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A Stochastic Volatility Alternative to SABR

Part of: Mathematical economics Special processes Applications Markov processes

Published online by Cambridge University Press:  14 July 2016

L. C. G. Rogers  and
L. A. M. Veraart
L. C. G. Rogers*
Affiliation:
University of Cambridge
L. A. M. Veraart*
Affiliation:
Princeton University
*
∗Postal address: Statistical Laboratory, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK. Email address: [email protected]
∗∗Current address: Institut für Stochastik, Universität Karlsruhe (TH), Kaiserstr. 89, 76133 Karlsruhe, Germany.
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Abstract

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We present two new stochastic volatility models in which option prices for European plain-vanilla options have closed-form expressions. The models are motivated by the well-known SABR model, but use modified dynamics of the underlying asset. The asset process is modelled as a product of functions of two independent stochastic processes: a Cox-Ingersoll-Ross process and a geometric Brownian motion. An application of the models to options written on foreign currencies is studied.

Keywords

SABREuropean optionvolatility smile

MSC classification

Secondary: 91B70: Stochastic models 60K99: None of the above, but in this section 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 4 , December 2008 , pp. 1071 - 1085
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau Stand. Appl. Math. Ser. 55). US Government Printing Office, Washington.Google Scholar
[2] Andersen, L. and Piterbarg, V. (2007). Moment explosions in stochastic volatility models. Finance Stoch. 11, 29–50.CrossRefGoogle Scholar
[3] Benaim, S. (2007). Regular variation and smile asymptotics. , University of Cambridge.Google Scholar
[4] Bisesti, L., Castagna, A. and Mercurio, F. (2005). Consistent pricing and hedging of an FX options book. Kyoto Econom. Rev. 74, 65–83.Google Scholar
[5] Cox, J. (1996). Notes on option pricing I: constant elasticity of variance diffusions. J. Portfolio Manag. 22, 15–17.Google Scholar
[6] Cox, J. C., Ingersoll, J. E. Jr. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53, 385–407.Google Scholar
[7] Delbaen, F. and Shirakawa, H. (2002). A note on option pricing for the constant elasticity of variance model. Asia-Pacific Financial Markets 9, 85–99.CrossRefGoogle Scholar
[8] Dupire, B. (1992). Arbitrage pricing with stochastic volatility. In Proc. AFFI Conf. (June 1992), Paris.Google Scholar
[9] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering (Appl. Math. 53). Springer, New York.Google Scholar
[10] Göing-Jaeschke, A. and Yor, M. (2003). A survey and some generalizations of Bessel processes. Bernoulli 9, 313–349.Google Scholar
[11] Hagan, P. S., Kumar, D., Lesniewski, A. S. and Woodward, D. E. (2002). Managing smile risk. Wilmott Magazine, pp. 84–108.Google Scholar
[12] Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343.CrossRefGoogle Scholar
[13] Hofmann, N., Platen, E. and Schweizer, M. (1992). Option pricing under incompleteness and stochastic volatility. Math. Finance 2, 153–187.Google Scholar
[14] Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities. J. Finance 42, 281–300.Google Scholar
[15] Hull, J. and White, A. (1988). An analysis of the bias in option pricing caused by a stochastic volatility. Adv. Futures Options Res. 3, 61.Google Scholar
[16] Jäckel, P. and Kahl, C. (2007). Hyp hyp hooray. Preprint. Available at http://www.jaeckel.org/.Google Scholar
[17] Johnson, H. and Shanno, D. (1987). Option pricing when the variance is changing. J. Financial Quant. Anal. 22, 143–151.Google Scholar
[18] Lamberton, D. and Lapeyre, B. (1996). Introduction to Stochastic Calculus Applied to Finance. Chapman and Hall, London.Google Scholar
[19] Nelder, J. and Mead, R. (1965). A simplex method for function minimization. Comput. J. 7, 308–313.Google Scholar
[20] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes and Martingales, Vol. 2. Cambridge University Press.Google Scholar
[21] Scott, L. (1987). Option pricing when the variance changes randomly: theory, estimation, and an application. J. Financial Quant. Anal. 22, 419–438.Google Scholar
[22] Sin, C. A. (1998). Complications with stochastic volatility models. Adv. Appl. Prob. 30, 256–268.Google Scholar
[23] Stein, E. and Stein, J. (1991). Stock price distributions with stochastic volatility: an analytic approach. Rev. Financial Studies 4, 727–752.CrossRefGoogle Scholar
[24] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Springer, New York.Google Scholar
[25] Wiggins, J. (1987). Option values under stochastic volatility: theory and empirical estimates. J. Financial Econom. 19, 351–372.Google Scholar