European call option issued on a bond governed by a geometric or a fractional geometric Ornstein-Uhlenbeck process

Abstract

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.

Keywords

• Objective option price
• fair option price
• modified geometric Ornstein-Uhlenbeck process
• modified geometric fractional Ornstein-Uhlenbeck process