Calibration method

Foundation of the method

Comment

1

Generalized method of moments

The unknown parameters are obtained minimizing the difference between the moments defined by the model of some random variable and the same moments estimated from the data.

This is the most popular calibration method. Option prices can be seen as moments of random variables. In this case the calibration can be reduced to the least squares fit of theoretical and observed option prices.

2

Maximum Entropy

The unknown parameters are obtained maximizing the entropy of the probability distribution of the asset price and of its stochastic volatility conditioned to the observations. The entropy is a measure of the randomness of a probability distribution.

This method is based on ideas taken from statistical mechanics. Both option prices and asset prices can be used as data. The entropy function can be substituted with an approximate entropy function to save computation.

3

Maximum Likelihood

The unknown parameters are obtained maximizing a likelihood function associated to the data. The likelihood function measures the probability of obtaining the observed data as a function of the parameter values.

Both option prices and asset prices can be used as data. The method exploits the fact that the joint probability distribution of the state variables of the models considered admits a simple integral representation formula.

4

Statistical Tests

The unknown parameters are determined using statistical hypothesis testing. Ad hoc statistical tests are developed to exploit the available data.

This method associates to the values of the parameters determined as solution of the calibration problem a statistical confidence. The ad hoc statistical tests built are based on sampling suitable random variables using numerical simulation.