0)+^𝑡0𝛽(𝑠)𝑑𝑊(𝑠),(2.30) where 𝛽={𝛽(𝑡),0≤𝑡≤𝑇} that is 𝐹𝑊-adapted and ∫^𝑇0𝛽²(𝑡)𝑑𝑡<∞ (𝐏 a.s.).

The following lemma and theorem provide the main results of [6, 10]. Let 𝜋^∗={(𝜋^∗1(𝑡),𝜋^∗2(𝑡)),0≤𝑡≤𝑇} be a trading strategy, where 𝜋^∗2(𝑡)=𝛽(𝑡)𝐵(𝑡)𝜋_∗1𝑁𝑆_𝑡,(𝑡)=𝑌(𝑡)−𝑆(𝑡)𝐵(𝑡)𝜋^∗2(𝑡),𝑡∈[0,𝑇].(2.31)

Lemma 2.6. 𝜋^∗∈SF(𝑀) and for each 𝑡∈[0,𝑇], 𝑌(𝑡)=𝑌^𝜋∗(𝑡) for each 𝑡∈[0,𝑇] where again 𝑌^𝜋∗ is the process defined in (2.22) with the minimal strategy 𝜋^∗ defined in (2.31).

Theorem 2.7. Let Λ be an 𝐹_𝑇-measurable random variable defined on the filtered probability space (Ω,𝐹,𝐏;𝐹^𝑊) that satisfies (2.24). Then the rational price 𝐶(𝑀;Λ) defined in (2.28) is given by 𝐶(𝑀;Λ)=𝐄𝑒^−𝑟𝑇Λ,(2.32) where 𝑟 is the positive constant that satisfies (2.12). Furthermore, there exists a minimal hedge 𝜋^∗={(𝜋^∗1(𝑡),𝜋^∗2(𝑡)),0≤𝑡≤𝑇}, where 𝜋^∗2(𝑡)=𝛽(𝑡)𝐵(𝑡)𝜋_∗1𝑁𝑆_𝑡,(𝑡)=𝑌^𝜋∗(𝑡)−𝜋^∗2(𝑡)𝑆(𝑡)𝐵(𝑡),(2.33) and the process 𝛽={𝛽(𝑡),0≤𝑡≤𝑇} is given by (2.30).
If in addition, the reward Λ is intrinsic, that is, Λ=Γ(𝑆(𝑀)) for some measurable function Γ∶𝐂₊→ℜ, then the rational price 𝐶(𝑀;Λ) does not depend on the mean growth rate 𝑀 of the stock and 𝐶(Λ)=𝐄𝑒^−𝑟𝑇Λ.(2.34)

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 𝐹_𝑇-measurable reward function Λ has the explicit expression Λ=𝑓(𝑆_𝑇), where again 𝑆_𝑇(𝜃)=𝑆(𝑇+𝜃), 𝜃∈[−ℎ,0] and 𝑆(𝐾)={𝑆(𝑡),𝑡≥0} is the unit price of the stock described by the following equation: 𝑑𝑆(𝑡)=𝑟𝑆(𝑡)𝑑𝑡+𝑁𝑆_𝑡𝑑𝑊(𝑡),𝑡≥0,(3.1) where 𝑆₀=𝜓∈𝐂₊. Throughout this section, we assume that 𝑆(𝑡), and therefore 𝑁(𝑆_𝑡), are uniformly bounded almost surely. This assumption is realistic for the price of a stock during time interval [0,𝑇] in a financial system with finite total wealth.

The remaining sections make extensive use of Fréchet derivatives. Let 𝐂^∗ be the space of bounded linear functionals Φ∶𝐂→ℜ. 𝐂^∗ is a real separable Banach space under the supremum operator norm ‖Φ‖=sup_𝜙≠0||Φ(𝜙)||‖𝜙‖.(3.2) For Ψ∶[0,𝑇]×𝐂→ℜ, we denote the Fréchet derivative of Ψ at 𝜙∈𝐂 by 𝐷Ψ(𝑡,𝜙). The second Fréchet derivative at 𝜙 is denoted as 𝐷²Ψ(𝑡,𝜙).

Let Γ be the vector space of all simple functions of the form 𝑣𝟏_{0}, where 𝑣∈𝑅 and 𝟏_{0}∶[−ℎ,0]→ℜ is defined by 𝟏_{0}(𝜃)=01,,ffoorr𝜃𝜃∈=0[.−ℎ,0),(3.3) Form the direct sum 𝐂⊕Γ and equip it with the complete norm ‖‖𝜙+𝑣𝟏_{0}‖‖=sup_{𝜃∈[−ℎ,0]}||𝜙(𝜃)||+|𝑣|,𝜙∈𝐂,𝑣∈ℜ.(3.4) Then 𝐷Ψ(𝑡,𝜙) has a unique continuous linear extension from 𝐂⊕Γ to ℜ which we will denote by 𝐷Ψ(𝑡,𝜙), and similarly for 𝐷²Ψ(𝑡,𝜙); see [16] or [17] for more details.

Finally, we define 𝐆(Ψ)𝑡,𝜓_𝑡=lim_𝑢→0+1𝑢Ψ𝑡,𝜓^𝑡+𝑢−Ψ𝑡,𝜓_𝑡(3.5) for all 𝑡∈[0,∞) and 𝜓∈𝐂₊, where 𝜓∶[−ℎ,∞)→ℜ is defined by 𝜓(𝑡)=𝜓𝜓((𝑡0))iiff𝑡𝑡∈≥0[−.ℎ,0)(3.6) Let 𝑓∶𝐂→ℜ. We say that 𝑓∈𝐶¹(𝐂) if 𝑓 has a continuous Fréchet derivative. Similarly, 𝑓∈𝐶^𝑛(𝐂) if 𝑓 has a continuous 𝑛th Fréchet derivative. For 𝑓∶𝑅₊×𝐂→ℜ, we say that 𝑓∈𝐶^∞,𝑛([0,∞)×𝐂) if 𝑓 is infinitely differentiable in its first variable and has a continuous 𝑛th partial derivative in its second variable.

Proposition 3.1. Let 𝜑∈𝐂 and 𝑓∶𝐂→ℜ with 𝑓∈𝐶²(𝐂). Define Ψ∶[0,𝑇]×𝐂→ℜ by Ψ(𝑡,𝜑)=𝑒^{−𝑟(𝑇−𝑡)}𝐄𝑓𝑆_𝑇∣𝑆_𝑡=𝜑.(3.7) Then Ψ∈𝐶^∞,2([0,∞)×𝐂).

Proof. That 𝑒^{−𝑟(𝑇−𝑡)} is 𝐶^∞[0,∞) is clear, so we have only to show that Υ∈𝐶²(𝐂), where Υ(𝜑)=𝐄[𝑓(𝑆_𝑇)∣𝑆_𝑡=𝜑] given that 𝑓∈𝐶²(𝐂).
We have that 𝑑𝑆(𝑡)=𝑟𝑆(𝑡)𝑑𝑡+𝑁𝑆_𝑡𝑑𝑊(𝑡),𝑡≥0,(3.8) with 𝑆₀=𝜓∈𝐂₊. Under Assumption 2.1 on 𝑁∶𝐂→ℜ and the properties of Υ, it can be shown that there exists 𝐻∶ℜ×ℜ×𝐂→𝐂 such that 𝑆_𝑡=𝐻(𝑡,𝑊(𝑡),𝜓). Therefore, Υ(𝜑)==𝐄1𝑓𝑆_𝑇∣𝑆_𝑡=𝜑√2𝜋^∞−∞𝑓(𝐻(𝑇−𝑡,𝑦,𝜑))𝑒^−𝑦2/2𝑑𝑦.(3.9) By Theorem 3.2, Chapter 2 of [16], 𝐻(𝑡,𝑦,⋅)∈𝐶¹(𝐂) as a function of 𝜓. By a second application of the same theorem (since 𝑓∈𝐶²(𝐂)), we have that 𝐻(𝑡,𝑦,⋅)∈𝐶²(𝐂) as a function of 𝜓. Define 𝑔∶ℜ×ℜ×𝐂→ℜ by 𝑔=𝑓∘𝐻. Since 𝑓∈𝐶²(𝐂) and 𝐻(𝑡,𝑦,⋅)∈𝐶²(𝐂) in its third variable, 𝑔(𝑡,𝑦,⋅)∈𝐶²(𝐂). Hence, for 𝜑,𝜙∈𝐂, 𝐄𝑓=𝑆_𝑇1∣𝑆_𝑡=𝜑+𝜙−𝐄𝑓𝑆_𝑇∣𝑆_𝑡=𝜑√2𝜋^∞−∞=1[𝑓(𝐻(𝑇−𝑡,𝑦,𝜑+𝜙))−𝑓(𝐻(𝑇−𝑡,𝑦,𝜑))]𝑒^−𝑦2/2𝑑𝑦√2𝜋^∞−∞𝐷𝑔(𝑇−𝑡,𝑦,𝜑)(𝜙)𝑒^−𝑦2/2𝑑𝑦+1√2𝜋^∞−∞𝑜(𝜙)𝑒^−𝑦2/2𝑑𝑦,(3.10) where 𝑜(𝜙) is a function mapping continuous functions into the reals such that 𝑜(𝜙)‖𝜙‖⟶0as‖𝜙‖⟶0.(3.11) The last integral is clearly 𝑜(𝜙) and 1√2𝜋^∞−∞𝐷𝑔(𝑇−𝑡,𝑦,𝜑)(𝜙)𝑒^−𝑦2/2𝑑𝑦(3.12) is bounded and linear in 𝜙, so this integral is the first Fréchet derivative with respect to 𝜑. Since 𝑔(𝑡,𝑦,⋅)∈𝐶²(𝐂), the process can be repeated, giving a second Fréchet derivative with respect to 𝜑 and so Υ∈𝐶²(𝐂).

Proposition 3.2. Let 𝜑∈𝐂 and 𝑓∶𝐂→ℜ. Further assume 𝑓∈𝐶²(𝐂) and let Ψ∶[0,𝑇]×𝐂→ℜ be defined by Ψ(𝑡,𝜑)=𝑒^{−𝑟(𝑇−𝑡)}𝐸𝑓𝑆_𝑇∣𝑆_𝑡=𝜑.(3.13) Then if 𝐷𝑓 and 𝐷²𝑓 are globally Lipschitz, then so is 𝐷²Ψ.

Recall from Proposition 3.1 that 𝑔∶𝑅×𝑅×𝐂→𝑅 is 𝑓∘𝐻 where 𝑆_𝑡=𝐻(𝑡,𝑊(𝑡),𝜓) with 𝑆₀=𝜓∈𝐂₊.

Proposition 3.3. Let 𝜑∈𝐂 and 𝑓∶𝐂→ℜ. Further assume 𝑓∈𝐶²(𝐂) and let Ψ∶[0,𝑇]×𝐂→ℜ be defined by Ψ(𝑡,𝜑)=𝑒^{−𝑟(𝑇−𝑡)}𝐸𝑓𝑆_𝑇∣𝑆_𝑡=𝜑.(3.14) Then if 𝑓 and 𝐆(𝑔)(𝑇−𝑡,𝑦,𝜓_𝑡) are globally bounded, then so is 𝐆(Ψ)(𝑡,𝜓_𝑡).

Proof. We have that 𝐆(Ψ)𝑡,𝜓_𝑡=lim_𝑢→0+1=lim_𝑢→0+𝑢Ψ𝑡,𝜓^𝑡+𝑢−Ψ𝑡,𝜓_𝑡1𝑢√12𝜋^∞−∞𝑔𝑇−𝑡,𝑦,𝜓_𝑡+𝑢−𝑔𝑇−𝑡,𝑦,𝜓_𝑡𝑒^−𝑦2/2=1𝑑𝑦√2𝜋^∞−∞lim_𝑢→0+1𝑢𝑔𝑇−𝑡,𝑦,𝜓^𝑡+𝑢−𝑔𝑇−𝑡,𝑦,𝜓_𝑡𝑒^−𝑦2/2≤1𝑑𝑦√2𝜋^∞−∞=𝑀<∞,𝑀𝑒^−𝑦2/2𝑑𝑦(3.15) where we used the assumption that 𝑓 and hence 𝑔 are globally bounded to move the limit inside the integral and 𝐆(𝑔)(𝑇−𝑡,𝑦,𝜓_𝑡)≤𝑀<∞.

Remark 3.4. Note that since 𝐆(Ψ)(𝜓_𝑠) is bounded for all 𝑠∈[0,𝑇], ∫^𝑡0𝐷Ψ(𝑠,𝜓_𝑠)(𝑑𝜓_𝑠) exits. Also, if 𝐷²𝑓 is bounded, ∫^𝑡0𝐷²Ψ(𝑠,𝜓_𝑠)(𝑑𝜓_𝑠,𝑑𝜓_𝑠) exits (see [18]).

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 [16]. With it, a generalized version of the classical Black-Scholes equation can be derived for when the (𝐵,𝑆)-market model is (2.7) and (2.8). The following theorem is a restatement of Theorem 3.1 in [10].

Theorem 4.1. Let Ψ(𝑡,𝜑)=𝑒^{−𝑟(𝑇−𝑡)}𝐄[𝑓(𝑆_𝑇)∣𝑆_𝑡=𝜑], where 𝑆₀=𝜓∈𝐂₊ and 𝑡∈[0,𝑇]. Let 𝑓 be a 𝐶²(𝐂) function with 𝐷𝑓 and 𝐷²𝑓 globally Lipschitz and let Λ=𝑓(𝑆_𝑇) and 𝑥=𝑋^𝜋∗(0). Finally, let 𝑓 and 𝐆(𝑔)(𝑇−𝑡,𝑦,𝜓_𝑡) be globally bounded. Then if 𝑋^𝜋∗(𝑡)=Ψ(𝑡,𝑆_𝑡) is the wealth process for the minimal (Λ,𝑥)-hedge, one has 𝑟Ψ(𝑡,𝜑)=𝜕𝜕𝑡Ψ(𝑡,𝜑)+𝐆(Ψ)𝑡,𝜑^𝑡++1𝐷Ψ(𝑡,𝜑)𝑟𝜑(0)𝟏_{0}2𝐷²Ψ(𝑡,𝜑)𝑁(𝜑)𝟏_{0},𝑁(𝜑)𝟏_{0},a.s.∀(𝑡,𝜑)∈[0,𝑇)×𝐂₊,(4.1) where Ψ(𝑇,𝜑)=𝑓(𝜑)∀𝜑∈𝐂₊,(4.2) and the trading strategy (𝜋^∗1(𝑡),𝜋^∗2(𝑡)) is defined by 𝜋𝜋^∗2(𝑡)=𝐷Ψ(𝑡,𝜑)𝟏_{0}a.s.,_∗1(𝑡)=1𝐵(𝑡)𝑋^𝜋∗(𝑡)−𝜑(0)𝜋^∗2(𝑡).(4.3) Furthermore, if (4.1) and (4.2) hold, then Ψ(𝑡,𝑆_𝑡) is the wealth process for the (Λ,𝑥)-hedge with 𝜋^∗2(𝑡)=𝐷Ψ(𝑡,𝑆_𝑡)(𝟏_{0}) and 𝜋^∗1(𝑡)=(1/𝐵(𝑡))[𝑋^𝜋∗(𝑡)−𝑆(𝑡)𝜋^∗2(𝑡)].

Proof. The theorem is a restatement of Theorem 3.1 in [10] and is therefore omitted.

Note
Equations (4.1) and (4.2) are the generalized Black-Scholes equation for the (𝐵,𝑆)-market with hereditary price structure as described by (2.7) and (2.8).

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 Ψ∶[0,𝑇]×𝐂→ℜ satisfying the following conditions:

(i)Ψ∈𝐶^1,2([0,𝑇]×𝐂),(ii)Ψ(𝑇,𝜑)=𝑓(𝜑) for all 𝜑∈𝐂,(iii)Ψ satisfies (4.1).

The sequence of approximate solutions is constructed by looking at finite-dimensional subspaces of 𝐂, 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 𝐂⊂𝐿²[−ℎ,0] where 𝐿²[−ℎ,0] is the space of all square-integrable functions on the interval [−ℎ,0]. Furthermore, 𝐂 is dense in 𝐿²[−ℎ,0]. It is well known (e.g., [19]) that even extensions of a function 𝜑 in 𝐿²[−ℎ,0] may be represented by a cosine Fourier series where ‖‖‖‖𝜑−^𝑁_𝑖=0𝑎_𝑖cos2𝜋𝑖⋅ℎ‖‖‖‖₂⟶0(5.1) as 𝑁→∞ where 𝑎₀=1ℎ^0−ℎ𝑎𝜑(𝜃)𝑑𝜃,_𝑖=2ℎ^0−ℎ𝜑(𝜃)cos2𝜋𝑖𝜃ℎ𝑑𝜃,𝑖=1,2,3,….(5.2) Here, ‖𝑓‖²²=^0−ℎ𝑓²(𝜃)𝑑𝜃(5.3) for 𝑓∈𝐿²[−ℎ,0]. If 𝜑 is Hölder-continuous, then the convergence is also point wise (see, e.g., [20]).

Throughout this section, we let 𝐿^2𝑛[−ℎ,0] be the subspace of 𝐿²[−ℎ,0] consisting of functions that can be represented as a finite Fourier series, that is, 𝜑^(𝑛)∈𝐿^2𝑛[−ℎ,0] if 𝜑^(𝑛)(𝜃)=^𝑛_𝑖=0𝑎_𝑖cos2𝜋𝑖𝜃ℎ(5.4) for all 𝜃∈[−ℎ,0].

We will see that it is convenient having a spanning set {𝑓_𝑖}^∞𝑖=0 for 𝐿^2𝑛[−ℎ,0] where 𝑓_𝑖∶[−ℎ,0]→ℜ for 𝑖=0,1,… such that 𝑓_𝑖(0)=1 for all 𝑖 and 𝑁(𝑓_𝑖)=∫^0−ℎ𝑓_𝑖(𝜃)𝑑𝜁(𝜃)=𝛿 for all 𝑖. Here, 𝛿=∫^0−ℎ𝑑𝜁(𝜃)∈ℜ. Let 𝑞∶[−ℎ,0]→ℜ be any function such that 𝑁(𝑞)=𝑞(0)≠0. For example, let 𝑞(𝜃)=1+1−𝛿−𝑑²𝑑₁𝜃+𝜃²,(5.5) where 𝑑₁=∫^0−ℎ𝜃𝑑𝜁(𝜃) and 𝑑₂=∫^0−ℎ𝜃²𝑑𝜁(𝜃). To this end, we define the following functions. Let 𝑓𝑓₀(𝜃)=1∀𝜃∈[−ℎ,0₁],𝑓(𝜃)=𝛼_1,1+𝛼_1,2𝑞(𝜃)∀𝜃∈[−ℎ,0₂],(𝜃)=𝛼_2,1𝑞(𝜃)+𝛼_2,2cos2𝜋𝜃ℎ∀𝜃∈[−ℎ,0],(5.6) and for 𝑖=3,4,…, 𝑓_𝑖(𝜃)=𝛼_𝑖,1cos2𝜋(𝑖−2)𝜃ℎ+𝛼^𝑖,2cos2𝜋(𝑖−1)𝜃ℎ∀𝜃∈[−ℎ,0].(5.7) Recall that 𝑁∶𝐿²[−ℎ,0]→ℜ is defined by 𝑁(𝜑)=^0−ℎ𝜑(𝜃)𝑑𝜁(𝜃),(5.8) and let 𝑐_𝑖=𝑁cos2𝜋𝑖⋅ℎ=^0−ℎcos2𝜋𝑖𝜃ℎ𝑑𝜁(𝜃).(5.9) Here again 𝑞∶[−ℎ,0]→ℜ is any function such that 𝑁(𝑞)=𝑞(0)≠0. For example, 𝑞 can be chosen as in (5.5). In this case, the constant 𝛼_1,2 is nonzero but otherwise arbitrary, 𝛼𝛼_1,1=1−𝛼_1,2𝑞(0),_2,1=𝛿−𝑐¹𝛼_2,2𝑞(0)1−𝑐₁,=1−𝛼_2,1𝑞(0),(5.10) and so on with 𝛼_𝑖,2=𝛿−𝑐^𝑖−2𝑐_𝑖−1−𝑐_𝑖−2𝛼_𝑖,1,=1−𝛼_𝑖,2(5.11) for 𝑖≥3.

Lemma 5.1. The set {𝑓_𝑖}^∞𝑖=0 defined in (5.6) and (5.7) forms a spanning set for 𝐿²[−ℎ,0] in the sense that ‖‖‖‖𝜑−^𝑛+1_𝑖=0𝑥_𝑖𝑓_𝑖‖‖‖‖₂⟶0(5.12) as 𝑛→∞, where the 𝑥_𝑖 are defined by 𝑥_𝑛+1=𝑎^𝑛𝛼_𝑛+1,2𝑥_𝑛,=𝑎^𝑛−1−𝑥_𝑛+1𝛼_𝑛+1,1𝛼_𝑛,2,(5.13) and continuing using 𝑥_𝑖=𝑎^𝑖−1−𝑥_𝑖+1𝛼_𝑖+1,1𝛼_𝑖,2(5.14) until 𝑥₁=−𝑥²𝛼_2,1𝛼_1,2𝑥₀,=𝑎₀−𝑥₁𝛼_1,1.(5.15) This set of functions has the properties that 𝑓_𝑖(0)=1 and 𝑁(𝑓_𝑖)=𝛿 for all 𝑖=0,1,….

Proof. For any 𝜑∈𝐿²[−ℎ,0], we can construct an even extension 𝜙∈𝐿²[−ℎ,ℎ] where 𝜙(𝜃)=𝜑(𝜃) for all 𝜃∈[−ℎ,0] and 𝜙(𝜃)=𝜑(−𝜃) for all 𝜃∈[0,ℎ]. The function 𝜙 may be represented by a Fourier series of cosine functions 𝜙(𝜃)∼^𝑁_𝑖=0𝑎_𝑖cos2𝜋𝑖𝜃ℎ,(5.16) where the “∼’’ is used to indicate that ‖‖‖‖𝜙−^𝑁_𝑖=0𝑎_𝑖cos2𝜋𝑖⋅ℎ‖‖‖‖₂⟶0(5.17) as 𝑁→∞. In what mentioned before, 𝑎₀=1ℎ^0−ℎ𝑎𝜙(𝜃)𝑑𝜃,_𝑖=2ℎ^0−ℎ𝜙(𝜃)cos2𝜋𝑖𝜃ℎ𝑑𝜃(5.18) for all 𝑖=1,2,…. For simplicity, we will replace the “∼’’ with an equality sign knowing that mean-square convergence is implied.
For the Fourier series, the basis is cos2𝜋ℎ𝑖𝜃^∞𝑖=0,(5.19) so the first term of this basis and {𝑓_𝑖}^∞𝑖=0 are the same, namely, the constant “1.’’ Clearly 𝑓₀(0)=1 and 𝑁(𝑓₀)=𝛿. The first part of this proof is to show that for all 𝑖=0,1,…, 𝑓_𝑖(0)=1 and 𝑁(𝑓_𝑖)=𝛿.
For 𝑓₁, we have that 𝑓₁(0)=𝛼_1,1+𝛼_1,2𝑞(0)=1 which implies that 𝛼_1,1=1−𝛼_1,2𝑞(0).(5.20) Also, 𝑁(𝑓₁)=𝛼_1,1𝛿+𝛼_1,2𝑁(𝑞)=𝛿. Since we do not want 𝛼_1,2=0, we require that 𝑁(𝑞)=𝑞(0).(5.21) There are no restrictions on 𝛼_1,2 other than 𝛼_1,2≠0.
For 𝑓₂, 𝛼_2,1𝑞(0)+𝛼_2,2=1 requires that 𝛼_2,2=1−𝛼_2,1𝑞(0).(5.22) Since we want 𝛼_2,1𝑁(𝑞)+𝛼_2,2𝑐₁=𝛿, then 𝛼_2,1=𝛿−𝑐¹𝑁(𝑞)−𝑞(0)𝑐₁=𝛿−𝑐¹𝑞(0)1−𝑐₁.(5.23)
The rest of the 𝑓_𝑖, that is, where 𝑖≥3, are handled alike. In order that 𝑓_𝑖(0)=1, we require that 𝛼_𝑖,1=1−𝛼_𝑖,2. To ensure that 𝑁(𝑓_𝑖)=𝛿, 𝛼_𝑖,2=𝛿−𝑐^𝑖−2𝑐_𝑖−1−𝑐_𝑖−2.(5.24) We have now shown that the sequence of functions {𝑓_𝑖}^∞𝑖=0 is such that 𝑓_𝑖(0)=1 and 𝑁(𝑓_𝑖)=𝛿 for all 𝑖=0,1,…. Now it must be shown that this sequence is a spanning set for 𝐿²[−ℎ,0]. To do this, we will compare this sequence of functions with the cosine Fourier sequence of functions.
Consider 𝜑^(𝑛)∶[−ℎ,0]→ℜ where 𝜑^(𝑛)(𝜃)=^𝑛_𝑖=0𝑎_𝑖cos2𝜋𝑖𝜃ℎ.(5.25) We would like 𝜑^(𝑛)(𝜃)=^𝑛+1_𝑖=0𝑥_𝑖𝑓_𝑖(𝜃)(5.26) for some set {𝑥_𝑖}^𝑛+1𝑖=0 of real numbers. By the Fourier expansion, 𝜑^(𝑛)(𝜃)=𝑎₀+𝑎₁cos2𝜋𝜃ℎ+⋯+𝑎^𝑛cos2𝜋𝑛𝜃ℎ.(5.27) We want {𝑥_𝑖}^𝑛+1𝑖=0 where 𝜑^(𝑛)(𝜃)=+𝑥₀𝑥+₂𝑥₁𝛼_1,1+𝛼_1,2𝑞(𝜃)+𝑥₃𝛼^2,1𝑞(𝜃)+𝛼_2,2cos2𝜋ℎ𝜃+⋯+𝛼𝑥^3,1_𝑛cos2𝜋ℎ𝜃+𝛼^3,2cos4𝜋ℎ𝜃𝛼^𝑛,1cos2𝜋(𝑛−2)𝜃ℎ+𝛼^𝑛,2cos2𝜋(𝑛−1)𝜃+𝑥_𝑛+1ℎ𝛼^𝑛+1,1cos2𝜋(𝑛−1)𝜃ℎ+𝛼^𝑛+1,2cos2𝜋𝑛𝜃=𝑥₀ℎ+cos+𝑥2₁𝛼𝜋𝜃_1,1+𝑞(𝜃)𝑥₁𝛼_1,2+𝑥₂𝛼_2,1ℎ𝑥²+⋯+cos2𝜋𝑖𝜃𝛼_2,2+𝑥₃𝛼_3,1ℎ𝑥^𝑖+1+⋯+cos2𝜋(𝑛−1)𝜃𝛼_𝑖+1,2+𝑥_𝑖+2𝛼_𝑖+2,1ℎ𝑥^𝑛+cos2𝜋𝑛𝜃𝛼_𝑛,2+𝑥_𝑛+1𝛼_𝑛+1,1ℎ𝑥^𝑛+1𝛼_𝑛+1,2.(5.28) Equating the last coefficients gives 𝑥_𝑛+1=𝑎^𝑛𝛼_𝑛+1,2𝑥_𝑛,=𝑎^𝑛−1−𝑥_𝑛+1𝛼_𝑛+1,1𝛼_𝑛,2.(5.29) Continuing, 𝑥_𝑖=𝑎^𝑖−1−𝑥_𝑖+1𝛼_𝑖+1,1𝛼_𝑖,2,(5.30) and finally 𝑥₁=−𝑥²𝛼_2,1𝛼_1,2𝑥₀,=𝑎₀−𝑥₁𝛼_1,1.(5.31) Hence, with the above choice of {𝑥_𝑖}^𝑛+1𝑖=0, ^𝑛_𝑖=0𝑎_𝑖cos2𝜋𝑖𝜃ℎ=^𝑛+1_𝑖=0𝑥_𝑖𝑓_𝑖(𝜃),(5.32) and so ‖‖‖‖𝜑−^𝑛+1_𝑖=0𝑥_𝑖𝑓_𝑖‖‖‖‖₂=‖‖‖‖𝜑−^𝑛+1_𝑖=0𝑥_𝑖𝑓_𝑖+^𝑛_𝑖=0𝑎_𝑖cos2𝜋𝑖⋅ℎ−^𝑛_𝑖=0𝑎_𝑖cos2𝜋𝑖⋅ℎ‖‖‖‖₂=‖‖‖‖𝜑−^𝑛_𝑖=0𝑎_𝑖cos2𝜋𝑖⋅ℎ+^𝑛_𝑖=0𝑎_𝑖cos2𝜋𝑖⋅ℎ−^𝑛+1_𝑖=0𝑥_𝑖𝑓_𝑖‖‖‖‖₂≤‖‖‖‖𝜑−^𝑛_𝑖=0𝑎_𝑖cos2𝜋𝑖⋅ℎ‖‖‖‖₂+‖‖‖‖^𝑛_𝑖=0𝑎_𝑖cos2𝜋𝑖⋅ℎ−^𝑛+1_𝑖=0𝑥_𝑖𝑓_𝑖‖‖‖‖₂=‖‖‖‖𝜑−^𝑛_𝑖=0𝑎_𝑖cos2𝜋𝑖⋅ℎ‖‖‖‖₂⟶0(5.33) as 𝑛→∞.

To find an approximate solution to the generalized Black-Scholes equation we start by letting 𝑋^𝜋∗(𝑡)=𝐄[𝑓(𝑆_𝑇)∣𝑆_𝑡=𝜑] (from [6]) and approximating 𝜑 by 𝜑^(𝑛)=^𝑛+1_𝑖=0𝑥_𝑖𝑓_𝑖.(5.34) We define the space 𝐂_𝑛 as the set of all continuous functions that can be represented by this summation for some {𝑥_𝑖}^∞𝑖=0. Note that 𝐂_𝑛⊂𝐿^2𝑛[−ℎ,0]. Also define 𝑒_𝑛∶ℜ^𝑛+2→ℜ by 𝑒_𝑛𝑥⃗=^𝑛+1_𝑖=0𝑥_𝑖𝑓_𝑖,(5.35) so that Ψ(𝑡,𝜑^(𝑛))=Ψ(𝑡,𝑒_𝑛(𝑥)⃗). Define Ψ_𝑛∶[0,𝑇]×ℜ^𝑛+2→ℜ by Ψ_𝑛(𝑡,𝑥)⃗=Ψ(𝑡,𝜑^(𝑛)) provided that the 𝑥⃗ is formed by the coefficients of 𝜑^(𝑛) in the spanning set {𝑓_𝑖}^∞𝑖=0. In general, 𝑥(⃗𝑡) is formed by the coefficients of 𝜑^𝑡(𝑛) in the spanning set {𝑓_𝑖}^∞𝑖=0. Also, define 𝑣_𝑛∶[−ℎ,0]→ℜ by 𝑣_𝑛(𝜃)=⎧⎪⎨⎪⎩0,for𝜃∈−ℎ,−1𝑛𝜃+1,for𝜃=−1𝑛,𝑛,0.(5.36) Last, let 𝑔_𝑛∶[0,𝑇]×𝐂_𝑛×ℜ^𝑛+1×𝐶^1,2([0,𝑇]×𝐂)→ℜ be defined by 𝑔_𝑛𝑡,𝜑^(𝑛),𝑥,⃗Ψ=𝑟^𝑛+1_𝑖=0𝑥_𝑖𝐷Ψ(𝑡,𝜑^(𝑛))𝟏_{0}−^𝑛+1_𝑖=0𝑘_𝑖𝜕𝜕𝑥_𝑖+𝛿Ψ_𝑛𝑡,𝑥₂⃗2^𝑛+1_𝑖=0𝑥_𝑖²𝐷²Ψ(𝑡,𝜑^(𝑛))𝟏_{0},𝟏_{0}−^𝑛+1_𝑖_,𝑗=0𝑘_𝑖𝑘_𝑗𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗Ψ_𝑛𝑡,𝑥⃗,(5.37) where the 𝑘_𝑖 are the coefficients of 𝑣_𝑛 using the spanning set {𝑓_𝑖}^∞𝑖=0. Finally, define the operator (⋅)_𝑛∶𝐂→𝐂_𝑛 by (𝜑)_𝑛=^𝑛+1_𝑖=0𝑥_𝑖𝑓_𝑖,(5.38) where the right-hand side is the first 𝑛+2 terms of the {𝑓_𝑖}-expansion of 𝜑.

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 𝐆(Ψ)(𝑡,𝜑_𝑡) term. The lack of this term allows approximate solutions to be found using traditional techniques.

Theorem 5.2. Let 𝑆₀=𝜓∈𝐂₊ and 𝑡∈[0,𝑇]. Let 𝑓 be a 𝐶²(𝐂) function satisfying the conditions of Theorem 4.1 and let Λ=𝑓(𝑆_𝑇). Then 𝑟Ψ𝑡,𝜑^(𝑛)=𝜕𝜕𝑡Ψ𝑡,𝜑^(𝑛)+𝐆(Ψ)𝑡,𝜑_𝑡_𝑛++1𝐷Ψ𝑡,𝜑^(𝑛)𝑟𝜑^(𝑛)(0)𝟏_{0}2𝐷²Ψ𝑡,𝜑^(𝑛)𝑁𝜑^(𝑛)𝟏_{0},𝑁𝜑^(𝑛)𝟏_{0},∀𝑡,𝜑^(𝑛)∈[0,𝑇)×𝐂_𝑛,(5.39) where Ψ𝑇,𝜑^(𝑛)=𝑓𝜑^(𝑛)∀𝜑^(𝑛)∈𝐂_𝑛(5.40) has a solution of the form 𝑉𝑡,𝜑_𝑡_𝑛𝐹𝑡,𝜑^(𝑛).(5.41) Here, 𝑉(𝑡,(𝜑_𝑡)_𝑛)=𝑤_𝑛(𝑡,0) is a solution to 𝐹𝑡,𝜑^(𝑛)𝜕𝑤^𝑛+𝐆(𝐹)𝜕𝑡𝑡,(𝑡𝜑,𝑢_𝑡)+𝐹𝑡,𝜑^𝑡_𝑛𝜕𝜕𝑤𝑢^𝑛(𝑡,𝑢)(5.42)_𝑛𝑤_𝑛(𝑡,𝑢)=0(5.43) for 𝑡∈[0,𝑇] and 𝑢∈[0,𝜖) for some 𝜖>0 and 𝑤_𝑛(𝑇,0)=1, and 𝐹∶ℜ₊×𝐂_𝑛→ℜ is a solution of 𝑟𝐹𝑡,𝜑^(𝑛)=𝜕𝜕𝑡𝐹𝑡,𝜑^(𝑛)++1𝐷𝐹𝑡,𝜑^(𝑛)𝑟𝜑^(𝑛)(0)𝟏_{0}2𝐷²𝐹𝑡,𝜑^(𝑛)𝑁𝜑^(𝑛)𝟏_{0},𝑁𝜑^(𝑛)𝟏_{0}∀𝑡,𝜑^(𝑛)∈[0,𝑇)×𝐂_𝑛,(5.44) where 𝐹𝑇,𝜑^(𝑛)=𝑓𝜑^(𝑛)∀𝜑^(𝑛)∈𝐂_𝑛,(5.45) and 𝑓 is a uniformly bounded 𝐶²(𝐂) function satisfying the conditions of Theorem 4.1.

Proof. We assume a solution of the form Ψ(𝑡,𝜑^(𝑛))=𝑉(𝑡,(𝜑_𝑡)_𝑛)𝐹(𝑡,𝜑^(𝑛)), then𝑟𝑉=𝑡,𝜕𝜑_𝑡_𝑛𝐹𝑡,𝜑^(𝑛)𝜕𝑡𝑉𝑡,𝜑^𝑡_𝑛𝐹𝑡,𝜑^(𝑛)+𝐆(𝑉𝐹)𝑡,𝜑_𝑡_𝑛++1𝐷(𝑉𝐹)𝑡,𝜑^(𝑛)𝑟𝜑^(𝑛)(0)𝟏_{0}2=𝐹𝐷𝑡,𝜑²(𝑉_(𝑛)𝐹)𝑡,𝜑^(𝑛)𝑁𝜑^(𝑛)𝟏_{0},𝑁𝜑^(𝑛)𝟏_{0}+𝐹𝑡,𝜑_(𝑛)𝜕𝜕𝑉𝑡𝑡,𝜑^𝑡_𝑛+𝑉𝑡,𝜑_𝑡_𝑛𝜕𝜕𝑡𝐹𝑡,𝜑^(𝑛)×𝐆(𝑉)𝑡,𝜑_𝑡_𝑛+𝑉𝑡,𝜑_𝑡_𝑛𝐆(𝐹)𝑡,𝜑^(𝑛)+𝑉𝑡,𝜑_𝑡_𝑛+𝐷1(𝐹)𝑡,𝜑^(𝑛)𝑟𝜑^(𝑛)(0)𝟏_{0}2𝐷²(𝐹)𝑡,𝜑^(𝑛)𝑁𝜑^(𝑛)𝟏_{0},𝑁𝜑^(𝑛)𝟏_{0},∀𝑡,𝜑^(𝑛)∈[0,𝑇)×𝐂_𝑛.(5.46) If 𝐹(𝑡,𝜑^(𝑛)) is the solution to (5.44), then 𝐹𝑡,𝜑^(𝑛)𝜕𝑉𝜕𝑡𝑡,𝜑^𝑡_𝑛+𝐹𝑡,𝜑_𝑡_𝑛𝐆(𝑉)𝑡,𝜑_𝑡_𝑛+𝐆(𝐹)𝑡,𝜑_𝑡_𝑛𝑉𝑡,𝜑_𝑡_𝑛=0.(5.47) Define 𝑇_𝑢∶𝐂→𝐂 by 𝑇_𝑢(𝜑)=𝜑_𝑢, that is, 𝑇_𝑢 is a shift operator. Now let 𝑉(𝑡,(𝜑_𝑡+𝑢)_𝑛)=𝑉(𝑡,(𝑇_𝑢(𝜑_𝑡))_𝑛)=𝑤_𝑛(𝑡,𝑢) for a fixed 𝜑∈𝐂. Then 𝐆(𝑉)𝑡,𝜑_𝑡_𝑛=𝜕𝑤^𝑛𝜕𝑢𝑡,0⁺,(5.48) where the superscript + denotes a right-hand derivative with respect to 𝑢. Then 𝐹𝑡,𝜑^(𝑛)𝜕𝑤^𝑛𝜕𝑡(𝑡,0)+𝐹𝑡,𝜑^𝑡_𝑛𝜕𝜕𝑤𝑢^𝑛𝑡,0⁺+𝐆(𝐹)𝑡,𝜑_𝑡_𝑛𝑤_𝑛(𝑡,0)=0.(5.49)
A slightly more restrictive, but more familiar form is 𝐹𝑡,𝜑^(𝑛)𝜕𝑤^𝑛𝜕𝑡(𝑡,𝑢)+𝐹𝑡,𝜑^𝑡_𝑛𝜕𝜕𝑤𝑢^𝑛(𝑡,𝑢)+𝐆(𝐹)𝑡,𝜑^𝑡_𝑛𝑤_𝑛(𝑡,𝑢)=0,(5.50) where 𝑡∈[0,ℎ] and 𝑢∈[0,𝜖) for some 𝜖>0. There is the additional requirement that 𝑤_𝑛(𝑇,0)=1 so that (5.44) holds.

Remark 5.3. It can be easily shown that 𝑆_𝑡 is 𝛼-Hölder continuous a.s. for 0<𝛼<1/2 provided that 𝑆₀ is 𝛼-Hölder continuous for the same 𝛼. Therefore, ||𝐹_𝑛𝑡,𝑥⃗−𝐹(𝑡,𝜑)||⟶0(5.51) for each 𝑡 as 𝑛→∞ where 𝐹(𝑡,𝜑) is a solution to (5.44) and 𝐹_𝑛(𝑡,𝑥)⃗=𝐹(𝑡,𝜑^(𝑛)) is an approximate solution, since 𝐹 is 𝐶²(𝐂) in its second variable and 𝐹_𝑛𝑡,𝑥⃗=𝐹𝑡,𝑒_𝑛𝑥⃗=𝐹𝑡,𝜑^(𝑛).(5.52)

The proof of the following corollary is identical to that of Theorem 5.2, with the use of Remark 5.3 to obtain Ψ(𝑡,𝜑).

Corollary 5.4. If 𝑆₀ is Hölder continuous, then 𝑟Ψ(𝑡,𝜑)=𝜕𝜕𝑡Ψ(𝑡,𝜑)+𝐆(Ψ)𝑡,𝜑^𝑡++1𝐷Ψ(𝑡,𝜑)𝑟𝜑(0)𝟏_{0}2𝐷²Ψ(𝑡,𝜑)𝑁(𝜑)𝟏_{0},𝑁(𝜑)𝟏_{0},∀(𝑡,𝜑)∈[0,𝑇)×𝐂₊,(5.53) where Ψ(𝑇,𝜑)=𝑓(𝜑)∀𝜑∈𝐂₊(5.54) has a solution of the form 𝑉(𝑡,𝜑_𝑡)𝐹(𝑡,𝜑). Here, 𝑉(𝑡,𝜑_𝑡)=𝑤(𝑡,0) is a solution to 𝐹(𝑡,𝜑)𝜕𝑤𝜕𝑡(𝑡,𝑢)+𝐹𝑡,𝜑^𝑡𝜕𝜕𝑤𝑢(𝑡,𝑢)+𝐆(𝐹)𝑡,𝜑^𝑡𝑤(𝑡,𝑢)=0(5.55) for 𝑡∈[0,𝑇] and 𝑢∈[0,𝜖) for some 𝜖>0. 𝐹(𝑡,𝜑) is the solution to (5.44) where one lets 𝑛→∞. In addition, 𝑤(𝑇,0)=1.

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.

Theorem 5.5. Let 𝑟Ψ𝑡,𝜑^(𝑛)=𝜕𝜕𝑡Ψ𝑡,𝜑^(𝑛)++1𝐷Ψ𝑡,𝜑^(𝑛)𝑟𝜑^(𝑛)(0)𝟏_{0}2𝐷²Ψ𝑡,𝜑^(𝑛)𝑁𝜑^(𝑛)𝟏_{0},𝑁𝜑^(𝑛)𝟏_{0}∀𝑡,𝜑^(𝑛)∈[0,𝑇)×𝐂_𝑛,(5.56) where Ψ𝑇,𝜑^(𝑛)=𝑓𝜑^(𝑛)∀𝜑^(𝑛)∈𝐂_𝑛,(5.57) and 𝑓 is a uniformly bounded 𝐶²(𝐂) function satisfying the conditions of Theorem 4.1. Let 𝑓_𝑛∶ℜ^𝑛+2→ℜ be defined by 𝑓_𝑛=𝑓∘𝑒_𝑛, then Ψ_𝑛𝑡,𝑥⃗=𝑒^{−𝑟(𝑇−𝑡)}√2𝜋^∞−∞𝑓_𝑛exp𝑟𝐵−𝛿2²𝐵²(𝑇−𝑡)+𝛿𝐵𝑦√+𝑇−𝑡𝑥⃗𝑒^−𝑦2/2𝑑𝑦_𝑇𝑡𝑔_𝑛𝑠,𝜑^𝑠(𝑛),𝑥⃗(𝑠),Ψ𝑒^{−𝑟(𝑠−𝑡)}𝑑𝑠.(5.58) Here, 𝐵=⎡⎢⎢⎢⎣𝑘⁰𝑘₀⋯𝑘₀𝑘₁𝑘₁⋯𝑘₁𝑘⋮⋮⋮⋮_𝑛+1𝑘_𝑛+1⋯𝑘_𝑛+1𝑣_𝑛⎤⎥⎥⎥⎦,=^𝑛+1_𝑖=0𝑘_𝑖𝑓_𝑖(5.59) from (5.36).

Proof. Since Ψ_𝑛∶[0,𝑇]×ℜ^𝑛+2→ℜ, the definition of the Fréchet derivatives and the properties of the set {𝑓_𝑖}^∞𝑖=0 give 𝑟Ψ_𝑛𝑡,𝑥⃗=𝜕𝜕𝑡Ψ^𝑛𝑡,𝑥⃗+𝑟^𝑛+1_𝑖=0𝑥_𝑖^𝑛+1_𝑖=0𝑘_𝑖𝜕𝜕𝑥_𝑖+𝛿Ψ_𝑛𝑡,𝑥₂⃗2^𝑛+1_𝑖=0𝑥_𝑖^2𝑛+1_𝑖,𝑗=0𝑘_𝑖𝑘_𝑗𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗+𝑔_𝑛Ψ_𝑛𝑡,𝑥⃗Ψ𝑡𝑇,𝜑,𝑥⃗(^𝑠(𝑛)𝑇,𝑥),⃗Ψ=𝑓,_𝑛∀𝑡,𝑥⃗∈[0,𝑇)×ℜ^𝑛+2,𝑥⃗(𝑇).(5.60) The 𝑥(⃗𝑇) consists of the first 𝑛+2 coefficients of 𝑆_𝑇 in the set of functions {𝑓_𝑖}. By the Feynman-Kac theorem (see [15, Theorem  5.7.6]), Ψ_𝑛𝑡,𝑥⃗=𝑒^{−𝑟(𝑇−𝑡)}𝐄𝑓𝑥⃗(𝑇)∣𝑥⃗(𝑡)=𝑥⃗+^𝑇𝑡𝑔_𝑛𝑠,𝜑^𝑠(𝑛),𝑥⃗(𝑠),Ψ𝑒^{−𝑟(𝑠−𝑡)}𝑑𝑠,(5.61) where 𝑥(⃗𝑡) is the solution to 𝑑𝑥_𝑖(𝑡)=𝑏_𝑖𝑡,𝑥⃗(𝑡)𝑑𝑡+^𝑛+1_𝑗=0𝜎_𝑖𝑗𝑡,𝑥⃗(𝑡)𝑑𝑊^(𝑗)(𝑡)(5.62) for 𝑖=0,1,…,𝑛+1. Noting that 𝑥(⃗𝑡)