Modeling feedback effects with stochastic liquidity

We model the interactions between the trading activities of a large investor, the stock price and the market liquidity. Our framework generalizes the model of Frey (2000), where liquidity is constant by introducing a stochastic liquidity factor. This innovation has two implications. First, we can analyse trading strategies for the large investor that are affected by a changing market depth. Second, the sensitivity of stock process to the trading strategy of the large investor can vary due to changes in liquidity. Features of our model are demonstrated using Monte Carlo simulation for different scenarios. The flexibility of our framework is illustrated by an application that deals with the pricing of a liquidity derivative. The claim under consideration compensates a large investor who follows a stop loss strategy for the liquidity risk that is associated with a stop loss order. The derivative matures when the asset price falls below a stop loss limit for the first time and then pays the price difference between the asset price immediately before and after the execution of the stop loss order. The setup to price the liquidity derivative is calibrated for one example using real world limit order book data so that one gets an impression about the order of magnitude of the liquidity effect.