Research On Pricing Of Volatility Index Futures Under Discrete Time Model

Volatility Index(VIX)is a market implied volatility index,which reflects investors'expectations of future market volatility.The volatility index is calculated from the option price,and it cannot be directly traded.VIX futures is the first tradable product derived from VIX.It plays a very important role in the allocation of market resources and investors' risk aversion.Based on the discrete-time model,this paper studies the pricing of VIX futures from two aspects:On the one hand,based on the GARCH model to price VIX futures,combined with high-frequency information separation to realized positive and negative semi-variance;on the other hand,based on the HAR model,directly model the logarithmic VIX information,and then derive the VIX futures pricing formula.In order to test the pricing power of these two types of VIX futures pricing models,this article introduces the classic Heston-Nandi GARCH model(HN-GARCH model)as a benchmark model,which is compared with the two pricing models proposed in this article.Firstly,based on the discrete-time GARCH model with high-frequency information,which is named RV-GARCH model,this paper decomposes the realized variance into realized positive and negative semi-variance,named RVu-RVd-GARCH model,and deduces the VIX pricing and VIX futures pricing formula under this model.The results show that when the market is in a crisis period,RV-GARCH model and RVu-RVd-GARCH model have the best performance under "VIX" method and "fut" method respectively;when the market fluctuation tends to be stable,the out of sample performance of RVu-RVd-GARCH model is unstable,which is related to the level of VIX.Secondly,this article proposes a VIX futures pricing method based on the HAR logarithmic VIX framework from a new perspective,which is recorded as the HARVIX model.The analytical expression of the VIX futures price under the model is deduced,and the maximum likelihood function is used to estimate the parameters.The empirical results show that whether it is in-sample or out-of-sample pricing,the HARVIX model outperforms the HN-GARCH model when the market is volatile.In contrast,when market volatility is relatively stable,the performance of the HARVIX model depends on the VIX level.Finally,this paper proposes a new hybrid pricing strategy(MOP),which combines the advantages of the HN-GARCH model and the RVu-RVd-GARCH model,the HN-GARCH model and the HARVIX model,and flexibly switches between different models according to the VIX level.Through the comparison of out-of-sample accuracy rates,error measures,and MCS test reports with different lookback periods,it is found that this strategy overcomes the instability of the RVu-RVd-GARCH model and HARVIX model during the plateau period and obtains more reliable VIX futures pricing results.