A Study On Variance Swaps With Stochastic Volatility

Essentially speaking, variance swaps are forward contracts. In theview of traders, one leg of the swap will pay an amount based upon therealized variance of the price changes of the underlying product. Theother leg of the swap will pay a fixed amount, which is the strike, quotedat the deal’s inception. Thus the net payoff to the counterparties will bethe difference between these two and will be settled in cash at theexpiration of the deal.On pricing variance swaps, considerable researchers tend to conducttheir investigation with a method of option pricing. In such way, thevariance swap can be regarded as a series of call and put options, whichreflects the essence of swap contract, the trade of options.In the literature, there have been two types of valuation approaches,numerical methods and analytical methods. The analytical methodstheoretically replicate a variance swap by a portfolio of standard options.The numerical methods price a variance swap by using the finite-difference method in an extended Black-Scholes framework or partial-integro differential equation approach.In this paper, we basically analyze the problem in Heston framework. Then, a new form of model is sited, in which the process of variance followed is modified. Furthermore, a numerical solution and its charts are presented.This paper is organized into four sections. In section2, the CIR interest process[16] is quoted and introduced in the form of dr=k(θ-r)dt+σ(?)rdW1(t),(0.1) where θ is the long-term mean of variance, κ is a mean-reverting speed parameter of the variance, σ is the so-called volatility of volatility.In section3, the analysis is conducted under the framework of Heston[17] model, which is where the two Wiener processes dWtS and dWtV describe the random noise in asset and variance, respectively. They are assumed to be correlated with a constant correlation coefficient p, which is (dWts,dWTV)=p.With the support of such hypothesis and model, the new model is presented in section4: The new model leads its correlated partial differential equation:In section5, the numerical solutions and charts of the PDE above are presented.