Jumps with a stochastic jump rate: An alternative option pricing model.
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文摘
In the past two decades financial economists have proposed several alternative option-pricing models in order to explain the “smile effect” associated with the widely used Black-Scholes option pricing formula. The “smile effect” indicates the phenomenon that far out-of-money options tend to be underpriced by B-S. Models which have been proposed to explain the smile include diffusions with stochastic volatility, discontinuous processes, and combinations thereof. The dissertation contributes to this literature in two ways. First, it presents an option-pricing model that extends models incorporating stochastic volatility and jumps by allowing the jump arrival rate itself to be a mean-reverting stochastic process. Second, a different line of option pricing models have recently been proposed that replace the Brownian motion with purely discontinuous Levy processes. We compare the predictive accuracies of two such models—hyperbolic and variance gamma—with those based on mixed jump-diffusions.;The first part of the dissertation is a through review and evaluation of different option pricing models. We focus on the statistical implications of each model, its strengths and weaknesses. In the second part of the dissertation, we set up a general model with stochastic volatility and jumps that allows for a stochastic jump rate. All previous models within the Brownian motion framework are nested within this. By making the jump rate stochastic, we hope to account for the time evolution in higher moments of stock returns that cannot be explained by stochastic volatility and jumps alone. This model captures the empirical observation that there are periods in which large price moves occur with high frequency and other periods of relative calm. A computationally feasible solution for the option price in this more elaborate model obtained by means of Fourier inversion of the characteristic function.;The third part of the dissertation estimates the parameters of the risk-neutral processes corresponding to the various models using tick data for four individual stock options and options on the S&P 500 index. This is done by minimizing the deviation of market option prices from those predicted by the model. Comparison of the in-sample fit and out-of-sample prediction of different sub-models shows that stochastic jump rate factor does improve the performance of models based on Brownian motions. The degree of improvement varies according to the nature of the options.;The last part of the dissertation is devoted to the theoretical formulation and empirical testing of the hyperbolic and variance-gamma models. Computational formulas are also presented. Our in-sample and out-of-sample results show that the models based on the two Levy processes do not outperform those of comparable complexity based on Brownian motions.