We consider the Black–Scholes model of financial market modified to capture the stochastic nature of volatility observed at real financial markets. For volatility driven by the Ornstein–Uhlenbeck process, we establish the existence of equivalent martingale measure in the market model. The option is priced with respect to the minimal martingale measure for the case of uncorrelated processes of volatility and asset price, and an analytic expression for the price of European call option is derived. We use the inverse Fourier transform of a characteristic function and the Gaussian property of the Ornstein–Uhlenbeck process.

One of the promising directions of enhancement of the classical Black–Scholes model is construction and research of diffusion models with volatility of risky asset governed by a stochastic process. Empirical studies [

Despite recent popularity of the stochastic volatility modification of the Black–Scholes theory, the range of models under consideration is quite narrow. One of the first models of such a type is presented in [

Nonnegativity is another desirable feature of the process modeling volatility. One of possible choices is to use the exponential function of the OU process (see [

Questions of existence of equivalent (local) martingale measures are investigated in different frameworks and different generality in [

A significant part of works (including aforementioned) use the Fourier transform to derive an analytical representation of the price of European call option. A great deal of information about developments in application of the Fourier transform to option pricing problems can be found in [

Our work investigates the market defined by a diffusion model with stochastic volatility being an arbitrary function governed by the Ornstein–Uhlenbeck process. Under general setting and quite mild assumptions, we prove that the market satisfies two distinct no-arbitrage properties for different classes of trading strategies. For the special case of uncorrelated Wiener processes, we derive an analytical expression for the price of European call option.

This paper is structured as follows: in Section

Let

Denote by

In Sections

The Wiener processes

the volatility function

the coefficients

For example, the conditions mentioned in assumption (A2) are satisfied for a measurable function

The unique solution of the Langevin equation (

Moreover, the OU process is Markov and admits the explicit representation

We can represent the process

Most of the information presented in this section can be found in more detail in [

We consider the market with one risky asset and one risk-free asset. Evolutions of prices of both assets are given by a semimartingale process

Agents acting in the market may buy or sell risky asset and make their decisions concerning the structure of their portfolios basing upon the information available at the moment of decision. This principle can be formalized by the following definition.

A trading strategy is a predictable process

Certain amount of preliminary concepts is necessary in order to introduce the essential notion of admissible self-financing strategy. Let a semimartingale

Let

A trading strategy is called admissible (relative to the price process

An admissible strategy is said to be self-financing (relative to the price process

Further, we define two particular classes of trading strategies along with the corresponding classes of

For each

Let

We denote the closures of the sets

Now following the notation presented in [

We say that the property

We say that the property

There are two theorems that establish necessary and sufficient conditions for the absence of arbitrage in the market in terms of equivalent (local) martingale measures. An important condition that will be addressed further is the local boundedness of the price process.

A probability measure

A stochastic process

([

([

The following theorem is a corollary of Proposition 6.1 from [

Denote by

Recall that there is a decomposition of a

A probability measure

A minimal martingale measure is unique (see [

In this section, we investigate the absence of arbitrage in the model (

(i) Since

Consider the process

It follows from the boundedness away from zero of the function

(ii) Now let us show that the measure

Hence, provided that assumption (A2), as mentioned before, yields the square integrability of

In order to prove

Since we have more than one equivalent martingale measure in the market, it is straightforward that the market is incomplete. Each EMM in the market is defined by the process

Under the EMM

In the risk-neutral model (

Let us define a modified set of assumptions:

(B1) The Wiener processes

(B2) = (A2);

(B3) = (A3).

Assumption (B1) simplifies the risk-neutral model to the following form:

Our purpose is to price a European call option in the model (

Suppose

Let

The process

The converse statement of the theorem comes straightforward from the uniqueness of MMM. □

The solution of the differential equation defining the evolution of the price of asset has the following representation:

For a fixed trajectory of

The value of European call option at time 0 w.r.t. the MMM is defined by the general formula

We apply the telescopic property of mathematical expectation to transform the previous expression as follows:

The inner expectation is conditional on the path of

Notice that the inner conditional expectation is an increasing function of

Taking into account the form of inner integral, in order to derive an analytic expression for the price of an option

From Eqs. (

We introduce the following deterministic functions

From (

The probabilities in the integrands may be represented as follows:

Similarly, for

Hence,

The probabilities from the integrands may be represented as follows:

Solutions of the quadratic equations, which correspond to the above quadratic inequalities, do not necessarily exist; therefore, we consider different cases:

The discriminant

The discriminant

Combining these cases, we get the following expressions for the option price:

for

for

Recalling that

Let

Assume that the probability density function of

We are now in a position to state the main result of this section.

If

Under the assumption that

Indeed, in this case,

The moments of the random variable

We have demonstrated that there is an analytic solution to the problem of pricing of European call option in the model. However, the resulting formula is complicated and cumbersome. Therefore, our further investigation will be aimed at comparison of numeric results produced by it with approximate calculations and possible simplifications.