The
Stein-Stein stochastic volatility model: transition probability density
function, moment formulae, option pricing, implied volatility, calibration
Lorella Fatone,
Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli
.
In [1] we give an integral representation formula of
the transition probability density function
of the Stein-Stein stochastic volatility model. Using this formula we
obtain explicit expressions for the moments of the log-price and of the
stochastic volatility, we price European options, we define implied volatility and we
formulate two calibration problems.
Examples of the calibration problems
involving real data are solved numerically. The real data considered are time series of daily data of the U.S.A. S&P
500 index and of the prices of some of
its European options or of the corresponding implied volatilities. The validity
of the Stein-Stein model and of the previous calibration problems is
established comparing the option prices
obtained with the calibrated models with the observed option prices. A 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.
[1] Fatone, L.,
Mariani, F., Recchioni, M.C., Zirilli, F: The Stein-Stein stochastic volatility
model: transition probability density function, moment formulae, option
pricing, implied volatility, calibration, submitted (2015).
[2] Stein, E.,
Stein, J.: Stock price distributions
with stochastic volatility: an analytic approach,- The Review of Financial
Studies, 4(4), (1991),
727-752.
[3] Fatone, L., Mariani, F., Recchioni, M.C., Zirilli, F.: The use of
statistical tests to calibrate the normal SABR model, Journal of Inverse
and Ill Posed Problems, 21(1),
(2013), 59-84, http://www.econ.univpm.it/recchioni/finance/w15/
.
[4] Fatone, L., Mariani, F., Recchioni, M.C.,
Zirilli, F.: Closed form moment formulae for
the lognormal SABR model and applications
to calibration problems, Open Journal of Applied Sciences, 3, (2013) , 345-359, http://www.econ.univpm.it/recchioni/finance/w16/
.
[5] Berestycky, H., Busca, J., Florent I.: Computing the implied volatility in stochastic
volatility models, Communications on Pure and Applied Mathematics, Vol. LVII,
(2004), 1352-1373.
[6] Fatone, L., Mariani, F. Recchioni, M.C., Zirilli, F.:
Some explicit formulae for the Hull and White stochastic volatility model,
International Journal of Modern Nonlinear Theory and Application 2, (2013), 14-33, http://www.econ.univpm.it/recchioni/finance/w17/