On the Schwartz Distribution Approach to Local Volatility for European Option in a Generalised Jump Model with Stochastic Interest Rate

The Dupire formula is commonly regarded as a convenient analytical tool for the valuation of financial derivatives. In this paper, we continue the analytical treatment developed in our previous papers, deriving the aforementioned formula for the European call option in the space of distributions. We consider asset prices driven by Lévy-type stochastic integrals in a framework that allows flexible jump structures and contains a class of stochastic short-rate models, including the Vasicek model. We prove existence and uniqueness for the underlying SDE and establish the Dupire formula in this general setting. More explicit versions of the Dupire formulas are obtained for two models: Merton’s model with the Vasicek short rate, and a model with time-dependent jump scaling.