Pricing and Optimal Liquidation of Immediacy

Abstract

A liquidity provider is approached by an impatient customer wishing to trade a fixed number of shares immediately. In determining whether to enter into the trade, the liquidity provider must consider how to price the initial trade and what to do with the position once it forms part of their inventory.Using techniques from two distinct areas of Mathematical Finance, namely option pricing and portfolio optimisation, we develop a model which addresses this problem. We introduce formulae to price the initial trade with the impatient customer and then develop trading strategies which maximise the mean of overall expected profits for a fixed level of variance. We go on to show that there is a minimum length of time, Tα, that the liquidity provider expects to bear a large proportion of risk associated with buying a fixed number of shares and then selling them optimally over time, that is unaffected by market or trade inputs. By extending Tα, expected profits rise at the expense of the associated variance. We construct an efficient frontier of all the attainable trading strategies, which are unique in their choice of Tα, where the curve defining the frontier maps levels of variance to the corresponding maximum expected overall profit of the trade described above.