Malliavin differentiability of the Heston volatility and applications to option pricing. (English) Zbl 1137.91422
Summary: We prove that the Heston volatility is Malliavin differentiable under the classical Novikov condition and give an explicit expression for the derivative. This result guarantees the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model. Furthermore, we derive conditions on the parameters which assure the existence of the second Malliavin derivative of the Heston volatility. This allows us to apply recent results of E. Alòs [Finance Stoch. 10, No. 3, 353–365 (2006; Zbl 1101.60044)] in order to derive approximate option pricing formulae in the context of the Heston model. Numerical results are given.
MSC:
| 91G80 | Financial applications of other theories |
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
Keywords:
Malliavin calculus; stochastic volatility model; Heston model; Cox-Ingersoll-Ross process; Hull and White formula; option pricingCitations:
Zbl 1101.60044References:
| [1] | Alòs, E. (2006). An extension of the Hull and White formula with applications to option pricing approximation. Finance Stoch. 10, 353–365. · Zbl 1101.60044 · doi:10.1007/s00780-006-0013-5 |
| [2] | Alòs, E. and Nualart, D. (1998). An extension of Itô’s formula for anticipating processes. J. Theoret. Prob. 11, 493–514. · Zbl 0914.60018 · doi:10.1023/A:1022692024364 |
| [3] | Alòs, E., Leon, J. A. and Vives, J. (2007). On the shorttime behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance Stoch. 11, 571–598. · Zbl 1145.91020 · doi:10.1007/s00780-007-0049-1 |
| [4] | Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae , 2nd edn. Birkhäuser, Basel. · Zbl 1012.60003 |
| [5] | Bossy, M. D. A. (2004). An efficient discretization scheme for one dimensional SDE’s with a diffusion coefficient function of the form \(| x |^\alpha\), \(\alpha \in [1/2,1)\). Res. Rep. 5396 INRIA. |
| [6] | Detemple, J., Garcia, R. and Rindisbacher, M. (2005). Representation formulas for Malliavin derivatives of diffusion processes. Finance Stoch. 9, 349–367. · Zbl 1088.60057 · doi:10.1007/s00780-004-0151-6 |
| [7] | Geman, H. and Yor, M. (1993). Bessel Processes, Asian Options and perpetuities. Math. Finance 3, 349–375. · Zbl 0884.90029 · doi:10.1111/j.1467-9965.1993.tb00092.x |
| [8] | Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343. · Zbl 1384.35131 |
| [9] | Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities. J. Finance 42, 281–300. · Zbl 1126.91369 |
| [10] | Jeanblanc, M. (2006). Cours de calcul stochastique. Master 2IF EVRY. Lecture Notes, University of Évry. Available at http://www.maths.univ-evry.fr/pages\(\_\)perso/jeanblanc/. |
| [11] | Karatzas, I. and Shreve, S.-E. (1988). Brownian Motion and Stochastic Calculus (Graduate Texts Math. 113 ). Springer, New York. · Zbl 0638.60065 |
| [12] | Nualart, D. (1995). The Malliavin Calculus and Related Topics . Springer, New York. · Zbl 0837.60050 |
| [13] | Yor, M. (1992). Some Aspects of Brownian Motion , Part 1, Some Special Functionals . Birkhäuser, Basel. · Zbl 0779.60070 |
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.
