SPX Index Option Pricing With Deep Learning Under Distorted Normal Distribution

With the development of the world economy,the proportion of derivatives markets in financial markets is becoming larger and larger,and the issue of pricing of derivatives has attracted more and more scholars' attention.As one of the most important derivatives,the precise pricing of options has always been a difficult problem in the financial field.The Black-Scholes-Merton(B-S-M)option pricing model has been widely recognized in the industry since its inception,and many researchers have further studied the issue of option pricing based on it.However,more accurate option pricing models have always been the goal of scholars.In order to improve the accuracy of option pricing,this article starts from two aspects:First,the existing option pricing model does not describe the psychology of various investors' decisions,so the model can be improved by describing abnormal investment decisions;Second,the machine learning technology is applied.In recent years,machine learning has developed rapidly,and more and more machine learning methods have achieved good results in the field of prediction.Therefore,this article introduces deep learning,which plays a significant role in maching learning realm,to fit the complex relationships contained in the options market.Based on the ideas of behavioral finance and deep learning technology,this paper deduces a new option pricing model based on the B-S-M model called Probabilistic distortion model based on deep learning,and conducts empirical research with SPX index options.Existing option pricing models often have errors that cannot be ignored.This article believes that these errors may be caused by investors' abnormal investment decisions.Therefore,the new model first uses the Esscher transformation to distort the existing normal distribution without changing its normal properties.Based on the B-S-M model,a normal probability distortion option pricing model that can better explain the volatility smile is constructed.Considering the complex non-linear relationship between the distortion factor and its influencing factors,this paper applies deep learning technology to build a multilayer neural network to predict the future distortion factor.Finally,based on the SPX index options,the empirical analysis of call options and put options is carried out,and four error indicators are used.The empirical results show that:(1)The results based on the pure neural network model are better than the classic B-S-M model,which shows that the neural network fits the complex nonlinear relationship between the option price and its influencing factors.(2)Under the four pricing error indicators of MSE,RMSE,MAE and MAPE,the performance of the normal probability distortion option pricing model based on deep learning has the smallest error.(3)Even if the training set and test set are divided by the sliding window method,the option pricing accuracy of the new model is always better than the classic B-S-M model and pure neural network model.