On the calculation of price sensitivities with jump-diffusion structure

Youssef, El-Khatib and Hatemi-J, Abdulnasser (2011): On the calculation of price sensitivities with jump-diffusion structure. Published in: Journal of Statistics Applications & Probability , Vol. 3, No. 1 (2012): pp. 171-182.

Abstract

We provide a new theoretical framework for estimating the price sensitivities of a trading position with regard to five underlying factors in jump-diffusion models using jump times Poisson noise. The proposition that results in a general solution is mathematically proved. The general solution that this paper offers can be applied to compute each price sensitivity. The suggested modeling approach deals with the shortcomings of the Black-Scholes formula such as the jumps that can occur at any time in the stock's price. Via the Malliavin calculus we show that differentiation can be transformed into integration, which makes the price sensitivities operational and more efficient. Thus, the solution that is provided in this paper is expected to make decision making under uncertainty more efficient.

• Item Type: MPRA Paper
• Language: English
• Keywords: Malliavin Calculus; Asset Pricing; Price Sensitivity; Jump-diffusion Models; Jump Times Poisson Noise; European Options.
• Subjects: G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates; G - Financial Economics > G1 - General Financial Markets > G10 - General; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C60 - General

References

[BB73] Black, F. and M. Scholes. The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81, 637-654.

[AOPU00] K. Aase, B. Øksendal, N. Privault, and J. Ubøe. White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance. Finance and Stochastics, 4(4) pages 465--496, 2000.

[BBM07] V. Bally, M. Bavouzet, and M. Messaoud. Integration by parts formula for locally smooth laws and applications to sensitivity computations. Annals of Applied probability, 17(1), 33-66, 2007.

[BM06] M. Bavouzet, and M. Messaoud. Computation of Greeks using Malliavin's calculus in jump type market models. Electronic Journal of Probability, 11, pages 276-300, 2006.

[BI83] J.M. Bismut. Calcul des variations stochastique et processus de sauts. Zeitschrift für Wahrscheinlichkeitstheories Verw. Gebiete, 63:147--235, 1983.

[CP90] E. Carlen and E. Pardoux. Differential calculus and integration by parts on Poisson space. In S. Albeverio, Ph. Blanchard, and D. Testard, editors, Stochastics, Algebra and Analysis in Classical and Quantum Dynamics (Marseille, 1988), volume 59 of Math. Appl., pages 63--73. Kluwer Acad. Publ., Dordrecht, 1990.

[DJ06] M. Davis and M. Johansson. Malliavin Monte Carlo Greeks for jump diffusions. Stochastic Processes and their Applications, 116, pages 101–129, 2006.

[D00] L. Denis. A criterion of density for solutions of Poisson-driven SDEs. Probab. Theory Relat. Fields, 118, pages 406--426, 2000.

[KP04] Y. El-Khatib and N. Privault. Computations of Greeks in a market with jumps via the Malliavin calculus. Finance and Stochastics, 8, 2 (May 2004), PP. 161-179.

[ET93] R.J. Elliott and A.H. Tsoi Integration by parts for Poisson processes. J. Multivariate Anal., 44(2), pages 179--190, 1993.

[FLLLT99] E. Fournié, J.M. Lasry, J. Lebuchoux, P.L. Lions, and N. Touzi. Applications of Malliavin calculus to Monte Carlo methods in finance. Finance and Stochastics, 3(4), pages 391--412, 1999.

[J79] Jacod, J. (1979): Calcul stochastique et problèmes de martingales. volume 714 of Lecture Notes in Mathematics. Springer Verlag.

[KK10] R. Kawai and A. Kohatsu-Higa. Computation of Greeks and multidimensional density estimation for asset price models with time-changed Brownian motion. To appear in Applied Mathematical Finance.

[KT10] R. Kawai and A. Takeuchi. Sensitivity analysis for averaged asset price dynamics with gamma processes. Statistics and Probability Letters, 80(1), pages 42-49, 2010.

[Nu95] D. Nualart (1995): The Malliavin Calculus and Related Topics. Springer-Verlag.

[NV90] D. Nualart and J. Vives. Anticipative calculus for the Poisson process based on the Fock space. In J. Azéma, P.A. Meyer, and M. Yor, editors, Séminaire de Probabilités XXIV, volume 1426 of Lecture Notes in Mathematics, pages 154--165. Springer Verlag, 1990.

[Øk] B. Øksendal. An introduction to Malliavin calculus with applications to economics. Working paper no. 3, Institute of Finance and Management Science, Norwegian School of Economics and Business Administration, 1996.

[Pr94] N. Privault. Chaotic and variational calculus in discrete and continuous time for the Poisson process. Stochastics and Stochastics Reports, 51, pages 83-109, 1994.

[Pr02] N. Privault. Distribution-valued gradient and chaotic decompositions of Poisson jump times functionals. Publicacions Matemàtiques, 46, pages 27-48, 2002.

[Pr09] N. Privault. Stochastic Analysis in discrete and Continuous settings: with normal martingales. Lecture Notes in Mathematics, Vol. 1982, 310 pages, Springer, 2009.