Modern Stochastics: Theory and Applications

We consider a discrete-time approximation of paths of an Ornstein–Uhlenbeck process as a mean for estimation of a price of European call option in the model of financial market with stochastic volatility. The Euler–Maruyama approximation scheme is implemented. We determine the estimates for the option price for predetermined sets of parameters. The rate of convergence of the price and an average volatility when discretization intervals tighten are determined. Discretization precision is analyzed for the case where the exact value of the price can be derived.

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.