Pricing VIX Derivatives Under A Series Of Stochastic Volatility Models

In line with the daily-growing breadth of financial derivative market and incessantly developing depth of the theoretical researches on mathematical financial,it seems that option products and the corresponding pricing theory is prone to be more and more complicate.Taking Chicago Board Options Exchange(CBOE)for example,it strategically launched CBOE Future Exchange(CFE)in year of 2004 which embarked on trading S&P500(SPX)Volatility Index(VIX)Futures,then it introduced VIX option to the market in year of 2006.In the following near two decades,volatility has been widely accepted by relevant investors,dealers and fund managers as an asset type and being deployed in concentration ratio and tail risk that are prevalent among investment,dispersive and hedging asset portfolio.In response to a sea of investors favor,the transaction volume of volatility derivatives embraced a steady growth which directly led to a blooming development and innovation of pricing theory of volatility option.With regard to the earlier models,this paper aims to further promote and develop the method of pricing VIX derivatives with stochastic volatility model.The traditional option pricing models in early stage stem from the Black-Scholes model which is co-founded and developed by Robert Carhart Merton and Myron Samuel Scholes,the two laureates of Nobel Economics Prize of 1997.In light of the relatively simplicity and efficiency that this model owns,it still tips the balance in the realm of option pricing.Moreover,it is the very basis upon which current frontier models are establish and stretched.Though there are diverse stochastic volatility modeling methods for VIX derivatives,they could basically divided into two types,i.e.consistency modeling and independence modeling.For the former,it starts with S&P500 index and its stochastic volatility modeling,then derives the expression of VIX index which is in accordance with the aforementioned dynamic stochasticpartial differential equation and the definition of VIX,and capable of pricing VIX derivatives.When it comes to the latter,the independence modeling,it refers to a VIX futures and Option pricing method that is built on the dynamic processes of VIX index,which is equivalent to the directly nominated stochastic partial differential equation.In consideration of the fact that VIX is the very invariable volatility index for S&P500,it is essential to take into account the internal connections between S&P500 and VIX.In other words,the independence modeling which possibly evades S&P500 owns impracticality to certain degree.Therefore,this paper accentuates on consistency modeling,joint with few mentions of independence modeling.The main contributions that this paper has made are detailed as below:1.The the application of Li symmetry theory over partial differential equation is initially made use of,followed by a conclusion of the known Bessel Process.Consecutively,a new 4/2 stochastic volatility framework is constructed,enabling the reputed Heston stochastic volatility and the 3/2 one to act as parametric examples.The instantaneous variance of Heston is a mean-reverting square root Bessel process(also known as CIR process or abbreviated as square process,considering that the index of its diffusion term is a square root).The inverse process of CIR process is still a mean-reverting process from which the 3/2 stochastic volatility model is derived.Because the extension index of this process is 3/2,it is reasonable to consider the linear superposition of these two mentioned processes,which also illustrates the reason why the volatility model is named 4/2(=1 /2+3/2).Based on this volatility,a brand-new stochastic volatility model is constructed and remarked as model 4/2SVJ,which owns not only the very advantages that those two classical models have,but also the jumps.To guarantee the reasonability of definitions,there are two points waiting for being proved.Firstly,it should be proved that the asset discounted process that this model has is a martingale rather than a local one.Secondly,it is necessary to prove that this partial differential equation shall not reach zero,neither blast,in parallel with timing.By defining real variance processas the quadratic variation processes of logarithmic assets,it is accessible to both asset and the closed-form solution of Fourier and Laplace transforms of real variance.Once the closed-form solution is accessible,either Fourier Transform Algorithm or Fourier COS Algorithm could be taken as rapid methods of calculating the derivatives of underlying asset and variance,for instance,European option,American option and variance swap.2.S&P500 data is estimated by applying the Generalized Method of Moments(GMM)Estimation which proposed by Hansen in year of 1982.Then,reflected by hypothesis tests,it is noticed that the index of stochastic volatility is not invariable.When the index of the fixed stochastic volatility term is 1/2 or3/2,it is reasonable to conclude that the models are over-constrained.Hence,under the conclusion of generalizing Bessel process,another brand-new model is accessible,Free Stochastic Volatility Model,here remarked as FSV Model.Unlike the general ones,this model sets its stochastic volatility index as a free variable inside a certain interval,letting the data approaching to the authentic value.The breadth of this interval takes the reputed Heston Model and 3/2model as the parameters for FSV model.Under this breadth,it is provable that the asset discounting process of FSV model has is a still a martingale and will not blast.Lastly,two free stochastic volatility models that respectively embrace drop jump and asymmetry jump are remarked as model FSV-DJ and model FSV-AJ.With these three models,it is accessible to the pricing formulas for VIX option and futures.3.In order to make asset model more conform to practical profit distribution,various pure jump Levy processes are constructed artificially.These pure jump processes are prone to produce infinite(high-frequent)jumps in finite time,namely,infinite activity.Through time-variant constructing,the 4/2 stochastic volatility is embedded into Levy processes so as to stimulate stochastic volatility.This new model is remarked as model NTS-4/2SV.Then,to reflect leverage effect,Heston model is deployed,that is adding 4/2-stochasticvolatility-relevant diffusion terms into model NTS-4/2SV so as to realize the relevance between asset price and volatility.This new model is remarked as model NTS-4/2SVR.In fact,this not constrained to 4/2-stochastic-volatilitycentered modeling environment,both 1/2 and 3/2 stochastic volatility,in conjunction with leverage effect,could also be introduced into Levy processes to stimulate asset stochastic volatility structure.The models are respectively remarked as NTSSV and NTSSVR with leverage effect,together with NTS-3/2SV and NTS-3/2SVR with leverage effect.What this paper finalizes in is not only the reduction of futures.and option pricing formulas applicable to various models,but also the variance-optimal hedging strategy numerical solution under the condition that transactions are continuously hedging.