Pricing the European call option in the model with stochastic volatility driven by Ornstein--Uhlenbeck process. Exact formulas

摘要

We consider the Black--Scholes model of financial market modified to capturethe stochastic nature of volatility observed at real financial markets. Forvolatility driven by the Ornstein--Uhlenbeck process, we establish theexistence of equivalent martingale measure in the market model. The option ispriced with respect to the minimal martingale measure for the case ofuncorrelated processes of volatility and asset price, and an analyticexpression for the price of European call option is derived. We use the inverseFourier transform of a characteristic function and the Gaussian property of theOrnstein--Uhlenbeck process.