SABR volatility model

The model is defined by the following equations

$$ dX_t = \sigma_t \left(X_t\right)^\beta dW_t $$ $$ d\sigma_t = \alpha \sigma_t d \hat Z_t $$ $$ \hat Z_t = \rho W_t + \sqrt{1-\rho^2} Z_t $$

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

The notations differ slightly from those used in the celebrated paper Managing Smile Risk, Wilmott, 2002 by P. S. Hagan et al..

The values of the pair $(X,\sigma)$ are obtained through Monte Carlo simulation using an Euler scheme to discretize the stochastic equations mentioned above, with a time step of $\Delta t = 0.001$. The derivative prices are computed by simulating a large number of trajectories. In the approximation of expectations, only the trajectories where $X_T < 15$ are considered.

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 distribution of $X_t$ requires appending the file name to the URL

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

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

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

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

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