Liquidity Risk, Price Impacts and the Replication Problem

TL;DR

This paper extends a liquidity risk model to include price impacts, analyzes the effects of transaction size and liquidity, and explores approximate replication of contingent claims using variance swaps within a stochastic volatility framework.

Contribution

It introduces a linear liquidity risk model with price impacts, characterizes self-financing strategies, and demonstrates approximate replication of asset-dependent payoffs via BSDEs with quadratic growth.

Findings

Price impacts depend on transaction size and liquidity level.

Contingent claims can be approximately replicated using variance swaps.

Analytical properties of BSDE solutions are investigated.

Abstract

We extend a linear version of the liquidity risk model of Cetin et al. (2004) to allow for price impacts. We show that the impact of a market order on prices depends on the size of the transaction and the level of liquidity. We obtain a simple characterization of self-financing trading strategies and a sufficient condition for no arbitrage. We consider a stochastic volatility model in which the volatility is partly correlated with the liquidity process and show that, with the use of variance swaps, contingent claims whose payoffs depend on the value of the asset can be approximately replicated in this setting. The replicating costs of such payoffs are obtained from the solutions of BSDEs with quadratic growth and analytical properties of these solutions are investigated.