ElKhatib, Youssef and Abdulnasser, HatemiJ (2011): On the calculation of price sensitivities with jumpdiffusion structure.
There is a more recent version of this item available. 

PDF
MPRA_paper_30596.pdf Download (322kB)  Preview 
Abstract
We provide a new theoretical framework for estimating the price sensitivities of a trading position with regard to five underlying factors in jumpdiffusion 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 BlackScholes 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 

Original Title:  On the calculation of price sensitivities with jumpdiffusion structure 
English Title:  On the calculation of price sensitivities with jumpdiffusion structure 
Language:  English 
Keywords:  Malliavin Calculus; Asset Pricing; Price Sensitivity; Jumpdiffusion 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 
Item ID:  30596 
Depositing User:  Abdulnasser HatemiJ 
Date Deposited:  03. May 2011 16:01 
Last Modified:  13. Feb 2013 18:08 
References:  [BB73] Black, F. and M. Scholes. The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81, 637654. [AOPU00] K. Aase, B. Øksendal, N. Privault, and J. Ubøe. White noise generalizations of the ClarkHaussmannOcone theorem with application to mathematical finance. Finance and Stochastics, 4(4) pages 465496, 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), 3366, 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 276300, 2006. [BI83] J.M. Bismut. Calcul des variations stochastique et processus de sauts. Zeitschrift für Wahrscheinlichkeitstheories Verw. Gebiete, 63:147235, 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 6373. 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 Poissondriven SDEs. Probab. Theory Relat. Fields, 118, pages 406426, 2000. [KP04] Y. ElKhatib and N. Privault. Computations of Greeks in a market with jumps via the Malliavin calculus. Finance and Stochastics, 8, 2 (May 2004), PP. 161179. [ET93] R.J. Elliott and A.H. Tsoi Integration by parts for Poisson processes. J. Multivariate Anal., 44(2), pages 179190, 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 391412, 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. KohatsuHiga. Computation of Greeks and multidimensional density estimation for asset price models with timechanged 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 4249, 2010. [Nu95] D. Nualart (1995): The Malliavin Calculus and Related Topics. SpringerVerlag. [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 154165. 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 83109, 1994. [Pr02] N. Privault. Distributionvalued gradient and chaotic decompositions of Poisson jump times functionals. Publicacions Matemàtiques, 46, pages 2748, 2002. [Pr09] N. Privault. Stochastic Analysis in discrete and Continuous settings: with normal martingales. Lecture Notes in Mathematics, Vol. 1982, 310 pages, Springer, 2009. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/30596 
Available Versions of this Item
 On the calculation of price sensitivities with jumpdiffusion structure. (deposited 03. May 2011 16:01) [Currently Displayed]