On a pricing problem for a multi-asset option with general transaction costs

TL;DR

This paper extends Leland's multi-asset option pricing model to include variable transaction costs, proving existence of solutions and developing a numerical scheme for practical computation.

Contribution

It generalizes Leland's condition to multi-asset settings with non-constant transaction costs and establishes the existence of viscosity solutions.

Findings

Proved existence of viscosity solutions for the generalized model.

Developed an ADI numerical scheme for approximation.

Applied the method to a specific multi-asset derivative.

Abstract

We consider a Black-Scholes type equation arising on a pricing model for a multi-asset option with general transaction costs. The pioneering work of Leland is thus extended in two different ways: on the one hand, the problem is multi-dimensional since it involves different underlying assets; on the other hand, the transaction costs are not assumed to be constant (i.e. a fixed proportion of the traded quantity). In this work, we generalize Leland's condition and prove the existence of a viscosity solution for the corresponding fully nonlinear initial value problem using Perron method. Moreover, we develop a numerical ADI scheme to find an approximated solution. We apply this method on a specific multi-asset derivative and we obtain the option price under different pricing scenarios.