Explicit Representation of the Minimal Variance Portfolio in Markets Driven by Lévy Processes

In a market driven by a Levy martingale, we consider a claim E. We study the problem of minimal variance hedging and we give an explicit formula for the minimal variance portfolio in terms of Malliavin derivatives. We discuss two types of stochastic (Malliavin) derivatives for E: one based on the chaos expansion in terms of iterated integrals with respect to the power jump processes and one based on the chaos expansion in terms of iterated integrals with respect to the Wiener process and the Poisson random measure components. We study the relation between these two expansions, the corresponding two derivatives, and the corresponding versions of the Clark-Haussmann-Ocone theorem.

[1]  Ken-iti Sato.  Lévy Processes and Infinitely Divisible Distributions , 1999 .

[2]  Cho-Hoi Hui,et al.  Comment on ‘Pricing double barrier options using Laplace transforms’ by Antoon Pelsser , 2000, Finance Stochastics.

[3]  Yu. A. Rozanov,et al.  On Stochastic Integration and Differentiation , 1999 .

[4]  Arne Løkka,et al.  Martingale Representation of Functionals of Lévy Processes , 2004 .

[5]  D. Sondermann.  Hedging of non-redundant contingent claims , 1985 .

[6]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[7]  H. Föllmer,et al.  Hedging of contingent claims under incomplete in-formation , 1991 .

[8]  Jorge A. León,et al.  On Lévy processes, Malliavin calculus and market models with jumps , 2002, Finance Stochastics.

[9]  池田 信行,et al.  Stochastic differential equations and diffusion processes , 1981 .

[10]  T. Chan.  Pricing contingent claims on stocks driven by Lévy processes , 1999 .

[11]  P. Protter.  Stochastic integration and differential equations , 1990 .

[12]  Stochastic integral representations, stochastic derivatives and minimal variance hedging , 2002 .

[13]  B. Øksendal.  AN INTRODUCTION TO MALLIAVIN CALCULUS WITH APPLICATIONS TO ECONOMICS , 1996 .

[14]  Paul Malliavin,et al.  Stochastic Analysis , 1997, Nature.

[15]  Bernt Øksendal,et al.  White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance , 2000, Finance Stochastics.

[16]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

[17]  Kiyosi Itô,et al.  SPECTRAL TYPE OF THE SHIFT TRANSFORMATION OF DIFFERENTIAL PROCESSES WITH STATIONARY INCREMENTS( , 1956 .

[18]  D. Nualart,et al.  Chaotic and predictable representation for L'evy Processes , 2000 .

[19]  D. Nualart,et al.  Backward stochastic differential equations and Feynman-Kac formula for Levy processes, with applications in finance , 2001 .

[20]  Ioannis Karatzas,et al.  An extension of clark' formula , 1991 .

[21]  M. Schweizer.  A guided tour through quadratic hedging approaches , 1999 .