A stochastic volatility model for option pricing.

摘要

Since 1970's, stochastic volatility models in which the conditional variance follows a certain stochastic process have been greatly developed. The initial interest in this type of model dates back to the work of Clark (1973), who proposed an iid mixture model for the distribution of stock-price changes. The Hull-White model (1987), the ARCH/GARCH model, and the variance gamma (VG) model (1987) are typical successful examples in this area. This dissertation proposes another stochastic volatility model to describe asset prices, called the Poisson Distributed Variance (PDV) model, which assumes the conditional variance of underlying asset price is a Poisson process with jump size σ 2 and intensity λ. The PDV model belongs to the subordinated process family and catches such empirical results as the fat tail of asset return distributions and implied volatility smiles. The lower moments of underling asset prices can be obtained through the characteristic function. At the same time, the implied volatility smile is proven to be symmetric around a point near the “at-the-money” position. Based on risk-neutral probability, we can also derive the European call/put options pricing formulas and an approximate expression.; In order to estimate the PDV model parameters using historic asset prices, several statistical inference tools are discussed: the moment-matching method, the estimation method using characteristic function and the Bayesian method. Among them, the Bayesian estimation is most recommended because it provides the most reliable results by our findings. We form a hierarchical structure for the PDV model and construct a simulation-based algorithm using the Markov-chain Monte Carlo method. The simulated series from the Markov chains converge in distribution to draws from posterior distributions enabling exact finite-sample inference. Sampling experiments are conducted to check the performance of the Bayes estimators in this paper. The results indicate that this algorithm for the PDV model is superior to other methods.; With historical data from the S&P 500 and Russell 2000 indices, we estimate the PDV model parameters and compare its performance to the Black-Scholes model through the Chi-square goodness of fit test. Moreover, we calculate the European option prices under the PDV model and perform residual analyses with the actual option prices. In both asset return distribution fitness and options pricing, the model discussed in this paper outperforms the Black-Scholes model.