Abstract:We study the problem of obtaining accurate estimates
of the parameters,of the initial stochastic variances and of
the risk premium parameteres of the risk neutral measure of
some stochastic volatility models used in mathematical
finance from the observations at discrete times of the asset
log return and of the prices of European call and put
options on the asset.The models considered include the
Heston stochastic volatility model and its
generalizations.This problem is an inverse problem for a
stochastic dynamical system known in the literature as
calibration problem.We formulate the calibration problem as
a constrained optimization problem where the objective
function is the logarithm of the likelihood associated to
the parameters,the initial stochastic variances and the risk
premium parameters of the model studied.The expression of
the likelihood function contains the solution of a filtering
problem.Form the solution of the calibration problem and of
the associated filtering problem we derive a tracking
procedure that is able to forecast prices for time values
where data are not available.We make numerical experiments
with synthetic data and with real data.The real data
considered are those relative to the S&P500 index in the
year 2005 and some electric power price data taken from the
U.S. market.In particular we compare forecasted prices
obtained with the tracking procedure mentioned previously
with prices actually observed. The results obtained are very
satisfactory.
Some auxiliary material useful to understand this work
including some animations can be found in the websites:
http://www.econ.univpm.it/recchioni/finance/w6
http://www.econ.univpm.it/recchioni/finance/w9
A more general reference to the work of the author and of
its coauthors in mathematical finance is the website:
http://www.econ.univpm.it/recchioni/finance