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A stylised model for extreme shocks: four moments of the apocalypse

Author

Listed:
  • Brace, Allan
  • Lauer, Mark
  • Rado, Milo

Abstract

We present a method for calculating the extreme tail quantiles, over arbitrary holding periods, of a continuous-time stochastic volatility model of the form proposed by Scott (1987) with correlation between the processes for volatility and price. The fat tails of this model enable a consistent, tuneable, stylised representation of non-normality in extreme moves of prices across differing markets. Because the model is analytically intractable, four moments are derived by numeric integration and matched to a one-period version of the model, whose quantiles are then found by further numeric integration. We also present a novel Monte-Carlo simulation scheme, which we have used to confirm the accuracy of the moment-matching approximation for quantiles as extreme as one-millionth. Two methods for calibrating the model to market data are also proposed. The model is used in production stress testing at nabCapital to define consistent real-world probabilities for extreme shocks over heterogeneous holding periods.

Suggested Citation

  • Brace, Allan & Lauer, Mark & Rado, Milo, 2007. "A stylised model for extreme shocks: four moments of the apocalypse," MPRA Paper 124773, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:124773
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    References listed on IDEAS

    as
    1. Jeremy Berkowitz, 1999. "A coherent framework for stress-testing," Finance and Economics Discussion Series 1999-29, Board of Governors of the Federal Reserve System (U.S.).
    2. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
    3. Carol Alexander & Elizabeth Sheedy, 2007. "Model-Based Stress Tests: Linking Stress Tests to VaR for Market Risk," ICMA Centre Discussion Papers in Finance icma-dp2007-02, Henley Business School, University of Reading.
    4. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(4), pages 419-438, December.
    5. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    Full references (including those not matched with items on IDEAS)

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    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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