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

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  and generalize similar formulae for the SABR and multiscale SABR models . 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 . 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)

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