State price densities are the probability densities to price financial options under the risk-neutral measures, and are particularly important for pricing and risk management. In this thesis, we compare four different methods to calibrate the state price densities using Taiwan stock index options data for years 2008 and 2012. The celebrated Black-Scholes model serves as our benchmark model. In addition, we consider its discretized version, the Cox, Ross and Rubinstein model (1979). However, to avoid model misspecification, we further study method proposed by the Derman and Kani (1994) which relaxes the assumptions on constant volatility, and a Bayesian nonparametric approach by Teng and Liechty (2009).
Our empirical results indicate that Cox, Ross and Rubinstein tree performs quite similar as the benchmark model because it is a discretized version of the Black-Scholes model. However, their pricing ability are not worse when the market changes dramatically, for example, during the 2008 financial crisis. Only Derman and Kani method perform worse during the period. Here, we use a standard interpolation to smooth the implied volatility. Derman and Kani tree does not produce good model fit, and appears to be unstable when the market changes significantly. Teng and Liechty method appears to be robust for options with all maturities. It outperforms all the other methods for options with longer time to maturities. The same conclusion holds for both 2008 and 2012.
