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

Brace, Allan and Lauer, Mark and Rado, Milo (2007): A stylised model for extreme shocks: four moments of the apocalypse. Published in: Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney No. 224 (August 2007)

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.

• Item Type: MPRA Paper
• Language: English
• Keywords: Stress testing, stochastic volatility
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C60 - General

References

[1] Alexander, C. and Sheedy, E. 2007 Model-Based Stress Tests: Linking Stress Tests to VaR for Market Risk. International Capital Markets Association Centre Discussion Papers in Finance DP2007-02

[2] Australian Prudential Regulation Authority (APRA) 2000. Guidance Note AGN 113.2 – The Internal Model Approach.

[3] Berkowitz, J. 1999. A Coherent Framework for Stress-testing. Finance and Economics Discussion Series 1999-29, Board of Governors of the Federal Reserve System (U.S.).

[4] Committee on the Global Financial System 2005. A Survey of Stress Tests and Current Practice at Major Financial Institutions.

[5] Embrechts, P., Klüppelberg, C. and Mikosch, T. 1997. Modelling Extremal Events for Insurance and Finance. Springer, Berlin.

[6] Gençay, R., Selçuk, F. and Ulugülyaǧci, A. 2003. High Volatility, Thick Tails and Extreme Value Theory in Value-at-risk Estimation. In Insurance: Mathematics and Economics, Vol. 33, issue 2, pp337-356.

[7] Hull, J. and White, A. 1987. The Pricing of Options on Assets with Stochastic Volatilities. In Journal Of Finance, Vol. 42, pp281-300.

[8] Scott, L. 1987. Option Pricing When the Variance Changes Randomly – Theory, Estimation, and an Application. In Journal Of Financial And Quantitative Analysis, Vol. 22, No. 4, pp419-438.

[9] Wiggins, J. 1987. Option Values Under Stochastic Volatility – Theory and Empirical Estimates. In Journal Of Financial Economics, Vol. 19, pp351-372.