A closed-form pricing formula for European options with market liquidity risk

Abstract

In this paper, the impact of liquidity on the underlying asset is taken into account when pricing European options through a discounting factor which depends on two factors, i.e., market liquidity risk modeled as a mean-reverting stochastic process and the sensitivity of the underlying to market liquidity. A closed-form pricing formula for liquidity-adjusted European options is derived in the form of an infinite series using Karhunen–Loève expansion for the Ornstein–Uhlenbeck process. The convergence of the series solution is theoretically proved to guarantee closedness so that market practitioners can adopt the new formula when they need to account in market liquidity risk. The speed of convergence is demonstrated through numerical experiments. Finally, the accuracy of the newly derived formula is shown by comparing option prices calculated with our formula and those obtained from Monte-Carlo simulation, and various properties of our formula are also investigated.