Research On VIX Time Series And Option Pricing Based On Mean Reverting Models With Tempered Stable Processes

With the continuous progress of economic globalization and integration,the financial markets in various countries have shown a strong linkage,which makes the measurement of systemic risk in financial markets more complicated,in addition,hedging systemic risks have become an issue for investors.In order to avoid systemic risks and find indicators that characterize systemic risks,scholars first developed derivatives based on the S&P100 index and the S&P500 index.However,it is only the current market rather than the future market expected by investors that could be reflected by the S&P100 index and S&P500 index.Thus,the VIX developed by scholars which based on S&P100 index and S&P500 index derivatives is not only contains information about the current market,but also reflects relevant information of investors expectation about the future.Therefore,as a market panic index,VIX could not only reflect the market fluctuations,but also could deeply reflect the degree of investors’ avoidance of systemic risks.Furthermore,the options based on VIX can be used not only as a derivative,but also as a tool for avoiding systemic risks.What mentioned above show that with the development of economy,VIX and its options play an increasingly important role in the financial market.Because VIX is calculated based on the S&P500 index and its derivatives,many studies about VIX and its options are researched abroad,and with rich results.According to the existing research,the following deficiency of the research on VIX and its options could be found:(1)In terms of model construction,in order to fit the characteristics of VIX spikes and thick tails,bias,mean reverting,and volatility clustering,the most studies have only built a mean reverting model based on simple jumps,or introduce a random volatility model based on this model.In addition,these articles give neither economic explanation with jumps,nor comparison of the advantages and disadvantages of the process with Lévy jumps and models with random volatility,that is,check whether the Lévy jump process could replace the stochastic volatility model.(2)As for stochastic simulation,there are many research results about the random simulation of the model,and most of them are generated separately by the jump term and the fluctuation term.However,if the current increment is related to the current state,the infinite separability of the Lévy process will fail.Thus,this method of generation will produce large errors.(3)In terms of option pricing,VIX is calculated based on the S&P500 index and its derivatives,and the S&P500 index is based on S&P500 stocks.Therefore,VIX is not a type of underlying assetss and cannot be directly used for trading.However,in some existing studies,VIX is regarded as a underlying assetss for arbitrage-free pricing.In order to obtain the martingale property under the risk-neutral measure,there are two following methods can be used: the one is to directly make it a Doob martingale under the risk-neutral measure;the other is to deduce the VIX model according to the S&P500 model under the risk-neutral measure,and add restrictions to make it become a martingale.Both of these methods affected the effect of the model on VIX option pricing.(4)In terms of the mean reverting model,there are many achievements in studying VIX and its options based on the stochastic mean model,but only the stochastic mean,and there is no discussion on the mean reversibility in combination with the economic background,nor a comparative analysis with other models.For the deficiencies in the above studies,the following researches are made in this paper:(1)In terms of model construction,in order to characterize VIX features,such as peaks and thick tails,bias,mean reverting,and volatility clustering,derived from the perspective of calendar time and intrinsic time,the tempered stable process is introduced into the mean reverting model,and the parameters in the model are randomized by applying the conditional expectation and heavy expectation formulas,and then the VIX and its option pricing analysis are performed based on the model.(2)In the aspect of stochastic simulation,the existing stochastic simulation method is improved by the method of inversely generating a sample of the density function,so that it can not only generate samples with density function perfectly,but also generate measures other than 1 and whose characteristic function has no explicit expression.(3)In terms of option pricing,under the premise that the underlying assets of VIX is S&P500 stock,removing the discount process of VIX under a risk-neutral measure is a constrain of martingale property.The option pricing model is constructed directly under the risk-neutral measure.(4)In terms of mean response model,considering the investor ’s cognitive sentiment to the financial market and its reflection in VIX,a mean reverting sentiment model is constructed.This interval mean reverting model is not only reflected the fact that the mean is a random value,but also that the mean reverting is determined based on the current value of VIX or the estimated value of its option expiration date.The research in this paper not only improve the existing studies about VIX and its options,but also enrich the existing time series analysis and option pricing theory,and provide some references for investors and system risk managers.According to the research,the following conclusions are drawn:(1)For a model,the more complex the better is not absolute.The CIR model with the classic tempered stable model(CIRCTS)is significantly better than other models in the pricing of VIX options.Considering the investor ’s response to the VIX mean reverting,the extended CIRCTST model based on the CIRCTS model performs best in the VIX time series analysis,while the CIRCTSP model has the best pricing among all option models.(2)When the stochastic differential equation is related to the current situation,the method of generating the fluctuation term and the jumping term cannot be used.The overall inverse generation method of the characteristic function can be used for effective random simulation.(3)Based on the premise that the underlying assets of VIX is S&P500 stocks,the pricing effect of building a modular option pricing model directly under a risk-neutral measure is better than other models.If the discount factor in the option pricing formula satisfies a certain relationship,in addition to the pre-judgment mean reverting model in Chapter 6,underlying asset pricing cannot be applied,other models can be applied to non-underlying asset pricing,but also to satisfy the martingale Apply option pricing for underlying assets.Based on the assumptions of non-underlying asset pricing,the conclusions obtained are more intuitive,and the formula for the discount factor is not so complicated.(4)The distribution of the long time series of VIX is different from the distribution of short time series of VIX options.The CIRCTST model works best when fitting the long time series of VIX,whilst in the short time sequential open analysis of VIX option pricing,the CIRCTSP model performs best,which means that VIX options cannot be priced with long time series correction parameters.The main innovations of this paper are mainly show in the following aspects:(1)In terms of model construction,from the perspective of calendar time and intrinsic time,a mean reverting model based on tempered stable state is constructed,at the same time,at the same time,this research use the conditional expectation and heavy expectation formula to randomize the important parameters of this kind of model,extend the model,and apply the model to the analysis of VIX time series as well as option pricing,which enriches the model of VIX time series and option pricing.Furthermore,it is found that when pricing VIX options,the pricing of CIR model which based on tempered stable state is better than it of the OU model with tempered stable state and random volatility,which means that the application of the tempered stable process in VIX option pricing can better simulate the phenomenon of "volatility smile" in the pricing of options.(2)In the aspect of stochastic simulation,when the fluctuation term is related to the current state,the method of separately generating the fluctuation term and the jump term cannot be applied.This paper proposes a method of inverse generation based on the overall characterestic function,which also improve the accuracy of the stochastic simulation.(3)In terms of option pricing,most of the current VIX option pricing models are based on the research that VIX is a underlying assets,however,VIX cannot trade directly.If the discount process is forced to be a martingale process under the risk-neutral measure,there will be many constraints.In this paper,the underlying assets of VIX,S&P500 stock based on is a martingale under the risk neutral measure,while the VIX is not necessarily a martingale.Therefore,the pricing effect of constructing the mean reverting model directly under the risk neutral measure is better than other models,which provides more reference for VIX option pricing.(4)In terms of the mean complex model,considering the investor’s cognitive sentiment towards the VIX mean reverting,this study attempts to build a mean reverting sentiment model,and conducts empirical research based on this model.Moreover,this research did some exploration for the combination of behavioral finance and mathematical finance,which also enriches the mean reverting model at the same time.