Stochastic Volatility Models

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In this chapter we discuss stochastic volatility models that play a crucial role in smile modeling not only on the theoretical side, but also on the practical side. Stochastic volatility models are extensively examined by financial researchers, and widely used in investment banks and financial institutions. In many trading floors, this type of option pricing model has replaced the Black-Scholes model to be the standard pricing engine, especially for exotic derivatives. Intuitively, it is a natural way to capture the volatility smile by assuming that volatility follows a stochastic process. Simple observations propose that the stochastic process for volatility should be stationary with some possible features such as mean-reverting, correlation with stock dynamics. In this sense, a mean-reverting square root process and a mean-reverting Ornstein-Uhlenbeck process adopted from the interest rate modeling are two ideal candidate processes for stochastic volatilities. Heston (1993) specified stochastic variances with a mean-reverting square root process and derived a pioneering pricing formula for options by using CFs. Stochastic volatility model with a mean-reverting Ornstein-Uhlenbeck process is examined by many researchers. Schobel and Zhu (1999) extended Stein and Stein’s (1991) solution for a zero-correlation case to a general non-zero correlation case. In this chapter, we will expound the basic skills to search for a closed-form formula for options in a stochastic volatility model. Generally, we have two approaches available: the PDE approach and the expectation approach, both are linked via the Feynman-Kac theorem. However, as demonstrated in following, the expectation approach utilizes the techniques of stochastic calculus and is then an easier and more effective way. To provide an alternative stochastic volatility model, we also consider a model with stochastic variances specified as a mean-reverting double square root process.