×

A stochastic volatility alternative to SABR. (English) Zbl 1152.91683

Summary: 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.

MSC:

91B70 Stochastic models in economics
60K99 Special processes
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
62P05 Applications of statistics to actuarial sciences and financial mathematics
PDF BibTeX XML Cite
Full Text: DOI

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. · Zbl 0171.38503
[2] Andersen, L. and Piterbarg, V. (2007). Moment explosions in stochastic volatility models. Finance Stoch. 11 , 29–50. · Zbl 1142.65004
[3] Benaim, S. (2007). Regular variation and smile asymptotics. Doctoral Thesis, University of Cambridge. · Zbl 1155.91377
[4] Bisesti, L., Castagna, A. and Mercurio, F. (2005). Consistent pricing and hedging of an FX options book. Kyoto Econom. Rev. 74 , 65–83.
[5] Cox, J. (1996). Notes on option pricing I: constant elasticity of variance diffusions. J. Portfolio Manag. 22 , 15–17.
[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. JSTOR: · Zbl 1274.91447
[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. · Zbl 1072.91020
[8] Dupire, B. (1992). Arbitrage pricing with stochastic volatility. In Proc. AFFI Conf. (June 1992), Paris.
[9] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering (Appl. Math. 53 ). Springer, New York. · Zbl 1038.91045
[10] Göing-Jaeschke, A. and Yor, M. (2003). A survey and some generalizations of Bessel processes. Bernoulli 9 , 313–349. · Zbl 1038.60079
[11] Hagan, P. S., Kumar, D., Lesniewski, A. S. and Woodward, D. E. (2002). Managing smile risk. Wilmott Magazine , pp. 84–108.
[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. · Zbl 1384.35131
[13] Hofmann, N., Platen, E. and Schweizer, M. (1992). Option pricing under incompleteness and stochastic volatility. Math. Finance 2 , 153–187. · Zbl 0900.90095
[14] Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities. J. Finance 42 , 281–300. · Zbl 1126.91369
[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.
[16] Jäckel, P. and Kahl, C. (2007). Hyp hyp hooray. Preprint. Available at http://www.jaeckel.org/.
[17] Johnson, H. and Shanno, D. (1987). Option pricing when the variance is changing. J. Financial Quant. Anal. 22 , 143–151.
[18] Lamberton, D. and Lapeyre, B. (1996). Introduction to Stochastic Calculus Applied to Finance . Chapman and Hall, London. · Zbl 0949.60005
[19] Nelder, J. and Mead, R. (1965). A simplex method for function minimization. Comput. J. 7 , 308–313. · Zbl 0229.65053
[20] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes and Martingales , Vol. 2. Cambridge University Press. · Zbl 0977.60005
[21] Scott, L. (1987). Option pricing when the variance changes randomly: theory, estimation, and an application. J. Financial Quant. Anal. 22 , 419–438.
[22] Sin, C. A. (1998). Complications with stochastic volatility models. Adv. Appl. Prob. 30 , 256–268. · Zbl 0907.90026
[23] Stein, E. and Stein, J. (1991). Stock price distributions with stochastic volatility: an analytic approach. Rev. Financial Studies 4 , 727–752. · Zbl 1458.62253
[24] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes . Springer, New York. · Zbl 0426.60069
[25] Wiggins, J. (1987). Option values under stochastic volatility: theory and empirical estimates. J. Financial Econom. 19 , 351–372.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.