Option Pricing Based On HMM And GARCH Model

Option,as one kind of derivatives,can help investors with effective risk man-agement.Proposed by Black and Scholes in 1973,Black-Scholes(B-S)option pricing model has been promoted by many researchers and widely applied in the market.On one hand,researchers loose up the limitations of B-S model and improve the estimation of the model parameters.On the other hand,researchers put forward new methods such as Binary Tree Method,Monte Carlo Method and Finate Difference Method based on B-S differential equation.The development of option pricing theory has directly en-hanced the derivative market prosperity in the past decades.On the basis of former research,this paper aims to improve the accuracy of option pricing by using the hidden Markov model(HMM)incorporated into B-S option pric-ing.We first consider preprocessing the historical time series of the underlying asset by normalization and Principle Component Analysis(PCA).Considering that volatility forecast is a key issue for option pricing,we use the processed time series to train a two-state HMM.Then we apply the model to recognize the hidden states behind the observable time series,dividing the entire time series into two regimes with differ-ent volatility levels.Within each regime,we train the corresponding generalized au-toregressive conditional heteroskedasticity(GARCH)model with different parameters.With the HMM parameters and the processed time series,we can predict the hidden state of next time.Then we can use the corresponding GARCH model to forecast the volatility next time.After obtaining the volatility sequence during the life of the op-tion,we can further make the forecast on option price based on B-S model.The basis idea of the method is applying the HMM to recognize the market states and training the GARCH model for each state to avoid overestimating the volatility.At last,we consider a practical case to illustrate the solution procedure given in previous sections.We choose the unique stock index option in China at present,i.e.,50ETF call option as our illustrative example.The result shows that our method is su-perior to traditional historical volatility and GARCH methods.Our method shows good applicability on options with different types,durations and strike prices.In one-option analysis,the forecast error of our method is significantly smaller than the traditional GARCH model for most of the time during the life of the option.We also find that the closer the expiration date is,the smaller the forecast error is.Moreover,no matter the option is in the money,out of the money or at the money,the advantage of our method won't disappear.In multi-option analysis,even if the durations,the strike prices or contrast months vary,the HMM-GARCH model is still superior to the other methods in option pricing.Though we have improved the accuracy of option pricing to a certain degree,there still exist certain research questions to be polished in the future.