Abstract
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.
Highlights
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
An expression for the price of European call option is derived under the following assumption: the volatility process is driven by a Brownian motion independent of the Brownian motion governing the price of risky asset
The authors of [16] describe the distribution of the price of risky asset and apply it to derive an estimate of the price of European call option
Summary
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. An expression for the price of European call option is derived under the following assumption: the volatility process is driven by a Brownian motion independent of the Brownian motion governing the price of risky asset. The OU process is mean-reverting, and there is a strong evidence that the volatility in real financial markets has such a feature [4, 3] Under this assumption, the authors of [16] describe the distribution of the price of risky asset and apply it to derive an estimate of the price of European call option. We consider the model of the market where one risky asset is traded, its price evolves according to the geometric Brownian motion {St , 0 ≤ t ≤ T }, and its volatility is driven by a stochastic process.
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