Pricing Asian options with stochastic volatility. (English) Zbl 1405.91615
Summary: In this paper, we generalize the recently developed dimension reduction technique of J. Vecer [“A new PDE approach for pricing arithmetic average Asian options”, J. Comput. Finance 4, No. 4, 105–113 (2001; doi:10.21314/jcf.2001.064); “Unified pricing of Asian Options”, Risk 15, No. 6, 113–116 (2002), https://www.researchgate.net/profile/Jan_Vecer/publication/239859350_Unified_Pricing_of_Asian_Options/links/00b49528a3cbb402e6000000.pdf] for pricing arithmetic average Asian options. The assumption of constant volatility in Vecer’s method will be relaxed to the case that volatility is randomly fluctuating and is driven by a mean-reverting (or ergodic) process. We then use the fast mean-reverting stochastic volatility asymptotic analysis introduced by J.-P. Fouque et al. [Int. J. Theor. Appl. Finance 3, No. 1, 101–142 (2000; Zbl 1153.91497)] to derive an approximation to the option price which takes into account the skew of the implied volatility surface. This approximation is obtained by solving a pair of one-dimensional partial differential equations.
MSC:
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
Citations:
Zbl 1153.91497References:
| [1] | Alizadeh, S, Brandt, M and Diebold, F. 2002. Range-based estimation of stochastic volatility models. J. Finance, 57: 1047-91. |
| [2] | Carr, P, Geman, H, Madan, D and Yor, M. The fine structure of asset returns: an empirical investigation. Working Paper. |
| [3] | Chernov, M, Gallant, R, Ghysels, E and Tauchen, G. 2001. Alternative models for stock price dynamics. J. Econometrics, at press · Zbl 1043.62087 |
| [4] | Eberlein, E and Prause, K. 1998. The generalized hyperbolic model: Financial derivatives and risk measures. : 56FDM Preprint · Zbl 0996.91067 |
| [5] | Fouque, J-P, Papanicolaou, G and Sircar, R. 2000. Derivatives in Financial Markets with Stochastic Volatility, Cambridge: Cambridge University Press. · Zbl 0954.91025 |
| [6] | Fouque, J-P, Papanicolaou, G and Sircar, R. 2000. Mean-reverting stochastic volatility. Int. J. Theor. Appl. Finance, 3: 101-42. · Zbl 1153.91497 |
| [7] | Fouque, J-P, Papanicolaou, G, Sircar, R and Solna, K. 2003. Short time-scales in S&P 500 volatility. J. Comput. Finance, 6: 1-23. |
| [8] | Fouque, J-P, Papanicolaou, G and Sircar, R. 2001. From the implied volatility skew to a robust correction to Black-Scholes American option prices. Int. J. Theor. Appl. Finance, 4: 651-75. · Zbl 1153.91495 |
| [9] | Fouque, J-P, Papanicolaou, G, Sircar, R and Solna, K. 2003. Singular perturbations in option pricing. SIAM J. Appl. Math., 63: 1648-65. · Zbl 1039.91024 |
| [10] | Fouque, J-P, Papanicolaou, G, Sircar, R and Solna, K. 2003. Multiscale stochastic volatility asymptotics. Preprint · Zbl 1074.91015 |
| [11] | Hoogland, J K and Neumann, C D D. 2001. Local scale invariance and contingent claim pricing. Int. J. Theor. Appl. Finance, 4: 1-21. · Zbl 1153.91509 |
| [12] | Hoogland, J K and Neumann, C D D. 2001. Local scale invariance and contingent claim pricing II. Int. J. Theor. Appl. Finance, 4: 23-43. · Zbl 1153.91510 |
| [13] | Hoogland, J K and Neumann, C D D. 2000. Asians and cash dividends: exploiting symmetries in pricing theory. Technical Report, MAS-0019, CWI |
| [14] | Ingersoll, J. 1987. Theory of Financial Decision Making, Oxford: Rowman and Littlefield. |
| [15] | Kamizono, K, Kariya, T, Liu, R and Nakatsuma, T. 1999. A new control variate estimator for an Asian option. Preprint · Zbl 1189.91208 |
| [16] | LeBaron, B. 2001. Stochastic volatility as a simple generator of apparent financial power laws and long memory. Quant. Finance, 1: 621-31. · Zbl 1405.91716 |
| [17] | Linetsky, V. 2002. “Spectral expansions for Asian (average price) options”. Northwestern University. Preprint |
| [18] | Rogers, L and Shi, Z. 1995. The value of an Asian option. J. Appl. Probab., 32: 1077-88. · Zbl 0839.90013 |
| [19] | Vecer, J. 2001. A new PDE approach for pricing arithmetic average Asian options. J. Comput. Finance, Summer: 105-13. |
| [20] | Vecer, J. 2002. Unified pricing of Asian options. Risk, June: 113-6. |
| [21] | Vecer, J and Xu, M. 2001. Pricing Asian options in a semimartingale model. Preprint · Zbl 1405.91652 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
