Heston volatility model

The model is defined by the following equations

$$ dX_t = X_t \left( \mu dt + \sqrt{\nu_t} dW_t \right) $$ $$ d\nu_t = \kappa (\theta - \nu_t) dt + \eta \sqrt{\nu_t} d \hat Z_t $$ $$ \hat Z_t = \rho W_t + \sqrt{1-\rho^2} Z_t $$

with $X_0 = 1$ and the risk-free rate $r = \mu = 0$.

The simuation is mainly based on the exact scheme presented in the paper Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes, OPERATIONS RESEARCH, 2006 by M, Broadie and Ö, Kaya. The only difference from this reference lies in the simulation of $\int \nu_t dt$ using a quadrature method, instead of the Fourier inversion method which involves the modified Bessel function of the first kind. The quadrature method provides sufficient accuracy, as it is implemented with a large number of time steps, with simulations performed using $\Delta t = 0.001$.

In addition to the information available in DataCarlo and AIvault, including details such as the name of each file to download, as well as the username and password, access to:

1. The neural network that generates the noncentral chi-square distribution requires appending the file name to the URL

   "mlp.lpma.math.upmc.fr/" + "AIvault/Heston/Asset/"

2. The neural network that generates the derivative price value requires appending the file name to the URL

   "mlp.lpma.math.upmc.fr/" + "AIvault/Heston/Derivatives/"

3. The training dataset requires appending the file name to the URL

   "mlp.lpma.math.upmc.fr/" + "DataCarlo/Heston/Training/"

4. The validation/training dataset requires appending the file name to the URL

   "mlp.lpma.math.upmc.fr/" + "DataCarlo/Heston/Testing/"