| Variance swaps is a financial instrument rising in the mid-1990 s.It is essentially a kind of forward contract.The parties to the contract agree to swap the realized variance of the price of the underlying asset during the contract period with a fixed value at a certain time in the future.In other words,it is to exchange fixed value for uncertain value in the future,so as to avoid volatility risk.According to the data of?Risk?,the trading volume of variance swaps is growing rapidly in recent years?The economic crisis in Southeast Asia,the subprime mortgage crisis since 2008,and the large-scale outbreak of COVID-19 have made the global financial market show more and more volatility risks.Therefore,it is necessary to use variance swap to provide more pure risk management of underlying assets.Variance swaps is one of the effective tools to obtain income or hedge volatility risk.At present,many research achievements have been made on the pricing of variance swaps under CIR(Cox-Ingersoll-Ross)-Heston hybrid model.However,due to the instantaneous interest rate and instantaneous volatility follow the Feller square root process,only a semi-closed solution can be obtained by solving the PDEs.A more simplified approach is proposed in this paper compared with other study,an important feature is that a closed-form solution is obtained by using martingale theory and It(?)'s lemma without solving PDEs.The closed-form solution is of great significance,which there is no need to use numerical simulation to adjust parameters.Another remarkable feature of this paper is that the influence of sampling frequency on the pricing formula is analyzed,and the closed-form solution is extended to an approximate solution.Finally,the price curves of the exact solution and the approximate solution are given by numerical simulation.It can be found that when the sampling frequency is large enough,the two curves almost coincide,which means that our approximate formula is simple and reliable. |
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