Liquidity risk, price impacts and the replication problem

We extend the model of liquidity risk of Çetin et al. [5] to allow for price impacts. Starting from simple principles, we show that the impact of a trade on prices is directly proportional to the size of the transaction and the amount of liquidity of the asset. This leads to a new characterization of self-financing trading strategies and a sufficient condition for no arbitrage. We show that, with the use of volatility swaps, contingent claims whose payoffs depend on the value of the asset can be approximately replicated. The replicating costs of such payoffs are obtained from the solutions of BSDEs with quadratic growth and analytical properties of these solutions are investigated.