Derivative pricing and optimal execution of portfolio transactions in finitely liquid markets

Abstract

In real markets, to some degree, every trade will incur a non-zero cost and will influence the price of the asset traded. In situations where a dynamic trading strategy is implemented these liquidity effects can play a significant role. In this thesis we examine two situations in which such trading strategies are inherent to the problem; that of pricing a derivative contingent on the asset and that of executing a large portfolio transaction in the asset. The asset's finite liquidity has been incorporated explicitly into its price dynamics using the Bakstein-Howison model [4]. Using this model we have derived the no-arbitrage price of a derivative on the asset and have found a true continuous-time equation when the bid-ask spread in the asset is neglected. Focussing on this pure liquidity case we then employ an asymptotic analysis to examine the price of a European call option near strike and expiry where the liquidity effects are shown to be most significant and closed-form expressions for the price are derived in this region. The asset price model is then extended to incorporate the empirical fact that an asset's liquidity mean reverts stochastically. In this situation the pricing equation is analyzed using the multiscale asymptotic technique developed by Fouque, Papanicolaou, and Sircar [22] and a simplified pricing and calibration framework is developed for an asset possessing liquidity risk. Finally, the derivative pricing framework (both with and without liquidity risk) is applied to a new contract termed the American forward which we present as a possible hedge against an asset's liquidity risk. In the second part of the thesis we investigate how to optimally execute a large transaction of a finitely liquid asset. Using stochastic dynamic programming and attempting only to minimize the transaction's cost, we first find that the optimal strategy is static and contains the naive strategy found in previous studies, but with an extra term to account for interest rates neglected by those studies. Including time risk into the optimization procedure we find expressions for the optimal strategy in the extreme cases when the trader's aversion to this risk is very small and very large. In the former case the optimal strategy is simply the cost-minimization strategy perturbed by a small correction proportional to the trader's level of risk aversion. In the latter case the problem is shown to be much more difficult; we analyze and derive implicit closed-form solutions to the much-simplified perfect liquidity case and show numerical results to demonstrate the agreement of the solution with our intuition