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HISTORICAL BACKTESTING OF LOCAL VOLATILITY MODEL USING AUD/USD VANILLA OPTIONS

Part of: Mathematical finance

Published online by Cambridge University Press:  17 February 2016

T. G. LING  and
P. V. SHEVCHENKO Open the ORCID record for P. V. SHEVCHENKO [Opens in a new window]
T. G. LING
Affiliation:
University of Technology, Sydney, NSW, Australia email [email protected]
P. V. SHEVCHENKO*
Affiliation:
CSIRO Computational Informatics and University of Technology Sydney, NSW, Australia email [email protected]
*
✉[email protected]
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Abstract

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The local volatility model is a well-known extension of the Black–Scholes constant volatility model, whereby the volatility is dependent on both time and the underlying asset. This model can be calibrated to provide a perfect fit to a wide range of implied volatility surfaces. The model is easy to calibrate and still very popular in foreign exchange option trading. In this paper, we address a question of validation of the local volatility model. Different stochastic models for the underlying asset can be calibrated to provide a good fit to the current market data, which should be recalibrated every trading date. A good fit to the current market data does not imply that the model is appropriate, and historical backtesting should be performed for validation purposes. We study delta hedging errors under the local volatility model using historical data from 2005 to 2011 for the AUD/USD implied volatility. We performed backtests for a range of option maturities and strikes using sticky delta and theoretically correct delta hedging. The results show that delta hedging errors under the standard Black–Scholes model are no worse than those of the local volatility model. Moreover, for the case of in- and at-the-money options, the hedging error for the Black–Scholes model is significantly better.

Keywords

local volatility modelbacktestingmodel validationforeign exchange optionsimplied volatility

MSC classification

Primary: 91G20: Derivative securities
Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 3 , January 2016 , pp. 319 - 338
Copyright
© 2016 Australian Mathematical Society 

References

Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Polit. Econ. 81 (1973) 637–654; doi:10.1086/260062.Google Scholar
Deelstra, G. and Rayee, G., “Local volatility pricing models for long-dated FX derivatives”, Appl. Math. Finance 20 (2013) 380–402; doi:10.1080/1350486X.2012.723516.Google Scholar
Derman, E. and Kani, I., “Riding on a smile”, Risk Mag. 7 (1994) 32–39 ;http://www.math.ku.dk/kurser/2005-1/finmathtowork/DermanKaniRISK.PDF.Google Scholar
Dupire, B., “Pricing with a smile”, Risk Mag. 7 (1994) 18–20 ;http://www.math.ku.dk/kurser/2005-1/finmathtowork/DupireRISK.PDF.Google Scholar
Feil, B., Kucherenko, S. and Shah, N., “Volatility calibration using spline and high dimensional model representation models”, Wilmott J. 1 (2009) 179–195; doi:10.1002/wilj.18.Google Scholar
Fengler, M. R., Semiparametric modelling of implied volatility (Springer, Berlin, 2005).Google Scholar
Glasserman, P., Monte Carlo methods in financial engineering, Volume 53 (Springer, New York, 2003); doi:10.1007/978-0-387-21617-1.CrossRefGoogle Scholar
Hagan, P. S., Kumar, D., Lesniewski, A. S. and Woodward, D. E., “Managing smile risk”, Wilmott Mag. (2002) 84–108 ; http://www.math.ku.dk/∼rolf/SABR.pdf.Google Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P., Numerical recipes, 3rd edn (Cambridge University Press, Cambridge, 2007).Google Scholar
Rebonato, R., Volatility and correlation: the perfect hedger and the fox, 2nd edn (John Wiley & Sons Ltd, Chichester, 2004) 836; doi:10.1002/9781118673539.CrossRefGoogle Scholar
Shevchenko, P. V., “Advanced Monte Carlo methods for pricing European-style options”, CMIS Technical Report CMIS 2001/148, CSIRO Mathematical and Information Sciences, 2001.Google Scholar
White, R., “Local volatility”, http://www.opengamma.com/sites/default/files/local-volatility- opengamma.pdf.Google Scholar
Wilmott, P., Paul Wilmott on quantitative finance, 2nd edn (John Wiley & Sons Ltd, Chichester, 2006) ; http://www.amazon.com/Wilmott-Quantitative-Finance-Volume-Edition/dp/0470018704.Google Scholar