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

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Abstract

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.

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  • Kristoffer Glover & Peter W Duck & David P Newton, 2010. "On nonlinear models of markets with finite liquidity: Some cautionary notes," Published Paper Series 2010-5, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
  • Handle: RePEc:uts:ppaper:2010-5
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    File URL: https://epubs.siam.org/doi/abs/10.1137/080736119
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    References listed on IDEAS

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    1. Eckhard Platen & Martin Schweizer, 1998. "On Feedback Effects from Hedging Derivatives," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 67-84, January.
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    4. Umut Çetin & Robert A. Jarrow & Philip Protter, 2008. "Liquidity risk and arbitrage pricing theory," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 8, pages 153-183, World Scientific Publishing Co. Pte. Ltd..
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    Cited by:

    1. Peter W. Duck & Geoffrey W. Evatt & Paul V. Johnson, 2014. "Perpetual Options on Multiple Underlyings," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(2), pages 174-200, April.
    2. Kevin S. Zhang & Traian A. Pirvu, 2020. "Numerical Simulation of Exchange Option with Finite Liquidity: Controlled Variate Model," Papers 2006.07771, arXiv.org.
    3. Ahmad Reza Yazdanian & T A Pirvu, 2014. "Numerical analysis for Spread option pricing model in illiquid underlying asset market: full feedback model," Papers 1406.1149, arXiv.org.

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