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A Discrete-Time Clark-Ocone Formula for Poisson Functionals

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Abstract

In this paper, we establish a discrete-time version of Clark(-Ocone-Haussmann) formula for Poisson functionals. The formula is applied to the estimation of “hedging error”.

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Notes

  1. On the space of all right-continuous functions with left-hand side limits, one can endow so-called the Skorohod topology which is metrizable and makes the space a complete separable metric space. For details, see Billingsley Billingsley (1999), Chapter 5.

References

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

    Article  Google Scholar 

  • Akahori, J., Amaba, T., & Okuma, K. (2013). A discrete-time Clark-Ocone formula and its application to an error analysis, arXiv:1307.0673, Submitted on 2 July 2013, last revised 30 August 2013.

  • Bertsimas, D., Kogan, L., & Lo, A. W. (2000). When is time continuous? Journal of Financial Economics, 55(2), 173–204.

    Google Scholar 

  • Bichteler, K., Gravereaux, J. B., & Jacod, J. (1987). Malliavin calculus for processes with jumps (p. 161). New York, ix: Gordon and Breach Science Publishers.

  • Billingsley, P. (1999). Convergence of probability measures (2nd ed.). New York: A Wiley-Interscience Publication.

    Book  Google Scholar 

  • Brodén, M., & Tankov, P. (2011). Tracking errors from discrete hedging in exponential lévy models. Journal of Theoretical and Applied Finance, 14(6), 803–837.

    Article  Google Scholar 

  • Clark, J. (1970). The representation of functionals of brownian motion by stochastic integrals. The Annals of Mathematical Statistics, 1282–1295.

  • Dermoune, A., Krée, P., & Wu, L. (1988). Calcul stochastique non adapté par rapport à la mesure aléatoire de poisson. Séminaire de Probabilités, XXII, 1321, 477–484.

    Article  Google Scholar 

  • Di Nunno, G., Øksendal, B., Proske, F. (2009). Malliavin calculus for Lévy processes with applications to finance (pp. xiv+413). Universitext. Berlin: Springer.

  • Elliott, R. J., & Tsoi, A. H. (1993). Integration by parts for poisson processes. Journal of Multivariate Analysis, 44(2), 179–190.

    Article  Google Scholar 

  • Geiss, C., & Geiss, S. (2004). On approximation of a class of stochastic integrals and interpolation. Stochastics and Stochastic Reports, 76(4), 339–362.

    Article  Google Scholar 

  • Gobet, E., & Temam, E. (2001). Discrete time hedging errors for options with irregular payoffs. Finance and Stochastics, 5(3), 357–367.

    Article  Google Scholar 

  • Haussmann, U. (1978). Functionals of ito processes as stochastic integrals. SIAM Journal on Control and Optimization, 16(2), 252–269.

    Article  Google Scholar 

  • Hayashi, T., & Mykland, P. A. (2005). Evaluating hedging errors: An asymptotic approach. Mathematical Finance, 15(2), 309–343.

    Article  Google Scholar 

  • Ikeda, N., & Watanabe, S. (1989). Stochastic differential equations and diffusion processes. Amsterdam: North-Holland.

    Google Scholar 

  • Itô, K. (1956). Spectral type of the shift transformation of differential processes with stationary increments. Transactions of the American Mathematical Society, 81(2), 253–263.

    Article  Google Scholar 

  • Lindstrøm, T. (2004). Hyperfinite lévy processes. Stochastics: An International Journal of Probability and Stochastic Processes, 76(6), 517–548.

    Google Scholar 

  • Malliavin, P., & Thalmaier, A. (2006). Stochastic calculus of variations in mathematical finance. Berlin: Springer.

    Google Scholar 

  • Nualart, D. (1988). Vives J (1990) Anticipative calculus for the poisson process based on the fock space. Séminaire de Probabilités XXIV, 89, 155–165.

    Google Scholar 

  • Nualart, D. (2006). The Malliavin calculus and related topics. In Probability and its applications (New York) (2nd ed., pp. xiv+382). Berlin: Springer-Verlag.

  • Ocone, D. (1984). Malliavin’s calculus and stochastic integral representations of functional of diffusion processes. Stochastics: An International Journal of Probability and Stochastic Processes, 12(3–4), 161–185.

    Article  Google Scholar 

  • Ocone, D. L., & Karatzas, I. (1991). A generalized clark representation formula, with application to optimal portfolios. Stochastics: An International Journal of Probability and Stochastic Processes, 34(3–4), 187–220.

    Google Scholar 

  • Picard, J. (1996). Formules de dualité sur l’espace de poisson. In Annales de l’IHP Probabilités et statistiques (vol. 32, pp. 509–548). Elsevier.

  • Privault, N. (1994). Chaotic and variational calculus in discrete and continuous time for the poisson process. Stochastics: An International Journal of Probability and Stochastic Processes, 51(1–2), 83–109.

    Google Scholar 

  • Privault, N. (2009). Stochastic analysis in discrete and continuous settings, vol. 200. Springer.

  • Privault, N., & Schoutens, W. (2002). Discrete chaotic calculus and covariance identities. Stochastics: An International Journal of Probability and Stochastic Processes, 72(3–4), 289–316.

    Google Scholar 

  • Tankov, P., & Voltchkova, E. (2009). Asymptotic analysis of hedging errors in models with jumps. Stochastic Processes and Their Applications, 119(6), 2004–2027.

    Article  Google Scholar 

  • Temam, E. (2003). Analysis of error with malliavin calculus: Application to hedging. Mathematical Finance, 13(1), 201–214.

    Article  Google Scholar 

  • Üstünel, A. S. (1987). Representation of the distributions on wiener space and stochastic calculus of variations. Journal of Functional Analysis, 70(1), 126–139.

    Article  Google Scholar 

  • Wu, L. (1987). Construction de l’opérateur de malliavin sur l’espace de poisson. Séminaire de Probabilités, XXI, 100–113.

    Article  Google Scholar 

  • Wu, L. (2000). A new modified logarithmic sobolev inequality for poisson point processes and several applications. Probability Theory and Related Fields, 118(3), 427–438.

    Article  Google Scholar 

  • Zhang, R. (1999). Couverture approchée des options européennes. PhD thesis, Ecole des Ponts ParisTech.

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Correspondence to Takafumi Amaba.

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Amaba, T. A Discrete-Time Clark-Ocone Formula for Poisson Functionals. Asia-Pac Financ Markets 21, 97–120 (2014). https://doi.org/10.1007/s10690-013-9178-z

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