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The Pricing Of Variance Swap Based On Stochastic Volatility Models

Posted on:2013-01-11Degree:MasterType:Thesis
Country:ChinaCandidate:Y J LiFull Text:PDF
GTID:2249330395453990Subject:Probability theory and mathematical statistics
Abstract/Summary:PDF Full Text Request
Variance swap is a forward contract which swaps the realized variance during a periodof time for the predetermined variance. When the contract expires, if the realized variance isgreater than the predetermined variance, the buyer will obtain a positive return, and the unit ofcontract revenue is the difference between the realized variance and the predetermined variance.As for the pricing of variance swap, a very popular method is the option replication strategywhich reveals the essential attribute of variance swap: variance swap can be replicated by aseries of standard European call and put options. However, in practice, as investors often faceissues such as implied volatility skew and the lack of market liquidity, the feasibility of thismethod is not very good. Therefore, we try to study the pricing of variance swap in a newtheoretical framework.Firstly, with the help of the EViews software, we examined a total of3629daily data ofthe Shaihai180index from December20,1996to December20,2011, and found that the dailylog returns exhibit a distinct pattern of volatility clustering which shows that the volatility or thesquare of the volatility is auto-correlated. Since the auto-correlation can be guaranteed by themean reversion hypothesis, we have used two stochastic models with a mean reversion natureto describe the volatility, namely the Stein&Stein model and the Heston model.Secondly, based on the Stein&Stein model and the Heston model, we have derived thepricing formulae Vtand Kvarrespectively by using the Ito integral and Ito’s formula. For theHeston model, we find that the fair exercise price Kvarfor the variance swap has the propertythat Kvaris a linear combination of σ02and θ2.Thirdly, based on the mathematical formula of Kvar, the sensitivity analysis for the Hestonmodel parameters: σ02, θ2and κ has been performed in our paper, which is the first study on thesensitivity analysis and allows us to have a more comprehensive and profound undstanding forvariance swap.Lastly, in the option replication theory, the fair exercise price Kvarfor the variance swapand the square of the implied volatility of an option struck at the forward σimp2(K, T)|K=FTsatisfy a linear relationship with a positive slope, which inspired us to study the relationshipbetween Kvarand σimp2(K, T)|K=FT in the Heston model. Our results show that, comparingwith the option replication theory, the relationship in the Heston model becomes more complex.In some cases, Kvarand σimp2(K, T)|K=FT satisfy a linear relationship with a positive slope,while in other cases, the conclusion does not hold. Therefore, it can be found that the result we have in the option replication theory is just a special case in the Heston model.
Keywords/Search Tags:variance swap, option replication strategy, Stein&Stein model, Heston model, sensitivity analysis
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