On nonlinear models of markets with finite liquidity: Some cautionary notes

The recent financial crisis and related liquidity issues have illuminated an urgent need for a better understanding of the effects of limited liquidity on all aspects of the financial system. This paper considers such effects on the Black-Scholes-Merton financial model, which for the most part result in highly nonlinear partial differential equations (PDEs). We investigate in detail a model studied by Schönbucher and Wilmott (2000) which incorporates the price impact of option hedging strategies. First, we consider a first-order feedback model, which leads to the exceptional case of a linear PDE. Numerical results, and more particularly an asymptotic approach close to option expiry, reveal subtle differences from the Black-Scholes-Merton model. Second, we go on to consider a full-feedback model in which price impact is fully incorporated into the model. Here, standard numerical techniques lead to spurious results in even the simplest cases. An asymptotic approach, valid close to expiry, is mounted, and a robust numerical procedure, valid for all times, is developed, revealing two distinct classes of behavior. The first may be attributed to the infinite second derivative associated with standard option payoff conditions, for which it is necessary to admit solutions with discontinuous first derivatives; perhaps even more disturbingly, negative option values are a frequent occurrence. The second failure (applicable to smoothed payoff functions) is caused by a singularity in the coefficient of the diffusion term in the option-pricing equation. Our conclusion is that several classes of model in the literature involving permanent price impact irretrievably break down (i.e., there is insufficient "financial modeling" in the pricing equation). Our analysis should provide the information necessary to avoid such pitfalls in the future. © 2010 Society for Industrial and Applied Mathematics

On the Resolution of Critical Flow Regions in Inviscid Linear And Nonlinear Instability Calculations

Numerical methods for tackling the inviscid instability problem are discussed. Convergence is demon- strated to be a necessary, but not a sufficient condition for accuracy. Inviscid flow physics set requirements regarding grid-point distribution in order for physically accurate results to be obtained. These requirements are relevant to the viscous problem also and are shown to be related to the resolution of the critical layers. In this respect, high-resolution nonlinear calculations based on the inviscid initial-boundary-value problem are presented for a model shear-layer flow, aiming at identification of the regions that require attention in the course of high-Reynolds-number viscous calculations. The results bear a remarkable resemblance with those pertinent to viscous flow, with a cascade of high-shear regions being shed towards the vortex-core centre as time progresses. In parallel, numerical instability related to the finite-time singularity of the nonlinear equations solved globally contaminates and eventually destroys the simulations, irrespective of resolution