# A closed-form representation of mean-variance hedging for additive processes via Malliavin calculus

## TL;DR

This paper derives a new explicit closed-form representation for mean-variance hedging strategies in exponential additive models using Malliavin calculus, facilitating efficient numerical computation via fast Fourier transforms.

## Contribution

It introduces a novel closed-form formula for mean-variance hedging strategies applicable to jump models, enabling practical numerical implementation.

## Key findings

- Effective numerical methods using FFT are developed.
- Numerical results demonstrate the approach's accuracy in exponential Lévy models.
- The representation simplifies the computation of hedging strategies in jump models.

## Abstract

We focus on mean-variance hedging problem for models whose asset price follows an exponential additive process. Some representations of mean-variance hedging strategies for jump type models have already been suggested, but none is suited to develop numerical methods of the values of strategies for any given time up to the maturity. In this paper, we aim to derive a new explicit closed-form representation, which enables us to develop an efficient numerical method using the fast Fourier transforms. Note that our representation is described in terms of Malliavin derivatives. In addition, we illustrate numerical results for exponential L\'evy models.