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Fast Monte Carlo Simulation for Pricing Covariance Swap Under Correlated Stochastic Volatility Models Junmei Ma, Ping He

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Fast Monte Carlo Simulation for Pricing Covariance Swap Under Correlated Stochastic Volatility Models Junmei Ma, Ping He IAENG International Journal of Applied Mathematics, 46:3, IJAM_46_3_09 ______________________________________________________________________________________ Fast Monte Carlo Simulation for Pricing Covariance Swap under Correlated Stochastic Volatility Models Junmei Ma, Ping He Abstract—The modeling and pricing of covariance swap with probability 1 − δ, where σn is the standard deviation derivatives under correlated stochastic volatility models are of estimation of V , δ is significance level and Z δ is the 2 studied. The pricing problem of the derivative under the case δ of discrete sampling covariance is mainly discussed, by efficient quantile of standard normal distribution under 2 . The error pσn is Z δ , which just rely on the deviation σ and sample Control Variate Monte Carlo simulation. Based on the closed n 2 form solutions derived for approximate models with correlated storage n.However, although Monte Carlo method is easy deterministic volatility by partial differential equation method, to implement and effective to solve the multi-dimensional two kinds of acceleration methods are therefore proposed when problem, it’s disadvantage is its slow convergence. The the volatility processes obey the Hull-White stochastic processes. − 1 Through analyzing the moments for the underlying processes, convergence rate of Monte Carlo method is O(n 2 ).We the efficient control volatility under the approximate model have to spend 100 times as much as computer time in order is constructed to make sure the high correlation between the to reduce the error by a factor of 10 [8]. control variate and the problem. The numerical results illustrate Many financial practitioners and researchers concentrate the high efficiency of the control variate Monte Carlo method; on the problem of computational efficiency, several ap- the results coincide with the theoretical results. The idea in the paper is also applicable for the valuation of other financial proaches to speed up Monte Carlo simulation, such as derivatives with discrete features under multi-factor models. control variate, antithetic variables, importance sampling and stratification[7] have been proposed over the last few years. Index Terms—Covariance swap, Stochastic volatility, Monte Carlo, Control variate, Variance reduction. These techniques aim to reduce the variance per Monte Carlo simulation so that a given level of accuracy can be achieved with a smaller number of simulations. Control variate is one I. INTRODUCTION of the most widely used variance reduction techniques, main- ly because of the simplicity of its implementations, and the ONTE Carlo method is a numerical method based on fact that it can be accommodated in an existing Monte Carlo M the probability theory. Monte Carlo method becomes calculator with a small effort. This paper mainly studies the more and more popular, as the rapid growth of the financial pricing of covariance derivatives under correlated stochastic derivative markets and the increase of the complexity of the volatility models through fast Monte Carlo simulation. pricing models. The advantage of Monte Carlo method is that Different control variates are proposed to accelerate the the efficiency of the method is independent of the number convergence rate of the simulation errors, based on the of state variables. If the number of state variables is greater closed form solutions of simplified models. The numerical than three, Monte Carlo method is suitably used to solve experiments show the high efficiency of our acceleration the high dimensional pricing problem and often becomes the methods, which can also be extended to the pricing of other only computationally feasible means of derivative pricing. path-dependent derivatives in an extended Black-Scholes Suppose the price of the derivativeP µ = E[V ] will be framework, such as Asian options, Lookback options and 1 n f gn estimated by mean value V n = n i=1 Vi, where Vi i=1 other variance derivatives. are independent and identical distribution (i:i:d) samples of The covariance swap is a covariance forward contact of the random variable V . Then by the Central Limit Theorem, the underlying assets S1 and S2, and its payoff at expiration the price µ asymptotically falls into the interval is equal to σn σn P = M × (Cov (S ;S ) − K2 ); [V n − p Z δ ; V n + p Z δ ]; R 1 2 cov n 2 n 2 where Kcov is a strike price, M is the notional amount, Manuscript received June 11, 2015; revised Oct 2, 2015. This work and CovR(S1;S2) is the realized covariance between two was supported by NSFC (NO.11226252, NO.11271243 and NO.11271240), assets S1 and S2. Therefore, in the risk-neutral world, the fair the Special Fund for Shanghai Colleges Outstanding Young Teacher- strike covariance can be attained by E[CovR(S1;S2)]. The s Scientific Research Project(No.ZZCD12007), the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Educa- procedure for calculating the realized covariance is usually tion of China(No.708040) and Research Projects of Young Teachers in clearly specified in the contract and includes details about SUFE(No.2011220656). the source and observation frequency of the price of the Junmei Ma is with the School of Mathematics, Shanghai University of Finance and Economics, Shanghai Key Laboratory of Financial Information underlying assets, and the method to calculate the covariance. Technology, 777 Guoding Road, Shanghai 200433, China(phone:+86-21- The covariance swap is the main representation of the 65903296;fax:+86-21-65904320;e-mail:[email protected]) new generation of variance and volatility swap derivatives, Ping He is with the School of Mathematics, Shanghai Universi- ty of Finance and Economics,777 Guoding Road, Shanghai 200433, and becoming very popular and actively traded in financial China(phone:+86-21-65908857;e-mail:[email protected]) practice in recent years. Variance and volatility derivatives (Advance online publication: 26 August 2016) IAENG International Journal of Applied Mathematics, 46:3, IJAM_46_3_09 ______________________________________________________________________________________ are based on a single underlying asset. Their payoffs just and discretely sampled covariance. For the continuously depend on the realized variance or volatility of the underlying sampled covariance derivatives, a closed form solution is asset or an index. The market for variance and covariance derived directly by the PDE method. For the discretely swaps has been growing, which are now actively quoted and sampled covariance derivatives, two kinds of fast Monte used to speculate on future (co)variance levels, to trade the Carlo simulation methods are developed and studied. The spread between implied and realized (co)variance levels, or discretely sampled covariance derivatives are more applicable to hedge the (co)variance exposures of other positions by in financial practice as the assets ”covariance” cannot be many financial institutions. The objective of this paper is observed directly in the market. The choice of efficient to demonstrate the modeling and the pricing of a covariance control variates is also discussed in the paper. swap. This contract pays the excess of the realized covariance The rest of the paper is arranged as follows. We begin by between two currencies over a constant specified at the outset modeling covariance swaps with the PDE method and ∆- of the contract. Such a contract is growing as a useful hedging principle under the stochastic volatility assumption complement for the variance contracts that trade OTC on in Section 2. In Section 3, the first control variate method of several currencies. By combining variance and covariance Monte Carlo simulation for the valuation of the covariance swaps, the realized variance of return on a portfolio of swap is proposed. The analytical solution for the control currencies can be locked in. variate is attained. Then, with the moment analysis for the Logically, a correlation swap is a correlation forward underlying and auxiliary stochastic processes, an approx- contract of two underlying rates S1 and S2, whose payoff imate “optimal volatility” is provided for developing the at expiration is equal to: highly efficient control variate in this framework. In Section 4, the second improved control variate method is constructed × − P = M (CorrR(S1;S2) Kcorr); based on the algorithm in Section 3 and the choice of where Kcorr is a strike correlation, M is the notional amount, the optimal control volatility constants is also considered. and CorrR(S1;S2) is the realized correlation of two under- In Section 5, the computational and analytical results are lying assets S1 and S2. Therefore the fair strike correlation illustrated and coincide with the theoretical analysis well. is E[CorrR(S1;S2)]. The correlation CorrR(S1;S2) can be Concluding remarks are given in Section 6. presented by the covariance, II. MODELING AND PRICING OF COVARIANCE SWAP CovR(S1;S2) CorrR(S1;S2) = : N this section, the partial differential equation pricing σR(S1)σR(S2) I model for the covariance swap under the correlated Therefore, the modeling and pricing of correlation swap will stochastic volatility assumption is obtained. The variance be attained with the research results of covariance swap. products are all derivatives based on stochastic volatility The research on the pricing of variance swap and volatility model. The research on the stochastic volatility starts from swap is very massive and we do not intend to give a list here. the early 1980’s. In 1973, Black
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