Market liquidity risk refers to the degree to which large size transactions can be carried out in a timely fashion with a minimal impact on prices. Emphasized by the G10 report in 1993 and the BIS report in 1997, it is viewed as one factor of destabilization in the financial markets, as illustrated recently by the Asian crisis, the faillure of the hedge fund LTCM during the Russian crisis. So in order to assess welfare implications of portfolio insurance strategies, it would be useful to estimate the dynamic hedging activity in securities markets through a specific parsimonious and realistic model. In the paper, large traders hold sufficient liquid assets to meet liquidity needs of other traders, and so bear the risk of their imbalanced derivatives portfolio. Their dynamic hedging strategies entail non-linear positive feedback effects, and in turn buying and selling derivatives at prices shifted by an amount that depends on their net holding. And therefore, the replicating equation turns to be a fully nonlinear parabolic PDE, as proposed by Frey [10]. It turns out that such a nonlinear PDE equation may be numerically unstable when using traditional finite-difference methods. Therefore we need some specific adequate numerical implementation in order to solve this equation with significant accuracy and flexibility, while keeping stability. In this respect paper we devised and customized two different numerical methods: one is a refined finite difference method; the other involves the probabilistic scheme proposed by Fahim and al. [9]. In contrast, another method based on Lie algebra and developed by Bordag and al. only provides a generic, albeit analytical, formulation of solutions, and not the specific one consistent with our payoff. Still, that method offers a reference for our proposed methods in terms of numerical accuracy. Using such a framework, a Large Player is then in a position to take into account those positive feedback effects in dynamic hedging. Lastly, we show how dynamic hedging may directly and endogenously give rise to empirically observed bid-offer spreads.
