Exchange options with stochastic liquidity risk

Abstract

In this article, we account for the liquidity risk in the underlying assets when pricing European exchange options, which has not been considered in the literature. An Ornstein–Uhlenbeck process with the mean-reversion property is selected to model the market liquidity risk, whose impacts on the underlying assets are assumed to be imposed with a liquidity discount factor. Under this framework, we develop a simplified approach to obtain the pricing formula, a distinguishing feature of which is that it does not require any numeraire change. We first transform the exchange option’s pricing PDE into a pricing PDE of the European vanilla option, which is then solved in closed-form using the characteristic function approach. Finally, we present accuracy tests and sensitivity analysis to demonstrate the correctness of the formula and the influence of introducing the liquidity risk on exchange options, respectively.