Summary |
In the past 30 years, the progress of option pricing theory and models are dramatic, from the classical Black-Scholes (1973) formula to the time-changed Lévy process model. The calibration of model parameters has become more and more important and difficult. In the first part of the thesis, the author will extend the concept, called implied volatility smirkness, introduced by Xiang and Zhang's (2004) paper, to a quartic polynomial fitting , which can match market information quite well and provide an intuitive and simple calibration methodology. This new concept will be called as the generalized implied volatility smirkness(GIVS) and defined as a quintuplet of at-the-money implied volatility, skewness(slope at the money), smileness(curvature at the money), curlness(third order derivative at the money) and twistness(fourth order derivative at the money) of implied volatility versus moneyness curve. Since these well-defined quantities are distillate of market information, one may calibrate any existing option pricing model based on GIVS. Furthermore, one can judge the model performance according to such GIVS based calibration. Empirical study shows that many existing option pricing models with constant parameter have difficulties to match the market information well. This reveals that an option pricing formula with time dependent parameter is more appropriately implied by the market. The calibration of some most popular option pricing models will be the second part of my thesis. Although the quaritc fitting matches market information quite well, it renders complicated calibration formulae(although they can be solved numerically), and makes them barely inapplicable in finance market. In the third part of this thesis, the author will provide a non-parametric calibration methodology based on Edgeworth expansion up to the second order and derive a beautiful and simple analytical approximation formula. Monte-Carlo simulation is done to compare the price difference between this methodology and Black-Scholes model. Numerical results shows that the skewness and kurtosis of return of underlying asset have a significant effect on option prices and implied volatility smirk. |