Note on multidimensional Breeden-Litzenberger representation for state price densities

TL;DR

This paper extends the Breeden-Litzenberger formula to multiple underlying assets, providing a multidimensional framework for relating derivatives of option prices to joint risk-neutral densities.

Contribution

It introduces a multidimensional version of the Breeden-Litzenberger result for European options on multiple assets, broadening the theoretical understanding of state price densities.

Findings

Derives a multidimensional relation between option price derivatives and joint densities.

Assumes absolute continuity of the pricing measure with respect to Lebesgue measure.

Provides a theoretical foundation for multi-asset option pricing analysis.

Abstract

In this note, we consider European options of type $h(X^1_T, X^2_T,\ldots, X^n_T)$ depending on several underlying assets. We give a multidimensional version of the result of Breeden and Litzenberger \cite{Breeden} on the relation between derivatives of the call price and the risk-neutral density of the underlying asset. The pricing measure is assumed to be absolutely continuous with respect to the Lebesgue measure on the state space.