Modern Stochastics: Theory and Applications

In this paper, we consider the Cox–Ingersoll–Ross (CIR) process in the regime where the process does not hit zero. We construct additive and multiplicative discrete approximation schemes for the price of asset that is modeled by the CIR process and geometric CIR process. In order to construct these schemes, we take the Euler approximations of the CIR process itself but replace the increments of the Wiener process with iid bounded vanishing symmetric random variables. We introduce a “truncated” CIR process and apply it to prove the weak convergence of asset prices. We establish the fact that this “truncated” process does not hit zero under the same condition considered for the original nontruncated process.

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.

European call option issued on a bond governed by a modified geometric Ornstein-Uhlenbeck process, is investigated. Objective price of such option as a function of the mean and the variance of a geometric Ornstein-Uhlenbeck process is studied. It is proved that the “Ornstein-Uhlenbeck” market is arbitrage-free and complete. We obtain risk-neutral measure and calculate the fair price of a call option. We consider also the bond price, governed by a modified fractional geometric Ornstein-Uhlenbeck process with Hurst index $H\in (1/2,1)$. Limit behaviour of the variance of the process as $H\to 1/2$ and $H\to 1$ is studied, the monotonicity of the variance and the objective price of the option as a function of Hurst index is established.