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This paper considers the pricing of a European
option using a

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., [

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

In [

In particular, [

This paper considers the pricing of a European option using a

This paper is organized as follows. Section

To describe the financial model with hereditary price structures, we start by defining our probability space. Let

Let

Let

Equivalently,

Consider the

The new model for the

The functions

We will, throughout the end, extend the domain of the above three functions to

Proposition

For the purpose of analyzing the discount rate for the bank account, let us assume that the solution process

Throughout the end, we will fix the initial unit price functions

In [

A

The

We will make the following basic assumption throughout this paper.

In the

Using the same notation as in [

For the unit price of the bank account

Define the process

The process

There exists a unique probability measure

The process

From the above, it has been shown (see [

Define the process

In [

Throughout, we assume the reward function

Let

Let

Let

The following lemma and theorem provide the main results of [

Let

If in addition, the reward

In this section, results are proven that allow the use of a Dynkins formula for stochastic functional differential equation as found in [

The remaining sections make extensive use of Fréchet derivatives. Let

Let

Finally, we define

Let

That

We have that

Let

Recall from Proposition

Let

We have that

Note that since

It is known (e.g., [

Let

The theorem is a restatement of Theorem

Equations (

In this section, we will solve the generalized Black-Scholes equation (

The sequence of approximate solutions is constructed by looking at finite-dimensional subspaces of

We start by noting that

Throughout this section, we let

We will see that it is convenient having a spanning set

The set

For any

For the Fourier series, the basis is

For

For

The rest of the

Consider

To find an approximate solution to the generalized Black-Scholes equation we start by letting

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

Let

We assume a solution of the form

A slightly more restrictive, but more familiar form is

It can be easily shown that

The proof of the following corollary is identical to that of Theorem

If

Now we must solve (

Let

Since

Let

It is clear that

Next, we will derive an approximate solution to (

Let

By Theorem

In this paper, we have continued [

Given

Use Theorem

Having found

In this appendix, we prove Proposition

Before proving the proposition, an additional result for Fréchet derivatives is needed.

Let

We start by considering

Since

From Proposition

By Lemma

For the next term,

Again using the operator define by

Since

We also have that