### PAPER:

### Some explicit
formulae for the Hull
and White stochastic volatility model

__Lorella Fatone__,
Francesca Mariani, __Maria
Cristina Recchioni__, Francesco
Zirilli

An explicit formula
for the transition probability density function of the Hull and White stochastic volatility model in
presence of nonzero correlation between the stochastic differentials of the
Wiener processes on the right hand side of the model equations is presented.
This formula gives the transition probability density function as a two
dimensional integral of an explicitly known integrand. Previously an explicit
formula for this probability density function was known only in the case of
zero correlation. In the case of nonzero correlation from the formula for the
transition probability density function we deduce explicit formulae for the
price of European call and put options and some formulae for the moments of the
asset price logarithm. These formulae are based on recent results on the
Whittaker functions [13] and generalize similar
formulae for the SABR and multiscale SABR models [4].
Using the option pricing formulae derived and the least squares method a
calibration problem for the Hull
and White model whose data are a set of option prices is formulated and solved
numerically. Experiments with real data are presented. The real data studied
are those belonging to a time series of the USA S&P 500 index and of the
prices of its European call and put options. The quality of the model and of
the calibration procedure is established in two ways:

i) comparing the forecast
option prices obtained with the calibrated model with the option prices
observed in the financial market,

ii) performing an hedging
experiment using asset and option prices observed in the financial market.

This website contains some auxiliary
material including animations and interactive applications that helps the understanding
of the results presented in [7]. 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.

**Keywords.** Stochastic
volatility models, option pricing, calibration problem.

**AMS (MOS) Subject
Classification.** 62F10, 35R30, 91B70.

**M.S.C.
classification.** 34M50, 60H10, 91B28.

**JEL classification:** *C53, G12, C61*

__Content__

1.
__Hull____
and White model with nonzero correlation: formulae for the transition
probability density function – __(click here)

2.
__Hull____
and White model with nonzero correlation: formulae for the moments of the logarithm
of the asset price - an interactive application __(click here)

3.
__Hull____
and White model with non zero correlation: option price formulae - __(click here)

4.
__Calibration problem and
forecasting experiment using the __U.S.A.__ S&P 500 index and the
corresponding European call and put option price data - __(click here)__ __**(movie)**

5.
__Hedging experiment using the U.S.A. S&P 500 index and the
corresponding European call option prices – an interactive application __(click here)

6.
References - (click here)

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