# A Clark-Ocone type formula via Ito calculus and its application to finance

## TL;DR

This paper extends the Clark-Ocone formula using Ito calculus to represent martingales related to Levy processes, enabling explicit strategies for digital options in finance without relying on Malliavin differentiability.

## Contribution

It introduces a new approach to martingale representation for non-Malliavin differentiable variables using Ito's formula, applied to financial derivatives.

## Key findings

- Explicit martingale representation for Levy process functionals.
- Representation of risk-minimizing strategies for digital options.
- Analysis of Malliavin differentiability of indicator functions.

## Abstract

An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which are not Malliavin differentiable. To this end, we make use of Ito's formula, instead of Malliavin calculus. As an application to mathematical finance, we shall give an explicit representation of locally risk-minimizing strategy of digital options for exponential Levy models. Since the payoff of digital options is described by an indicator function, we also discuss the Malliavin differentiability of indicator functions with respect to Levy processes.