Variance reduction for Monte Carlo methods to evaluate option prices under multi-factor stochastic volatility models. (English) Zbl 1405.91692
Summary: We present variance reduction methods for Monte Carlo simulations to evaluate European and Asian options in the context of multiscale stochastic volatility models. European option price approximations, obtained from singular and regular perturbation analysis [the first author et al., Multiscale Model. Simul. 2, No. 1, 22–42 (2004; Zbl 1074.91015)], are used in importance sampling techniques, and their efficiencies are compared. Then, we investigate the problem of pricing arithmetic average Asian options (AAOs) by Monte Carlo simulations. A two-step strategy is proposed to reduce the variance where geometric average Asian options (GAOs) are used as control variates. Due to the lack of analytical formulas for GAOs under stochastic volatility models, it is then necessary to consider efficient Monte Carlo methods to estimate the unbiased means of GAOs. The second step consists in deriving formulas for approximate prices based on perturbation techniques, and in computing GAOs by using importance sampling. Numerical results illustrate the efficiency of our method.
MSC:
91G60 | Numerical methods (including Monte Carlo methods) |
91G20 | Derivative securities (option pricing, hedging, etc.) |
65C05 | Monte Carlo methods |
Citations:
Zbl 1074.91015References:
[1] | Alizadeh, S, Brandt, M and Diebold, F. 2002. Range-based estimation of stochastic volatility models. Journal of Finance, 57: 1047-91. |
[2] | Boyle, P, Broadie, M and Glasserman, P. 1997. Monte Carlo methods for security pricing. Journal of Economic and Control, 21: 1267-321. · Zbl 0901.90007 |
[3] | Chernov, M, Gallant, R, Ghysels, E and Tauchen, G. 2003. Alternative models for stock price dynamics. J. of Econometrics, 116: 225-57. · Zbl 1043.62087 |
[4] | Fouque, J-P and Han, C-H. 2003. Pricing asian options with stochastic volatility. Quantitative Finance, 3: 353-62. · Zbl 1405.91615 |
[5] | Fouque, J-P and Han, C-H. 2004. Asian options under multiscale stochastic volatility. AMS Contemporary Mathematics: Mathematics of Finance, 351: 125-38. · Zbl 1120.91013 |
[6] | Fouque J-PHan C-H2004Geometric average asian options under multiscale stochastic volatility (Preprint) |
[7] | Fouque J-PPapanicolaou GSircar R2000Derivatives in Financial Markets with Stochastic VolatilityCambridgeCambridge University Press · Zbl 0954.91025 |
[8] | Fouque, J-P, Papanicolaou, G, Sircar, R and Solna, K. 2003. Multiscale stochastic volatility asymptotics. SIAM Journal on Multiscale Modeling and Simulation, 2(1): 22-42. · Zbl 1074.91015 |
[9] | Fouque, J-P and Tullie, T. 2002. Variance reduction for Monte Carlo simulation in a stochastic volatility environment. Quantitative Finance, 2: 24-30. · Zbl 1405.91693 |
[10] | Glasserman P2003Monte Carlo Methods in Financial EngineeringNew YorkSpringer Verlag |
[11] | Molina GHan C-HFouque J-P2003MCMC Estimation of Multiscale Stochastic Volatility Models (Preprint) |
[12] | Newton, N. 1994. Variance Reduction for Simulated Diffusions. SIAM J. Appl. Math, 54: 1780-805. · Zbl 0811.60046 |
[13] | Wong, H-Y and Cheung, Y-L. 2004. Geometric Asian Options: valuation and calibration with stochastic volatility. Quantitative Finance, 4: 301-14. · Zbl 1405.91655 |
[14] | Vecer, J and Xu, M. 2004. Pricing Asian options in a semimartingale model. Quantitative Finance, 4: 170-5. · Zbl 1405.91652 |
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