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

Authors: Mou-Hsiung Chang; Roger K. Youree
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2009)

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## 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.


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## Body

## 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, as(1.1)dB(t)=rB(t)dt,B(0)=x,dS(t)=αS(t)dt+σS(t)dW(t),S(0)=y,
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 therational 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. Section2 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. Let0<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
(2.1)C+[a,b]={ϕ∈C[a,b]∣ϕ(θ)≥0∀θ∈[a,b]}.
Note that C[a,b] is a real separable Banach space equipped with the uniform topology defined by the sup-norm ∥ϕ∥=supt∈[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
(2.2)C+={ϕ∈C∣ϕ(θ)≥0∀θ∈[-h,0]}
for simplicity. If ψ∈C[-h,∞) and t∈[0,∞), let ψt∈C be defined by ψt(θ)=ψ(t+θ), θ∈[-h,0].LetΩ=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, where(2.3)d(ω,ω′)=∑n=1∞12nsup-h≤t≤n|ω(t)-ω′(t)|1+sup-h≤t≤n|ω(t)-ω′(t)|.LetP be the Wiener measure defined on (Ω,F) with
(2.4)P{ω∈Ω∣ω(θ)=0∀θ∈[-h,0]}=1.
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 FW={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
(2.5)F(t)=σ(W(s),0≤s≤t),t≥0.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 FW defined above is right continuous in the sense of [15].Consider theC-valued process {Wt,t≥0}, where W0=0 and Wt(θ)=W(t+θ), θ∈[-h,0] for all t≥0. That is, for each t≥0, Wt(ω)=ωt and W0=0. In [10], it is shown that F0=F(t) for t∈[-h,0] and F(t)=Ft for t≥0, where
(2.6)Ft=σ(Ws,0≤s≤t),t≥0.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:
(2.7)dB(t)=L(Bt)dt,t≥0,(2.8)dS(t)=M(St)dt+N(St)dW(t),t≥0,
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])
(2.9)L(ϕ)=∫-h0ϕ(θ)dη(θ),M(ϕ)=∫-h0ϕ(θ)dξ(θ),(2.10)N(ϕ)=∫-h0ϕ(θ)dζ(θ),ϕ∈C,
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 functionsη,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 ∫-h0ϕ(θ)dζ(θ)≥σ>0 for every ϕ∈C+.We will, throughout the end, extend the domain of the above three functions toR by defining η(θ)=η(-h) for θ≤-h and η(θ)=η(0) for θ≥0, and so forth.Proposition2.3 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 processB(L;ϕ)={B(t),-h≤t<∞} of (2.7) with the initial function ϕ∈C+ takes the following form:
(2.11)B(t)=ϕ(0)ert,t≥0,
and B0=ϕ∈C+. Then the constant r satisfies the following equation:
(2.12)r=∫-h0erθdη(θ).
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 functionsϕ, 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(ϕt)=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.Atrading strategy in the (B,S)-market is a progressively measurable vector process π={(π1(t),π2(t)),0≤t<∞} defined on (Ω,F,P;FW) such that for each a>0,
(2.13)∫0aE[πi2(t)]dt<∞,i=1,2,
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.Thewriter's total asset is described by the wealth process Xπ(M)={Xπ(t),0≤t<∞} defined by
(2.14)Xπ(t)=π1(t)B(t)+π2(t)S(t),0≤t<∞,
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:
(2.15)Xπ(t)=Xπ(0)+∫0tπ1(s)dB(s)+∫0tπ2(s)dS(s),0≤t<∞,a.s.
or equivalently,
(2.16)dXπ(t)=π1(t)dB(t)+π2(t)dS(t),0≤t<∞.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 accountB(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
(2.17)W̃(t)=W(t)+∫0tγ(Bs,Ss)ds,t≥0,
where γ:C+×C+→ℜ is defined by
(2.18)γ(ϕ,ψ)=ϕ(0)M(ψ)-ψ(0)L(ϕ)ϕ(0)N(ψ).Define the processZ(L,M,N;ϕ,ψ)={Z(t),t≥0} by
(2.19)Z(t)=exp{∫0tγ(Bs,Ss)dW(s)-12∫0t|γ(Bs,Ss)|2ds},t≥0.
The following results are proven in [6].Lemma 2.3.
The processZ(L,M,N;ϕ,ψ)={Z(t),t≥0} defined by (2.19) is a martingale defined on (Ω,F,P;FW).Lemma 2.4.
There exists a unique probability measureP̃ defined on the canonical measurable space (Ω,F) such that
(2.20)P̃(A)=E[1AZ(T)]∀A∈FT,0<T<∞,
where 1A is the indicator function of A∈FT.Lemma 2.5.
The processW̃ defined by (2.17) is a standard Brownian motion defined on the filtered probability space (Ω,F,P̃;FW).From the above, it has been shown (see [6, equation (14)]) that
(2.21)dS(t)=rS(t)dt+N(St)dW̃(t),
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 P̃; that is, they have the same distribution.Define the processYπ(L,M,N;ϕ,ψ)={Yπ(t),t≥0}, called the discounted wealth process, by
(2.22)Yπ(t)=Xπ(t)B(t),t≥0.
We say that a trading strategy π from SF(L,M,N;ϕ,ψ) belongs to a subclass SFς⊂SF if P̃ a.s.
(2.23)Yπ(t)≥-Ẽ[ς∣Ft],t≥0,
where Ẽ is the expectation with respect to P̃, ς is a nonnegative F-measurable random variable such that Ẽ[ς]<∞. We say that π belongs to SF+⊂SF if ς≥0.In [6, 10], it is shown that Yπ is a local martingale; for π∈SFς, Yπ is a supermartingale, and is a nonnegative supermartingale if π∈SF+.Throughout, we assume the reward functionΛ is an FT-measurable nonnegative random variable satisfying the following condition:
(2.24)E[Λ1+ϵ]<∞,
for some ϵ>0. Here, T>0 is the expiration time. (Note that the above condition on Λ implies that Ẽ[Λ]<∞.)LetΛ be a nonnegative FT-measurable random variable satisfying (2.24). A trading strategy π∈SF is a (M;Λ,x)-hedge of European type if
(2.25)Xπ(0)=π1(0)ϕ(0)+π2(0)ψ(0)=x
and P̃ a.s.
(2.26)Xπ(T)≥Λ.
We say that a (M;Λ,x)-hedge trading strategy π*∈SF(M) is minimal if
(2.27)Xπ(T)≥Xπ*(T)
for any (M;Λ,x)-hedge strategy π∈SF(M).LetΠ(M;Λ,x) be the set of (M;Λ,x)-hedge strategies from SF+(M). Define
(2.28)C(M;Λ)=inf{x≥0:Π(M;Λ,x)≠∅}.
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 P̃ a.s. Xπ(T)≥Λ.LetY(M)={Y(t),0≤t≤T} be defined by
(2.29)Y(t)=Ẽ[ΛB(T)∣F̃t],0≤t≤T,
where F̃t=σ(W̃s,0≤s≤t). In [10], it is shown that the process Y(M) is a martingale defined on (Ω,F,P̃;FW̃) and can be represented by
(2.30)Y(t)=Y(0)+∫0tβ(s)dW̃(s),
where β={β(t),0≤t≤T} that is FW̃-adapted and ∫0Tβ2(t)dt<∞ (P a.s.).The following lemma and theorem provide the main results of [6, 10]. Let π*={(π1*(t),π2*(t)),0≤t≤T} be a trading strategy, where
(2.31)π2*(t)=β(t)B(t)N(St),π1*(t)=Y(t)-S(t)B(t)π2*(t),t∈[0,T].Lemma 2.6.
π*∈SF(M) and for each t∈[0,T], Y(t)=Yπ*(t) for each t∈[0,T] where again Yπ* is the process defined in (2.22) with the minimal  strategy π* defined in (2.31).Theorem 2.7.
LetΛ be an FT-measurable random variable defined on the filtered probability space (Ω,F,P;FW) that satisfies (2.24). Then the rational price C(M;Λ) defined in (2.28) is given by
(2.32)C(M;Λ)=Ẽ[e-rTΛ],
where r is the positive constant that satisfies (2.12). Furthermore, there exists a minimal hedge π*={(π1*(t),π2*(t)),0≤t≤T}, where
(2.33)π2*(t)=β(t)B(t)N(St),π1*(t)=Yπ*(t)-π2*(t)S(t)B(t),
and the process β={β(t),0≤t≤T} is given by (2.30).
If in addition, the rewardΛ 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
(2.34)C(Λ)=Ẽ[e-rTΛ].


## 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 FT-measurable reward function Λ has the explicit expression Λ=f(ST), where again ST(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:
(3.1)dS(t)=rS(t)dt+N(St)dW̃(t),t≥0,
where S0=ψ∈C+. Throughout this section, we assume that S(t), and therefore N(St), 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. LetC* be the space of bounded linear functionals Φ:C→ℜ. C* is a real separable Banach space under the supremum operator norm
(3.2)∥Φ∥=supϕ≠0|Φ(ϕ)|∥ϕ∥.
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).LetΓ 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
(3.3)1{0}(θ)={0,forθ∈[-h,0),1,forθ=0.
Form the direct sum C⊕Γ and equip it with the complete norm
(3.4)∥ϕ+v1{0}∥=supθ∈[-h,0]|ϕ(θ)|+|v|,ϕ∈C,v∈ℜ.
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(3.5)G(Ψ)(t,ψ̃t)=limu→0+1u[Ψ(t,ψ̃t+u)-Ψ(t,ψ̃t)]
for all t∈[0,∞) and ψ∈C+, where ψ̃:[-h,∞)→ℜ is defined by
(3.6)ψ̃(t)={ψ(t)ift∈[-h,0)ψ(0)ift≥0.
Let f:C→ℜ. We say that f∈C1(C) if f has a continuous Fréchet derivative. Similarly, f∈Cn(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.
Letφ∈C and f:C→ℜ with f∈C2(C). Define Ψ:[0,T]×C→ℜ by
(3.7)Ψ(t,φ)=e-r(T-t)Ẽ[f(ST)∣St=φ].
Then Ψ∈C∞,2([0,∞)×C).Proof.
Thate-r(T-t) is C∞[0,∞) is clear, so we have only to show that Υ∈C2(C), where Υ(φ)=Ẽ[f(ST)∣St=φ] given that f∈C2(C).
We have that(3.8)dS(t)=rS(t)dt+N(St)dW̃(t),t≥0,
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 St=H(t,W̃(t),ψ). Therefore,
(3.9)Υ(φ)=Ẽ[f(ST)∣St=φ]=12π∫-∞∞f(H(T-t,y,φ))e-y2/2dy.
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,
(3.10)Ẽ[f(ST)∣St=φ+ϕ]-Ẽ[f(ST)∣St=φ]=12π∫-∞∞[f(H(T-t,y,φ+ϕ))-f(H(T-t,y,φ))]e-y2/2dy=12π∫-∞∞Dg(T-t,y,φ)(ϕ)e-y2/2dy+12π∫-∞∞o(ϕ)e-y2/2dy,
where o(ϕ) is a function mapping continuous functions into the reals such that
(3.11)o(ϕ)∥ϕ∥→0as∥ϕ∥→0.
The last integral is clearly o(ϕ) and
(3.12)12π∫-∞∞Dg(T-t,y,φ)(ϕ)e-y2/2dy
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.
Letφ∈C and f:C→ℜ. Further assume f∈C2(C) and let Ψ:[0,T]×C→ℜ be defined by(3.13)Ψ(t,φ)=e-r(T-t)Ẽ[f(ST)∣St=φ].
Then if Df and D2f are globally Lipschitz, then so is D2Ψ.Recall from Proposition3.1 that g:R×R×C→R is f∘H where St=H(t,W̃(t),ψ) with S0=ψ∈C+.Proposition 3.3.
Letφ∈C and f:C→ℜ. Further assume f∈C2(C) and let Ψ:[0,T]×C→ℜ be defined by
(3.14)Ψ(t,φ)=e-r(T-t)Ẽ[f(ST)∣St=φ].
Then if f and G(g)(T-t,y,ψ̃t) are globally bounded, then so is G(Ψ)(t,ψ̃t).Proof.
We have that(3.15)G(Ψ)(t,ψ̃t)=limu→0+1u[Ψ(t,ψ̃t+u)-Ψ(t,ψ̃t)]=limu→0+1u12π∫-∞∞[g(T-t,y,ψ̃t+u)-g(T-t,y,ψ̃t)]e-y2/2dy=12π∫-∞∞limu→0+1u[g(T-t,y,ψ̃t+u)-g(T-t,y,ψ̃t)]e-y2/2dy≤12π∫-∞∞Me-y2/2dy=M<∞,
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,ψ̃t)≤M<∞.Remark 3.4.
Note that sinceG(Ψ)(ψ̃s) is bounded for all s∈[0,T], ∫0tDΨ(s,ψ̃s)(dψ̃s) exits. Also, if D2f is bounded, ∫0tD2Ψ(s,ψ̃s)(dψ̃s,dψ̃s) 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.
LetΨ(t,φ)=e-r(T-t)Ẽ[f(ST)∣St=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(ST) and x=Xπ*(0). Finally, let f and G(g)(T-t,y,ψ̃t) be globally bounded. Then if Xπ*(t)=Ψ(t,St) is the wealth process for the minimal (Λ,x)-hedge, one has
(4.1)rΨ(t,φ)=∂∂tΨ(t,φ)+G(Ψ)(t,φ̃t)+DΨ(t,φ)¯(rφ(0)1{0})+12D2Ψ(t,φ)¯(N(φ)1{0},N(φ)1{0}),a.s.∀(t,φ)∈[0,T)×C+,
where
(4.2)Ψ(T,φ)=f(φ)∀φ∈C+,
and the trading strategy (π1*(t),π2*(t)) is defined by
(4.3)π2*(t)=DΨ(t,φ)¯(1{0})a.s.,π1*(t)=1B(t)[Xπ*(t)-φ(0)π2*(t)].
Furthermore, if (4.1) and (4.2) hold, then Ψ(t,St) is the wealth process for the (Λ,x)-hedge with π2*(t)=DΨ(t,St)¯(1{0}) and π1*(t)=(1/B(t))[Xπ*(t)-S(t)π2*(t)].Proof.
The theorem is a restatement of Theorem3.1 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∈C1,2([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 ofC, 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 thatC⊂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
(5.1)∥φ-∑i=0Naicos(2πi·h)∥2→0
as N→∞ where
(5.2)a0=1h∫-h0φ(θ)dθ,ai=2h∫-h0φ(θ)cos(2πiθh)dθ,i=1,2,3,….
Here,
(5.3)∥f∥22=∫-h0f2(θ)dθ
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 letLn2[-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)∈Ln2[-h,0] if
(5.4)φ(n)(θ)=∑i=0naicos(2πiθh)
for all θ∈[-h,0].We will see that it is convenient having a spanning set{fi}i=0∞ for Ln2[-h,0] where fi:[-h,0]→ℜ for i=0,1,… such that fi(0)=1 for all i and N(fi)=∫-h0fi(θ)dζ(θ)=δ for all i. Here, δ=∫-h0dζ(θ)∈ℜ. Let q:[-h,0]→ℜ be any function such that N(q)=q(0)≠0. For example, let
(5.5)q(θ)=1+1-δ-d2d1θ+θ2,
where d1=∫-h0θdζ(θ) and d2=∫-h0θ2dζ(θ). To this end, we define the following functions. Let
(5.6)f0(θ)=1∀θ∈[-h,0],f1(θ)=α1,1+α1,2q(θ)∀θ∈[-h,0],f2(θ)=α2,1q(θ)+α2,2cos(2πθh)∀θ∈[-h,0],
and for i=3,4,…,
(5.7)fi(θ)=αi,1cos(2π(i-2)θh)+αi,2cos(2π(i-1)θh)∀θ∈[-h,0].
Recall that N:L2[-h,0]→ℜ is defined by
(5.8)N(φ)=∫-h0φ(θ)dζ(θ),
and let
(5.9)ci=N(cos(2πi·h))=∫-h0cos(2πiθh)dζ(θ).
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 α1,2 is nonzero but otherwise arbitrary,
(5.10)α1,1=1-α1,2q(0),α2,1=δ-c1q(0)(1-c1),α2,2=1-α2,1q(0),
and so on with
(5.11)αi,2=δ-ci-2ci-1-ci-2,αi,1=1-αi,2
for i≥3.Lemma 5.1.
The set{fi}i=0∞ defined in (5.6) and (5.7) forms a spanning set for L2[-h,0] in the sense that
(5.12)∥φ-∑i=0n+1xifi∥2→0
as n→∞, where the xi are defined by
(5.13)xn+1=anαn+1,2,xn=an-1-xn+1αn+1,1αn,2,
and continuing using
(5.14)xi=ai-1-xi+1αi+1,1αi,2
until
(5.15)x1=-x2α2,1α1,2,x0=a0-x1α1,1.
This set of functions has the properties that fi(0)=1 and N(fi)=δ for all i=0,1,….Proof.
For anyφ∈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
(5.16)ϕ(θ)~∑i=0Naicos(2πiθh),
where the “~’’ is used to indicate that
(5.17)∥ϕ-∑i=0Naicos(2πi·h)∥2→0
as N→∞. In what mentioned before,
(5.18)a0=1h∫-h0ϕ(θ)dθ,ai=2h∫-h0ϕ(θ)cos(2πiθh)dθ
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(5.19){cos(2πiθh)}i=0∞,
so the first term of this basis and {fi}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,…,  fi(0)=1 and N(fi)=δ.
Forf1, we have that f1(0)=α1,1+α1,2q(0)=1 which implies that
(5.20)α1,1=1-α1,2q(0).
Also, N(f1)=α1,1δ+α1,2N(q)=δ. Since we do not want α1,2=0, we require that
(5.21)N(q)=q(0).
There are no restrictions on α1,2 other than α1,2≠0.
Forf2, α2,1q(0)+α2,2=1 requires that
(5.22)α2,2=1-α2,1q(0).
Since we want α2,1N(q)+α2,2c1=δ, then
(5.23)α2,1=δ-c1N(q)-q(0)c1=δ-c1q(0)(1-c1).
The rest of thefi, that is, where i≥3, are handled alike. In order that fi(0)=1, we require that αi,1=1-αi,2. To ensure that N(fi)=δ,
(5.24)αi,2=δ-ci-2ci-1-ci-2.
We have now shown that the sequence of functions {fi}i=0∞ is such that fi(0)=1 and N(fi)=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.
Considerφ(n):[-h,0]→ℜ where
(5.25)φ(n)(θ)=∑i=0naicos(2πiθh).
We would like
(5.26)φ(n)(θ)=∑i=0n+1xifi(θ)
for some set {xi}i=0n+1 of real numbers. By the Fourier expansion,
(5.27)φ(n)(θ)=a0+a1cos(2πθh)+⋯+ancos(2πnθh).
We want {xi}i=0n+1 where
(5.28)φ(n)(θ)=x0+x1(α1,1+α1,2q(θ))+x2(α2,1q(θ)+α2,2cos(2πθh))+x3(α3,1cos(2πθh)+α3,2cos(4πθh))+⋯+xn(αn,1cos(2π(n-2)θh)+αn,2cos(2π(n-1)θh))+xn+1(αn+1,1cos(2π(n-1)θh)+αn+1,2cos(2πnθh))=(x0+x1α1,1)+q(θ)(x1α1,2+x2α2,1)+cos(2πθh)(x2α2,2+x3α3,1)+⋯+cos(2πiθh)(xi+1αi+1,2+xi+2αi+2,1)+⋯+cos(2π(n-1)θh)(xnαn,2+xn+1αn+1,1)+cos(2πnθh)(xn+1αn+1,2).
Equating the last coefficients gives
(5.29)xn+1=anαn+1,2,xn=an-1-xn+1αn+1,1αn,2.
Continuing,
(5.30)xi=ai-1-xi+1αi+1,1αi,2,
and finally
(5.31)x1=-x2α2,1α1,2,x0=a0-x1α1,1.
Hence, with the above choice of {xi}i=0n+1,
(5.32)∑i=0naicos(2πiθh)=∑i=0n+1xifi(θ),
and so
(5.33)∥φ-∑i=0n+1xifi∥2=∥φ-∑i=0n+1xifi+∑i=0naicos(2πi·h)-∑i=0naicos(2πi·h)∥2=∥φ-∑i=0naicos(2πi·h)+∑i=0naicos(2πi·h)-∑i=0n+1xifi∥2≤∥φ-∑i=0naicos(2πi·h)∥2+∥∑i=0naicos(2πi·h)-∑i=0n+1xifi∥2=∥φ-∑i=0naicos(2πi·h)∥2→0
as n→∞.To find an approximate solution to the generalized Black-Scholes equation we start by lettingXπ*(t)=Ẽ[f(ST)∣St=φ] (from [6]) and approximating φ by
(5.34)φ(n)=∑i=0n+1xifi.
We define the space Cn as the set of all continuous functions that can be represented by this summation for some {xi}i=0∞. Note that Cn⊂Ln2[-h,0]. Also define en:ℜn+2→ℜ by
(5.35)en(x⃗)=∑i=0n+1xifi,
so that Ψ(t,φ(n))=Ψ(t,en(x⃗)). Define Ψn:[0,T]×ℜn+2→ℜ by Ψn(t,x⃗)=Ψ(t,φ(n)) provided that the x⃗ is formed by the coefficients of φ(n) in the spanning set {fi}i=0∞. In general, x⃗(t) is formed by the coefficients of φt(n) in the spanning set {fi}i=0∞. Also, define vn:[-h,0]→ℜ by
(5.36)vn(θ)={0,forθ∈[-h,-1n),nθ+1,forθ=[-1n,0].
Last, let gn:[0,T]×Cn×ℜn+1×C1,2([0,T]×C)→ℜ be defined by
(5.37)gn(t,φ(n),x⃗,Ψ)=r(∑i=0n+1xi)[DΨ(t,φ(n))¯(1{0})-∑i=0n+1ki∂∂xiΨn(t,x⃗)]+δ22(∑i=0n+1xi)2[D2Ψ(t,φ(n))¯(1{0},1{0})-∑i,j=0n+1kikj∂2∂xi∂xjΨn(t,x⃗)],
where the ki are the coefficients of vn using the spanning set {fi}i=0∞. Finally, define the operator (·)n:C→Cn by
(5.38)(φ)n=∑i=0n+1xifi,
where the right-hand side is the first n+2 terms of the {fi}-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 theG(Ψ)(t,φ̃t) term. The lack of this term allows approximate solutions to be found using traditional techniques.Theorem 5.2.
LetS0=ψ∈C+ and t∈[0,T]. Let f be a C2(C) function satisfying the conditions of Theorem 4.1 and let Λ=f(ST). Then
(5.39)rΨ(t,φ(n))=∂∂tΨ(t,φ(n))+G(Ψ)(t,(φ̃t)n)+DΨ(t,φ(n))¯(rφ(n)(0)1{0})+12D2Ψ(t,φ(n))¯(N(φ(n))1{0},N(φ(n))1{0}),∀(t,φ(n))∈[0,T)×Cn,
where
(5.40)Ψ(T,φ(n))=f(φ(n))∀φ(n)∈Cn
has a solution of the form
(5.41)V(t,(φ̃t)n)F(t,φ(n)).
Here, V(t,(φ̃t)n)=wn(t,0) is a solution to
(5.42)F(t,φ(n))∂wn∂t(t,u)+F(t,(φ̃t)n)∂wn∂u(t,u)(5.43)+G(F)(t,(φ̃t)n)wn(t,u)=0
for t∈[0,T] and u∈[0,ϵ) for some ϵ>0 and wn(T,0)=1, and F:ℜ+×Cn→ℜ is a solution of
(5.44)rF(t,φ(n))=∂∂tF(t,φ(n))+DF(t,φ(n))¯(rφ(n)(0)1{0})+12D2F(t,φ(n))¯(N(φ(n))1{0},N(φ(n))1{0})∀(t,φ(n))∈[0,T)×Cn,
where
(5.45)F(T,φ(n))=f(φ(n))∀φ(n)∈Cn,
and f is a uniformly bounded C2(C) function satisfying the conditions of Theorem 4.1.Proof.
We assume a solution of the formΨ(t,φ(n))=V(t,(φ̃t)n)F(t,φ(n)), then(5.46)rV(t,(φ̃t)n)F(t,φ(n))=∂∂t(V(t,(φ̃t)n)F(t,φ(n)))+G(VF)(t,(φ̃t)n)+D(VF)(t,φ(n))¯(rφ(n)(0)1{0})+12D2(VF)(t,φ(n))¯(N(φ(n))1{0},N(φ(n))1{0})=F(t,φ(n))∂V∂t(t,(φ̃t)n)+V(t,(φ̃t)n)∂∂tF(t,φ(n))+F(t,φ(n))G(V)(t,(φ̃t)n)+V(t,(φ̃t)n)G(F)(t,φ(n))+V(t,(φ̃t)n)×{12D(F)(t,φ(n))¯(rφ(n)(0)1{0})+12D2(F)(t,φ(n))¯(N(φ(n))1{0},N(φ(n))1{0})},∀(t,φ(n))∈[0,T)×Cn.
If F(t,φ(n)) is the solution to (5.44), then
(5.47)F(t,φ(n))∂V∂t(t,(φ̃t)n)+F(t,(φ̃t)n)G(V)(t,(φ̃t)n)+G(F)(t,(φ̃t)n)V(t,(φ̃t)n)=0.
Define Tu:C→C by Tu(φ)=φu, that is, Tu is a shift operator. Now let V(t,(φ̃t+u)n)=V(t,(Tu(φ̃t))n)=wn(t,u) for a fixed φ∈C. Then
(5.48)G(V)(t,(φ̃t)n)=∂wn∂u(t,0+),
where the superscript + denotes a right-hand derivative with respect to u. Then
(5.49)F(t,φ(n))∂wn∂t(t,0)+F(t,(φ̃t)n)∂wn∂u(t,0+)+G(F)(t,(φ̃t)n)wn(t,0)=0.
A slightly more restrictive, but more familiar form is(5.50)F(t,φ(n))∂wn∂t(t,u)+F(t,(φ̃t)n)∂wn∂u(t,u)+G(F)(t,(φ̃t)n)wn(t,u)=0,
where t∈[0,h] and u∈[0,ϵ) for some ϵ>0. There is the additional requirement that wn(T,0)=1 so that (5.44) holds.Remark 5.3.
It can be easily shown thatSt is α-Hölder continuous a.s. for 0<α<1/2 provided that S0 is α-Hölder continuous for the same α. Therefore,
(5.51)|Fn(t,x⃗)-F(t,φ)|→0
for each t as n→∞ where F(t,φ) is a solution to (5.44) and Fn(t,x⃗)=F(t,φ(n)) is an approximate solution, since F is C2(C) in its second variable and
(5.52)Fn(t,x⃗)=F(t,en(x⃗))=F(t,φ(n)).The proof of the following corollary is identical to that of Theorem5.2, with the use of Remark 5.3 to obtain Ψ(t,φ).Corollary 5.4.
IfS0 is Hölder continuous, then
(5.53)rΨ(t,φ)=∂∂tΨ(t,φ)+G(Ψ)(t,φ̃t)+DΨ(t,φ)¯(rφ(0)1{0})+12D2Ψ(t,φ)¯(N(φ)1{0},N(φ)1{0}),∀(t,φ)∈[0,T)×C+,
where
(5.54)Ψ(T,φ)=f(φ)∀φ∈C+
has a solution of the form V(t,φ̃t)F(t,φ). Here, V(t,φ̃t)=w(t,0) is a solution to
(5.55)F(t,φ)∂w∂t(t,u)+F(t,φ̃t)∂w∂u(t,u)+G(F)(t,φ̃t)w(t,u)=0
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(5.56)rΨ(t,φ(n))=∂∂tΨ(t,φ(n))+DΨ(t,φ(n))¯(rφ(n)(0)1{0})+12D2Ψ(t,φ(n))¯(N(φ(n))1{0},N(φ(n))1{0})∀(t,φ(n))∈[0,T)×Cn,
where
(5.57)Ψ(T,φ(n))=f(φ(n))∀φ(n)∈Cn,
and f is a uniformly bounded C2(C) function satisfying the conditions of Theorem 4.1. Let fn:ℜn+2→ℜ be defined by fn=f∘en, then
(5.58)Ψn(t,x⃗)=e-r(T-t)2π∫-∞∞fn(exp[(rB-δ22B2)(T-t)+δByT-t]x⃗)e-y2/2dy+∫tTgn(s,φs(n),x⃗(s),Ψ)e-r(s-t)ds.
Here, (5.59)B=[k0k0⋯k0k1k1⋯k1⋮⋮⋮⋮kn+1kn+1⋯kn+1],vn=∑i=0n+1kifi
from (5.36).Proof.
SinceΨn:[0,T]×ℜn+2→ℜ, the definition of the Fréchet derivatives and the properties of the set {fi}i=0∞ give
(5.60)rΨn(t,x⃗)=∂∂tΨn(t,x⃗)+r(∑i=0n+1xi)∑i=0n+1ki∂∂xiΨn(t,x⃗)+δ22(∑i=0n+1xi)2∑i,j=0n+1kikj∂2∂xi∂xjΨn(t,x⃗)+gn(t,φs(n),x⃗,Ψ),∀(t,x⃗)∈[0,T)×ℜn+2,Ψ(T,x⃗(T))=fn(x⃗(T)).
The x⃗(T) consists of the first n+2 coefficients of ST in the set of functions {fi}. By the Feynman-Kac theorem (see [15, Theorem  5.7.6]),
(5.61)Ψn(t,x⃗)=e-r(T-t)Ẽ[f(x⃗(T))∣x⃗(t)=x⃗]+∫tTgn(s,φs(n),x⃗(s),Ψ)e-r(s-t)ds,
where x⃗(t) is the solution to
(5.62)dxi(t)=bi(t,x⃗(t))dt+∑j=0n+1σij(t,x⃗(t))dW̃(j)(t)
for i=0,1,…,n+1. Noting that x⃗(t)=x⃗,
(5.63)bi(t,x⃗)=rki(∑i=0n+1xi),δ2kikj(∑i=0n+1xi)2=∑k=0n+1σik(t,x⃗)σjk(t,x⃗)
with 0≤i,j≤n+1. Hence,
(5.64)b⃗(t,x⃗)=r[k0(∑i=0n+1xi)k1(∑i=0n+1xi)⋮kn+1(∑i=0n+1xi)],
and so
(5.65)b⃗(t,x⃗)=rB[x0x1⋮xn+1].
Also,
(5.66)σ(t,x⃗)σ⊤(t,x⃗)=δ2(∑i=0n+1xi)2[k02k0k1⋯k0kn+1k0k1k12⋯k1kn+1⋮⋮⋮⋮k0kn+1k1kn+1⋯kn+12].
Therefore,
(5.67)σ(t,x⃗)=δB[x0x1⋮xn+1]
as well. Thus,
(5.68)dx⃗(t)=rBx⃗(t)dt+δBx⃗(t)dW̃(t)
which has the solution
(5.69)x⃗(t)=exp[(rB-δ22B2)t+δBW̃(t)]x⃗(0).
Notice that b⃗(t,x⃗) and σ(t,x⃗) satisfy the conditions necessary for applying the Feynman-Kac formula. Also note that (5.58) satisfies the polynomial growth condition
(5.70)max0≤t≤T|Ψn(t,x⃗)|≤M(1+∥x⃗∥2μ)
for M>0 and μ≥1 due to the boundedness of fn and gn for n sufficiently large. Therefore, by (5.61), equation (5.58) follows.Lemma 5.6.
LetA:[0,T]×ℜ→ℜ(n+2)×(n+2) be defined by
(5.71)A(t,y)=(rB-δ22B2)(T-t)+δByT-t,
where
(5.72)B=[k0k0⋯k0k1k1⋯k1⋮⋮⋮⋮kn+1kn+1⋯kn+1].
Then A(t,y)=QD(t,y)Q-1 where Q is the (n+2)×(n+2) matrix defined by
(5.73)Q=[k0α1212⋯12k1α00⋯-12⋮⋮⋮⋮⋮kn+1α-120⋯0],Q-1=[111⋯12kn+1α2kn+1α2kn+1α⋯-2α-kn+1α2knα2knα2knα⋯2knα⋮⋮⋮⋮⋮2k1α-2α-k1α2k1α⋯2k1α],D(t,y)=[(r-δ22α)α(T-t)+δαyT-t00⋯0000⋯0⋮⋮⋮⋮⋮000⋯0].
Here, α=∑i=0n+1ki.Proof.
It is clear thatB2=αB, so
(5.74)A(t,y)=[(r-δ22α)(T-t)+δyT-t]B.
The matrix A(t,y) is an (n+2)×(n+2) matrix with n+2 eigenvalues. The n+1 eigenvectors associated with the eigenvalue 0 are
(5.75)[12-1200⋮0],[120-120⋮0],
and so forth. The remaining eigenvalue is
(5.76)[(r-δ22α)(T-t)+δyT-t]α
with the associated eigenvector of
(5.77)1α[k0k1k2⋮kn+1].
The matrix Q is simply a matrix with these eigenvectors for columns, the expression for Q-1 can easily be verified, and D(t,y)=Q-1A(t,y)Q.Next, we will derive an approximate solution to (5.44) with the auxiliary condition being that for the standard European call option. By using Theorem 5.2, a significant piece of the approximate solution is then known.Proposition 5.7.
Let(5.78)rΨ(t,φ)=∂∂tΨ(t,φ)+DΨ(t,φ)¯(rφ(0)1{0})+12D2Ψ(t,φ)¯(N(φ)1{0},N(φ)1{0}),∀(t,φ)∈[0,T)×C+,
where
(5.79)Ψ(T,φ)=(φ(0)-K)+∀φ∈C+,
and K is the strike price of the option contract. Then
(5.80)Ψn(t,x⃗)=e-r(T-t){kn+1αexp[rα(T-t)]∑i=0n+1xi(Φ(δαT-t-Y0))-KΦ(-Y0)}+∫tTgn(s,φs(n),x⃗(s),Ψ)e-r(s-t)ds,
where Ψn(t,x⃗)→Ψ(t,φ) pointwise as n→∞. Here,
(5.81)Φ(u)=12π∫-∞ue-y2/2dy,Y0=ln[αK/kn+1∑i=0n+1xi]δαT-t-(rδ-δα2)T-t.Proof.
By Theorem5.5, the equation
(5.82)rΨ(t,φ(n))=∂∂tΨ(t,φ(n))+DΨ(t,φ(n))¯(rφ(n)(0)1{0})+12D2Ψ(t,φ(n))¯(N(φ(n))1{0},N(φ(n))1{0}),∀(t,φ(n))∈[0,T)×Cn,
with
(5.83)Ψ(T,φ(n))=f(φ(n))∀φ(n)∈Cn,
and f being a C2(C) function has the solution
(5.84)Ψn(t,x⃗)=e-r(T-t)2π∫-∞∞fn(exp[(rB-δ22B2)(T-t)+δByT-t]x⃗)e-y2/2dy+∫tTgn(s,φs(n),x⃗(s),Ψ)e-r(s-t)ds.
By Lemma 5.6,
(5.85)A(t,y)=(rB-δ22B2)(T-t)+δByT-t
may be expressed as A(t,y)=QD(t,y)Q-1 where Q, Q-1, and D(t,y) are defined in the lemma. Since D(t,y) is diagonal, eD(t,y) is straightforward to find and eA(t,y)=QeD(t,y)Q-1. Let v⃗(t,y) be the (n+2)th row of eA(t,y), expressed as a column vector. Then
(5.86)v⃗(t,y)=kn+1αexp[(r-δ22α)α(T-t)+δαyT-t][111⋮1].
Let un+1(t,y,x⃗) be the (n+2)th component of eA(t,y)x⃗, we have that
(5.87)un+1(t,y,x⃗)=v⃗(t,y)·x⃗=kn+1α(exp[(r-δ22α)α(T-t)+δαyT-t])∑i=0n+1xi.
Therefore,
(5.88)Ψn(t,x⃗)=e-r(T-t)2π∫Y0∞[un+1(t,y,x⃗)-K]e-y2/2dy+∫tTgn(s,φs(n),x⃗(s),Ψ)e-r(s-t)ds
with
(5.89)kn+1α(exp[(r-δ22α)α(T-t)+δαY0T-t])∑i=0n+1xi=K
since [un+1(t,y,x⃗)-K]+ is C2 in x⃗ on the interval (Y0,∞), with the necessary Lipschitz properties. Solving for Y0 we have
(5.90)Y0=ln[αK/kn+1∑i=0n+1xi]δαT-t-(rδ-δα2)T-t.
So,
(5.91)Ψn(t,x⃗)=e-r(T-t)2π∫Y0∞[kn+1αexp[(r-δ22α)α(T-t)+δαyT-t]∑i=0n+1xi-K]e-y2/2dy+∫tTgn(s,φs(n),x⃗(s),Ψ)e-r(s-t)ds=e-r(T-t)2π∫Y0∞[kn+1αexp[(r-δ22α)α(T-t)+δαyT-t]∑i=0n+1xi]e-y2/2dy-Ke-r(T-t)(1-Φ(Y0))+∫tTgn(s,φs(n),x⃗(s),Ψ)e-r(s-t)ds=e-r(T-t){kn+1αexp[(r-δ22α)α(T-t)]×∑i=0n+1xi12π∫Y0∞e-y2/2+δαyT-tdy-K(1-Φ(Y0))}+∫tTgn(s,φs(n),x⃗(s),Ψ)e-r(s-t)ds=e-r(T-t){(1-Φ(Y0))(-K)+kn+1αexp[(r-δ22α)α(T-t)]×∑i=0n+1xieδ2α2(T-t)/2(1-Φ(Y0-δαT-t))}+∫tTgn(s,φs(n),x⃗(s),Ψ)e-r(s-t)ds=e-r(T-t){kn+1αexp[rα(T-t)]∑i=0n+1xi(Φ(δαT-t-Y0))-KΦ(-Y0)}+∫tTgn(s,φs(n),x⃗(s),Ψ)e-r(s-t)ds.
This result is the solution given φ(n)∈Cn. Notice that
(5.92)gn(s,φs(n),x⃗(s),Ψ)→0
pointwise as n→∞ since
(5.93)limn→∞DΨ(t,φ(n))¯(1{0})=limn→∞DΨn(t,x⃗)(vn)=limn→∞∑i=0n+1ki∂∂xiΨn(t,x⃗),
and similarly for the second derivatives. By dominated convergence, we have that
(5.94)∫tTgn(s,φs(n),x⃗(s),Ψ)e-r(s-t)ds→0
pointwise as n→∞. By Remark 5.3, we have that Ψn(t,x⃗)→Ψ(t,φ) pointwise as n→∞.


## 6. Summary and Conclusions

In this paper, we have continued [6] by deriving an infinite-dimensional Black-Scholes equation for the European option problem, where the (B,S)-market model is given by (2.7) and (2.8). The resulting deterministic partial differential equation is a new type of equation, one where the partial differentiation contains extended Fréchet derivatives. Given the (B,S)-market model equations, a spanning set for the space of square-integrable function is developed which simplifies finding an approximate solution to this equation. The solution method detail in this paper consists of the following steps.Step 1.
Givenr and N(·), use (5.6)-(5.7) with (5.5) and (5.10)-(5.11) to find the spanning set {fi}i=0n+1 for n sufficiently large. Coefficients in this spanning set of functions are found using (5.13)–(5.15).Step 2.
Use Theorem5.5 or Proposition 5.7, depending on the reward function, to find Ψn(t,x⃗). The term
(6.1)∫tTgn(s,φs(n),x⃗(s),Ψ)e-r(s-t)ds
approaches zero as n approaches infinity, so this term may be assumed small for sufficiently large n. The vector x⃗ is found from (5.69).Step 3.
Having foundΨn(t,x⃗), solve (5.43) for wn(t,0), then wn(t,0)Ψn(t,x⃗) is an approximate solution to the generalized Black-Scholes equation. By Corollary 5.4, wn(t,0)Ψn(t,x⃗)→Ψ(t,φ) pointwise as n→∞.



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*Source: 782572-2010-01-14.xml*