Explicit Representation of the Minimal Variance Portfolio in Markets Driven by Lévy Processes
Tóm tắt
In a market driven by a Lévy martingale, we consider a claim ξ. 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 ξ: 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.
Từ khóa
Tài liệu tham khảo
Bertoin J., 1996, Lévy Processes
Di Nunno G., 2002, Stochastic Integral Representations, Stochastic Derivatives and Minimal Variance Hedging, Preprint series in Pure Mathematics, 73, 181
Föllmer H. andM.Schweizer(1991):Hedging of Contingent Claims under Incomplete Information; in Applied Stochastic Analysis: Stochastics Monograp5 eds.M. H. A.DavisandA.Mas‐Colell. New York :Gordon and Breach 389–414.
Föllmer H. andD.Sondermann(1986):Hedging of Non‐Redundant Contingent Claims in Contributions to Mathematical Economics eds.W.HildenbrandandA.Mas‐Colell. Amsterdam : North Holland 205–223.
Ikeda N., 1989, Stochastic Differential Equations and Diffusion Processes
Lamberton D., 1996, Introduction to Stochastic Calculus Applied to Finance
Løkka A.(2001):Martingale Representation of Functionals of Lévy Processes.Preprint series in Pure Mathematics University of Oslo 21.
Øksendal B.(1996):An Introduction to Malliavin Calculus with Applications to Economics Working paper 3 Norwegian School of Economics and Business Administration Bergen .
Sato K., 1999, Lévy Processes and Infinitely Divisible Distributions