Closed Form Moment Formulae for the Lognormal SABR Model and Applications to Calibration Problems

Affiliation(s)

Dipartimento di Matematica e Informatica Università di Camerino via Madonna delle Carceri 9, Camerino, Italy.

Dipartimento di Scienze Economiche Università degli Studi di Verona Vicolo Campo_ore 2, Verona, Italy.

Dipartimento di Management Università Politecnica delle Marche Piazza Martelli 8, Ancona, Italy.

Dipartimento di Matematica “G. Castelnuovo” Università di Roma “La Sapienza” Piazzale Aldo Moro 2, Roma, Italy.

Dipartimento di Matematica e Informatica Università di Camerino via Madonna delle Carceri 9, Camerino, Italy.

Dipartimento di Scienze Economiche Università degli Studi di Verona Vicolo Campo_ore 2, Verona, Italy.

Dipartimento di Management Università Politecnica delle Marche Piazza Martelli 8, Ancona, Italy.

Dipartimento di Matematica “G. Castelnuovo” Università di Roma “La Sapienza” Piazzale Aldo Moro 2, Roma, Italy.

ABSTRACT

We study two calibration problems for the lognormal SABR model using
the moment method and some new formulae for the moments of the logarithm of the
forward prices/rates variable. The lognormal SABR model is a special case of
the SABR model [1]. The acronym “SABR” means “Stochastic-*α**β**ρ*” and comes from the original names of the model parameters (*i.e*., *α*,*β*,*ρ*) [1]. The SABR model is a system of two stochastic
differential equations widely used in mathematical finance whose independent
variable is time and whose dependent variables are the forward prices/rates and
the associated stochastic volatility. The lognormal SABR model corresponds to
the choice *β *= 1 and depends on three
quantities: the parameters *α*,*ρ* and the initial
stochastic volatility. In fact the initial stochastic volatility cannot be
observed and can be regarded as a parameter. A calibration problem is an
inverse problem that consists in determineing the values of
these three parameters starting from a set of data. We consider two different
sets of data, that is: i) the set of the forward prices/rates observed at a
given time on multiple independent trajectories of the lognormal SABR model,
ii) the set of the forward prices/rates observed on a discrete set of known
time values along a single trajectory of the lognormal SABR model. The
calibration problems corresponding to these two sets of data are formulated as
constrained nonlinear least-squares problems and are solved
numerically. The formulation of these nonlinear least-squares problems is based
on some new formulae for the moments of the logarithm of the forward
prices/rates. Note that in the financial markets the first set of data
considered is hardly available while the second set of data is of common use
and corresponds simply to the time series of the observed forward prices/rates.
As a consequence the first calibration problem although
realistic in several contexts of science and engineering is of limited interest
in finance while the second calibration problem is of practical use in finance
(and elsewhere). The formulation of these calibration problems and the methods
used to solve them are tested on synthetic and on real data. The real data
studied are the data belonging to a time series of exchange rates between
currencies (euro/U.S. dollar exchange rates).

Cite this paper

L. Fatone, F. Mariani, M. Recchioni and F. Zirilli, "Closed Form Moment Formulae for the Lognormal SABR Model and Applications to Calibration Problems,"*Open Journal of Applied Sciences*, Vol. 3 No. 6, 2013, pp. 345-359. doi: 10.4236/ojapps.2013.36045.

L. Fatone, F. Mariani, M. Recchioni and F. Zirilli, "Closed Form Moment Formulae for the Lognormal SABR Model and Applications to Calibration Problems,"

References

[1] P. S. Hagan, D. Kumar, A. S. Lesniewski and D. E. Woodward, “Managing Smile Risk,” Wilmott Magazine, 2002, pp. 84-108. http://www.wilmott.com/pdfs/021118_smile.pdf

[2] J. C. Cox, J. E. Ingersoll and S. A. Ross, “A Theory of the Term Structure of Interest Rates,” Econometrica, Vol. 53, No. 2, 1985, pp. 385-407.

[3] F. Black and M. Scholes, “c,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-659.

[4] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “The Use of Statistical Tests to Calibrate the Normal SABR Model,” Journal of Inverse and IllPosed Problems, Vol. 21, No. 1, 2013, pp. 59-84. http://www.econ.univpm.it/recchioni/finance/w15

[5] S. Mergner, “Application of State Space Models in Finance,” Universitätsverlag Göttingen, Göttingen, 2004.

[6] F. Mariani, G. Pacelli and F. Zirilli, “Maximum Likelihood Estimation of the Heston Stochastic Volatility Model Using Asset and Option Prices: An Application of Nonlinear Filtering Theory,” Optimization Letters, Vol. 2, No. 2, 2008, pp. 177-222. http://www.econ.univpm.it/pacelli/mariani/finance/w1.

[7] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “Maximum Likelihood Estimation of the Parameters of a System of Stochastic Differential Equations That Models the Returns of the Index of Some Classes of Hedge Funds,” Journal of Inverse and Ill-Posed Problems, Vol. 15, No. 5, 2007, pp. 329-362. http://www.econ.univpm.it/recchioni/finance/w5

[8] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “Some Explicitly Solvable SABR and Multiscale SABR Models: Option Pricing and Calibration,” Journal of Mathematical Finance, Vol. 3, No. 1, 2013, pp. 10-32. http://dx.doi.org/10.4236/jmf.2013.31002 http://www.econ.univpm.it/recchioni/finance/w14

[9] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, “Higher Trascendental Functions,” McGraw-Hill Book Company, New York, 1953.

[10] J. Hull and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” The Journal of Finance, Vol. 42, No. 2, 1987, pp. 281-300. http://dx.doi.org/10.1111/j.1540-6261.1987.tb02568.x

[11] B. A. Surya, “Two-Dimensional Hull-White Model for Stochastic Volatility and Its Nonlinear Filtering Estimation,” Procedia Computer Science, Vol. 4, 2011, pp. 1431-1440. http://dx.doi.org/10.1016/j.procs.2011.04.154

[12] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, “Tables of Integral Transforms,” McGraw-Hill Book Company, New York, 1954.

[13] P. L. Lions and M. Musiela, “Correlations and Bounds for Stochastic Volatility Models,” Annales de l’Institut Henri Poincaré (C) Analyse Non Linèaire, Vol. 24, No. 1, 2007, pp. 1-16.

[14] P. Wilmott, “Paul Wilmott Introduces Quantitative Finance,” Wiley Desktop Editions, Chichester, 2007.

[15] http://www.econ.univpm.it/recchioni/finance/w16

[1] P. S. Hagan, D. Kumar, A. S. Lesniewski and D. E. Woodward, “Managing Smile Risk,” Wilmott Magazine, 2002, pp. 84-108. http://www.wilmott.com/pdfs/021118_smile.pdf

[2] J. C. Cox, J. E. Ingersoll and S. A. Ross, “A Theory of the Term Structure of Interest Rates,” Econometrica, Vol. 53, No. 2, 1985, pp. 385-407.

[3] F. Black and M. Scholes, “c,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-659.

[4] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “The Use of Statistical Tests to Calibrate the Normal SABR Model,” Journal of Inverse and IllPosed Problems, Vol. 21, No. 1, 2013, pp. 59-84. http://www.econ.univpm.it/recchioni/finance/w15

[5] S. Mergner, “Application of State Space Models in Finance,” Universitätsverlag Göttingen, Göttingen, 2004.

[6] F. Mariani, G. Pacelli and F. Zirilli, “Maximum Likelihood Estimation of the Heston Stochastic Volatility Model Using Asset and Option Prices: An Application of Nonlinear Filtering Theory,” Optimization Letters, Vol. 2, No. 2, 2008, pp. 177-222. http://www.econ.univpm.it/pacelli/mariani/finance/w1.

[7] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “Maximum Likelihood Estimation of the Parameters of a System of Stochastic Differential Equations That Models the Returns of the Index of Some Classes of Hedge Funds,” Journal of Inverse and Ill-Posed Problems, Vol. 15, No. 5, 2007, pp. 329-362. http://www.econ.univpm.it/recchioni/finance/w5

[8] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “Some Explicitly Solvable SABR and Multiscale SABR Models: Option Pricing and Calibration,” Journal of Mathematical Finance, Vol. 3, No. 1, 2013, pp. 10-32. http://dx.doi.org/10.4236/jmf.2013.31002 http://www.econ.univpm.it/recchioni/finance/w14

[9] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, “Higher Trascendental Functions,” McGraw-Hill Book Company, New York, 1953.

[10] J. Hull and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” The Journal of Finance, Vol. 42, No. 2, 1987, pp. 281-300. http://dx.doi.org/10.1111/j.1540-6261.1987.tb02568.x

[11] B. A. Surya, “Two-Dimensional Hull-White Model for Stochastic Volatility and Its Nonlinear Filtering Estimation,” Procedia Computer Science, Vol. 4, 2011, pp. 1431-1440. http://dx.doi.org/10.1016/j.procs.2011.04.154

[12] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, “Tables of Integral Transforms,” McGraw-Hill Book Company, New York, 1954.

[13] P. L. Lions and M. Musiela, “Correlations and Bounds for Stochastic Volatility Models,” Annales de l’Institut Henri Poincaré (C) Analyse Non Linèaire, Vol. 24, No. 1, 2007, pp. 1-16.

[14] P. Wilmott, “Paul Wilmott Introduces Quantitative Finance,” Wiley Desktop Editions, Chichester, 2007.

[15] http://www.econ.univpm.it/recchioni/finance/w16