Our goal in this paper is to give a representation of the mean-variance hedging strategy for models whose asset price process is discontinuous as an extension of Gouriéroux, Laurent and Pham (1998) and Rheinländer and Schweizer (1997). However, we have to impose some additional assumptions related to the variance-optimal martingale measure. Copyright Springer-Verlag Berlin/Heidelberg 2005Mean-variance hedging, incomplete market, variance-optimal martingale measure, reverse Hölder inequality,

In this paper, we aim to obtain explicit representations of locally risk-minimizing by using Malliavin calculus for Lévy processes. For incomplete market models whose asset price is described by a solution to a stochastic differential equation driven by a Lévy process, we derive general formulas of locally risk-minimizing including Malliavin derivatives; and calculate its concrete expressions for call options, Asian options and lookback options.

We illustrate how to compute local risk minimization (LRM) of call options for exponential Lévy models. We have previously obtained a representation of LRM for call options; here we transform it into a form that allows use of the fast Fourier transform method suggested by Carr & Madan.In particular, we consider Merton jump-diffusion models and variance gamma models as concrete applications.and θ x := µ S (e x − 1) σ 2 + R 0 (e y − 1) 2 ν(dy) for x ∈ R 0 . In the development of our approach, we rely on the following: Assumption 1.1.1. R 0 (|x| ∨ x 2 )ν(dx) < ∞, and R 0 (e x − 1) n ν(dx) < ∞ for n = 2, 4.

We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no-arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized by several equivalent properties and in particular, we see that a convex risk measure is a good deal valuation only if it is given as a risk indifference price. An application to shortfall risk measure is given. In addition, we show that the no-free-lunch (NFL) condition is equivalent to the existence of a relevant convex risk measure which is a good deal valuation. The relevance turns out to be a condition for a good deal valuation to be reasonable. Further we investigate conditions under which any good deal valuation is relevant.