European Option Pricing With Liquidity Shocks

Abstract

Abstract. We study the valuation and hedging problem of European options in a market subject to liquidity shocks. Working within a Markovian regime-switching setting, we model illiquidity as the inability to trade. To isolate the impact of such liquidity constraints, we focus on the case where the market is completely static in the illiquid regime. We then consider derivative pricing using either equivalent martingale measures or exponential indifference mechanisms. Our main results concern the analysis of the semi-linear coupled HJB equation satisfied by the indifference price, as well as its asymptotics when the probability of a liquidity shock is small. We then present several numerical studies of the liquidity risk premia obtained in our models leading to practical guidelines on how to adjust for liquidity risk in option valuation and hedging.

Well posedness and comparison principle for option pricing with switching liquidity

“…The integro-differential equation in (1) is derived from a system of coupled parabolic PDE and ODE which is suggested by M. Ludkovski and Q. Shen [6] in European option pricing in a financial market switching between two states -a liquid state (0) and an illiquid (1) one. We briefly describe their model.…”

“…The supremum above is taken over all admissible trading strategies (π t ) and the function h(S) denotes the terminal payoff of a contingent claim. Standard stochastic control methods and the properties of the exponential utility function imply that the value functions can be presented by Û i (t, X, S) = −e −γX e −γR i (t,S) , i = 0, 1, where R i (t, S) are the unique viscosity solutions of the system ( [6])…”

IMEX schemes for a parabolic-ODE system of European options with liquidity shocks

“…Here σ is volatility of the underlying, ν 01 , ν 10 are transition intensities from state (0) to state (1) and vice versa, respectively, µ is drift of the underlying and d 0 = µ 2 /2σ 2 , see [5] for more details.…”

“…The presence of liquidity shocks is a source of non-liquidity risk and makes this market incomplete. Ludkowsky and Shen [5] investigate a nonlinear pricing mechanism based on utility maximization. They consider the investor whose utility is described by an exponential utility function…”

“…Here σ is volatility of the underlying, ν 01 , ν 10 are transition intensities from state (0) to state (1) and vice versa, respectively, µ is drift of the underlying and d 0 = µ 2 /2σ 2 , see [5] for more details. Using U i and V i , the buyer's indifference price p (initial state 0) and q (initial state 1) are defined via U 0 (t, X − p, S) = V 0 (t, X), U 1 (t, X − q, S) = V 1 (t, X), (5) where U , V are the optimal solutions for terminal wealth with and without options respectively. The value functions V i , i = 0, 1 are given by V i = e −γX F i (t) and V i (t, X, S) = e −γR i (t,S) , i = 1, 2…”