Empirical Comparison Of Numerical Solutions Of Two SABR Option Pricing Models

The random alpha-betarrho(SABR)model first proposed by Hagan,Kumar,Lesnieewski and Woodward provides a very popular tool for modeling the implied volatility[1].However,existing analytical methods for pricing derivatives under the SABR model generally ignored the fact that the forward prices of underlying assets may hit zero,which leads to the possibility of arbitrage,In order to eliminate this arbitrage opportunity,Nian Yang et al.proposed a SABR model with a absorption boundary zero[2],The author made a closed approximation to the price of ordinary options and considered the influence of such boundary conditions,and gave the explicit solutions with second-order precision.The empirical results show that SABR model can well fit the volatility smile in the foreign exchange and interest rate options market,but its practicability in the index options market is controversial.This paper is mainly based on the SPX500 index option data of 2017,and the required model parameter values are obtained by the method of minimizing variance optimization,Then the empirical compare of the SABR model proposed by Hagen et al.and the SABR model with an absorption boundary is carried out.We classify the acquired option data according to different expiration date and moneyness,and compared the prediction effects of the two models on the call option and put option respectively from the prediction errors of implied volatility and option price.The results showed that for both the call and the put options,the error of the prediction of the option price with short maturity time and small moneyness range is small,and the error of the model's price prediction increases with the increase of the maturity time and moneyness range.However,the prediction error of the model for the option volatility does not change significantly with the change of expiration time and the moneyness.In general,the prediction effect of SABR model proposed by Hagen et al.was better than that of the SABR model with an absorption boundary regardless of the length of maturity time and different moneyness,and the prediction effect of call options of both models are better than that of put options.At the same time,we point out that although the prediction of SABR model with an absorption boundary is not as good as the method proposed by Hagen et al.,its relatively continuous peak error curve makes it possible to calculate the explicit expression of the error term of the model,which also provides an idea for the further study of this model.Finally,we verified the empirical reliability by comparing the prediction results of SABR model under different parameter estimation methods with the calculation method proposed by Hagen et al.