The SABR Model: Explicit Formulae of the Moments of the Forward Prices/Rates Variable and Series Expansions of the Transition Probability Density and of the Option Prices
The SABR stochastic volatility model with β-volatility β ? (0,1)and an
absorbing barrier in zero imposed to the forward prices/rates stochastic
process is studied. The presence of (possibly) nonzero correlation between the
stochastic differentials that appear on the right hand side of the model
equations is considered. A series expansion of the transition probability
density function of the model in powers of the correlation coefficient of these
stochastic differentials is presented. Explicit formulae for the first three
terms of this expansion are derived. These formulae are integrals of known
integrands. The zero-th order term of the expansion is a new integral formula
containing only elementary functions of the transition probability density
function of the SABR model when the correlation coefficient is zero. The
expansion is deduced from the final value problem for the backward Kolmogorov
equation satisfied by the transition probability density function. Each term of
the expansion is defined as the solution of a final value problem for a partial
differential equation. The integral formulae that give the solutions of these
final value problems are based on the Hankel and on the Kontorovich-Lebedev
transforms. From the series expansion of the probability density function we
deduce the corresponding expansions of the European call and put option prices.
Moreover we deduce closed form formulae for the moments of the forward
prices/rates variable. The moment formulae obtained do not involve integrals or
series expansions and are expressed using only elementary functions. The option
pricing formulae are used to study synthetic and real data. In particular we
study a time series (of real data) of futures prices of the EUR/USD currency's
exchange rate and of the corresponding option prices. The website:
References
[1]
Hagan, P.S., Kumar, D., Lesniewski, A.S. and Woodward, D.E. (2002) Managing Smile Risk. Wilmott Magazine, September, 84-108. http://www.wilmott.com/pdfs/021118_smile.pdf
[2]
Chen, B., Oosterlee, C.W. and van Weeren, S. (2010) Analytical Approximation to Constant Maturity Swap Convexity Corrections in a Multi-Factor SABR Model. International Journal of Theoretical and Applied Finance, 13, 1019-1046. http://dx.doi.org/10.1142/S0219024910006091
[3]
Doust, P. (2012) No-Arbitrage SABR. The Journal of Computational Finance, 15, 3-31.
[4]
Fatone, L., Mariani, F., Recchioni, M.C. and Zirilli, F. (2013) Some Explicitly Solvable SABR and Multiscale SABR Models: Option Pricing and Calibration. Journal of Mathematical Finance, 3, 10-32. http://dx.doi.org/10.4236/jmf.2013.31002
[5]
Fatone, L., Mariani, F., Recchioni, M.C. and Zirilli, F. (2013) The Use of Statistical Tests to Calibrate the Normal SABR Model. Journal of Inverse and Ill-Posed Problems, 21, 59-84. http://dx.doi.org/10.1515/jip-2012-0093
[6]
Fatone, L., Mariani, F., Recchioni, M.C. and Zirilli, F. (2013) Closed Form Moment Formulae for the Lognormal SABR Model Variables and Application to Calibration Problems. Open Journal of Applied Sciences, 3, 345-359. http://dx.doi.org/10.4236/ojapps.2013.36045
[7]
Forde, M. (2011) Exact Pricing and Large-Time Asymptotics for the Modified SABR Model and the Brownian Exponential Functional. International Journal of Theoretical and Applied Finance, 14, 559-578. http://dx.doi.org/10.1142/S0219024911006735
[8]
Hagan, P.S., Lesniewski, A.S. and Woodward, D.E. (2005) Probability Distribution in the SABR Model of Stochastic Volatility. http://lesniewski.us/papers/working/ProbDistrForSABR.eps
[9]
Islah, O. (2009) Solving SABR in Exact Form and Unifying It with LIBOR Market Model. http://papers.ssrn.com/sol3/papers.cfm abstract id=1489428 http://dx.doi.org/10.2139/ssrn.1489428
[10]
Andersen, L. and Andreasen, J. (2002) Volatile Volatilities. Risk, 15, 163-168.
[11]
Andersen, L. and Piterbarg, V.V. (2007) Moment Explosions in Stochastic Volatility Models. Finance and Stochastics, 11, 29-50. http://dx.doi.org/10.1007/s00780-006-0011-7
[12]
West, G. (2005) Calibration of the SABR Model in Illiquid Markets. Applied Mathematical Finance, 12, 371-385. http://dx.doi.org/10.1080/13504860500148672
[13]
Johnson, S. and Nonas, B. (2009) Arbitrage-Free Construction of the Swaption Cube. http://papers.ssrn.com/sol3/papers.cfm abstract id=1330869
[14]
Obloj, J. (2008) Fine-Tune Your Smile: Correction to Hagan et al. http://arxiv.org/abs/0708.0998
[15]
Forde, M. and Pogudin, A. (2013) The Large-Maturity Smile for the SABR and CEV-Heston Models. International Journal of Theoretical and Applied Finance, 16, Article ID: 1350047-1-1350047-20.
[16]
Wu, Q. (2012) Series Expansion of the SABR Joint Density. Mathematical Finance, 22, 310-345. http://dx.doi.org/10.1111/j.1467-9965.2010.00460.x
[17]
Antonov, A. and Spector, M. (2012) Advanced Analytics for the SABR Model. http://papers.ssrn.com/sol3/papers.cfm abstract id=2026350
[18]
Yakubovich, S.B. (2011) The Heat Kernel and Heisenberg Inequalities Related to the Kontorovich-Lebedev Transform. Communications on Pure and Applied Analysis, 10, 745-760. http://dx.doi.org/10.3934/cpaa.2011.10.745
[19]
Yakubovich, S.B. (2009) Beurling’s Theorems and Inversion Formulas for Certain Index Transforms. Opuscula Mathematica, 29, 93-110.
[20]
Ishiyama, K. (2005) Methods for Evaluating Density Functions of Exponential Functions Represented as Integrals of Geometric Brownian Motion. Methodology and Computing in Applied Probability, 7, 271-283. http://dx.doi.org/10.1007/s11009-005-4517-9
[21]
Abramowitz, M. and Stegun, I.A. (1970) Handbook of Mathematical Functions. Dover, New York.
[22]
Arfken, G.B. and Weber, H.J. (2005) Mathematical Methods for Physicists. 6th Edition, Academic Press, San Diego.
[23]
Szmytkowski, R. and Bielski, S. (2010) Comment on the Orthogonality of the Macdonald Functions of Imaginary Order. Journal of Mathematical Analysis and Applications, 365, 195-197. http://dx.doi.org/10.1016/j.jmaa.2009.10.035
[24]
Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. (1954) Tables of Integral Transforms. Vol. 1, McGrawHill Book Company, New York.
[25]
Bjrk, T. and Landén, C. (2002) On the Term Structure of Futures and Forward Prices. In: Geman, H., Madan, D., Pliska, S. and Vorst, T., Eds., Mathematical Finance—Bachelier Congress 2000, Springer Verlag, Berlin, 111-150. http://dx.doi.org/10.1007/978-3-662-12429-1_7
[26]
Glasser, M.L. and Montaldi, E. (1994) Some Integrals Involving Bessel Functions. Journal of Mathematical Analysis and Applications, 183, 577-590. http://dx.doi.org/10.1006/jmaa.1994.1164
[27]
Fatone, L., Mariani, F., Recchioni, M.C. and Zirilli, F. (2009) An Explicitly Solvable Multi-Scale Stochastic Volatility Model: Option Pricing and Calibration. Journal of Futures Markets, 29, 862-893. http://dx.doi.org/10.1002/fut.20390
