Stock Price Models with Stochastic Volatility

Stock price models with stochastic volatility have been developed in the last decades to improve the performance of the celebrated Black-Scholes model. The volatility of the stock in such a model is described by a nonnegative stochastic process. For instance, in the Hull-White model, a geometric Brownian motion plays the role of stochastic volatility, in the Stein-Stein model, the volatility is represented by an Ornstein-Uhlenbeck process, or by the absolute value of this process, while in the Heston model, the volatility process is the square root of a CIR-process. Chapter 2 focuses on stochastic volatility models. In addition, it presents Girsanov’s theorem, risk-neutral measures, and market prices of risk. It is explained in Chap. 2 how to use Girsanov’s theorem to find risk-neutral measures in uncorrelated stochastic volatility models, and how to overcome complications, arising in the case of a non-zero correlation between the stock price and the volatility. The chapter presents results of C. Sin, concerning risk-neutral measures in the correlated Hull-White model. Sin’s results show that the existence of such measures is determined by the possibility of explosions in finite time for solutions of certain auxiliary stochastic differential equations.