TITLE:
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
AUTHORS:
Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli
KEYWORDS:
SABR Stochastic Volatility Models, Option Pricing, Spectral Decomposition, FX Data
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.2 No.7,
June
13,
2014
ABSTRACT: 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:
http://www.econ.univpm.it/recchioni/finance/w18 contains material including
animations, an interactive application and an app that helps the understanding
of the paper. A more general reference to the work of the authors and of their
coauthors in mathematical finance is the website:http://www.econ.univpm.it/recchioni/finance.