Pricing Variance Swaps Under Bi-Heston-CIR Hybrid Model

In this paper,we discuss the problem of pricing discretely-sampled variance swaps under a Bi-Heston-CIR hybrid stochastic model.Variance swaps are volatility derivatives which have been playing an important role in the financial markets.Today,variance swaps have become the most widespread accepted and popular traded products.Our attention focuses on the pricing discrete sampling variance swaps among underlying asset,interest rate,and volatility.The Bi-Heston-CIR hybrid stochastic model will be presented in this paper,which can better fit the real situation of the financial market.Different from variance swap pricing based on Heston model,we can only obtain an efficient semi-closed form of pricing formula for variance swaps instead of a closed-form solution based on the derivation of characteristic functions.This semi-closed form solution will be analyzed by numerical analysis.The main research results of this paper are as follows:(1)Because of the CIR interest rate model in the model,we use the T-forward measure method here,which directly leads to the semi-closed pricing formula.(2)The difference of variance swap pricing between the Bi-Heston-CIR mixed stochastic model and Heston model is compared.In this paper,the analytical solution based on Heston model is briefly introduced at first,and then the semi-closed solution based on Bi-Heston-CIR hybrid stochastic model is given in detail.There is a clear difference in the process of proofs.(3)Firstly,the pricing formula of variance swap under hybrid stochastic model,Monte Carlo simulations and continuous model are compared by numerical analysis.Then the effect of several parameters of hybrid stochastic model on the pricing formula is studied.