# Distribution Approach to Local Volatility for European Options in the Merton Model with Stochastic Interest Rates

## TL;DR

This paper derives a formula for pricing European call options using a rigorous mathematical approach in a financial model that includes jumps and stochastic interest rates.

## Contribution

The paper extends the Dupire formula to the Merton model with stochastic interest rates using a distribution approach.

## Key findings

- The Dupire formula is derived for European call options in the Merton jump-diffusion model with stochastic interest rates.
- The approach uses a mathematically rigorous framework previously applied to the Margrabe option.
- The minimal entropy martingale measure is shown to be applicable as a risk-neutral measure in this context.

## Abstract

The Dupire formula is a very useful tool for pricing financial derivatives. This paper is dedicated to deriving the aforementioned formula for the European call option in the space of distributions by applying a mathematically rigorous approach developed in our previous paper concerning the case of the Margrabe option. We assume that the underlying asset is described by the Merton jump-diffusion model. Using this stochastic process allows us to take into account jumps in the price of the considered asset. Moreover, we assume that the instantaneous interest rate follows the Merton model (1973). Therefore, in contrast to the models combining a constant interest rate and a continuous underlying asset price process, frequently observed in the literature, applying both stochastic processes could accurately reflect financial market behaviour. Moreover, we illustrate the possibility of using the minimal entropy martingale measure as the risk-neutral measure in our approach.