파라미트릭 점프 프로세스를 위한 최적의 모수 추정 방법론

Abstract

According to numerous empirical evidences observed in option markets, it is clear that the celebrating Black-Scholes-Merton option pricing model can not explain the intrinsic properties of option prices in real markets such as the implied volatility smile behavior. To capture the smile effect many option pricing models or methods have been developed in a non-parametric and parametric way. In non-parametric approaches they do not rely on pre-assumed models but instead try to uncover/induce the model. There is a weak point with non-parametric approaches which it cannot applied to pricing path-dependent exotic options due to its lack of underlying dynamics.Recently in financial literature parametric methods, such as exponential L´evy models and affine jump-diffusion models, have been widely adopted as alternative models that explain stylized facts of asset returns and volatility smile effects of traded option prices. Hence if the parameters are calibrated reasonably the parametric models can be very powerful. Unfortunately the number of parameters is a lot and it’s hard to estimate parameters from the information in financial market. To calibrate them we use cross-sectional data of option prices. Least-squaresense is usually employed to calibrate them in finance, although it is well-known ill-posed inverse problem. To conquer the ill-posed inverse problem we propose a derivative-free calibration method constrained by four observable statistical moments (mean, variance, skewness and kurtosis) from underlying time series and so-called multi-basin system which consists of three sequential phases to expedite the search for agood parameter set.To verify the performance of the proposed methods, we conduct simulations on some model-generated option prices data and real-world option market data. The simulation results show that the proposed methods fit the option ranges well and calibrate the parameter set ofexponential L´evy models and affine jump-diffusion models reasonably and robustly.In this thesis we also give a modularized summary of all the detailed equations relevant to all exponential L´evy models and affine jumpdiffusion models in a consistent way by using the unified notations.