# Calibration: Comparison of deterministic Optimizers

The gaol is to calibrate a time dependent Heston model defined by the following SDE



The parameter set $\{\kappa_t, \theta_t, ,\sigma_t, \rho_t\}$ is supposed to be piecewise constant in time. This model has a semi-closed solution for plain vanilla European put/call options based on the characteristic function method [1].

The DAX implied volatility surface based on July 5, 2002 data taken from [2] defines the “benchmark” calibration problem. The benchmark model parameters for the optimization problem are given by

$\Theta = \{\nu_0, \kappa_{0\leq t < 0.25}, \kappa_{0.25 \leq t},\theta, \sigma, \rho\}$.

The non-linear least square optimization problem is defined by the goodness of fit measure

$\zeta=\min_{\Theta} \sum_{i=1}^N \left( \sigma_i^{market}(K,T) - \sigma_i^{model}(K,T, \Theta) \right)^2$

where $\sigma_i^{market}(K,T)$ is the market implied volatility for strike K and maturity T and $\sigma_i^{model}(K,T, \Theta)$ is the corresponding Black-Scholes volatility implied from the model price. The optimal solution for this problem is

$\Theta_{min} = \{ 0.2231, 39.651, 7.546, 0.0954, 5.1865, -0.5004 \}$

leading to a goodness of fit measure of $\zeta_{min}=74.4731$ (Please keep in mind that this result is the outcome of a naive calibration procedure. Due to the large $\kappa$ and $\sigma$ values I’d not use these parameters to price a derivative.).

The diagram above shows the “goodness of fit”-surface for the parameter sets in

$\Theta = \{ 0.2231, 39.651, \kappa_{0.25 \leq t} \in [0, 16], 0.0954, \sigma \in [0, 16], -0.5004 \}$

To be able to compare a larger number of deterministic optimizers the model calibration will be carried out using R and with help of the additional packages minpack.lm and minqa.

Non-linear Least Square Optimization:

1. nls.lm: Levenberg-Marquardt algorithm(based on MINPACK, also available in QuantLib)
2. nls: Gauss-Newton algorithm
3. nl2sol:based on the PORT library.

Non-linear (trusted region) Minimization:

1. nlm:Newton style minizer
3. l-bfgs-b: limited memory BFGS algorithm incl.box constraints