neuralSDE

Target data

In both approaches target data is produced by Monte Carlo simulations under SABR model

$\text{d}F_t = \sigma_t F_t^{\beta}\text{d}W_t, F_0 = f$

$\text{d}\sigma_t = \alpha \sigma_t \text{d}B_t, \sigma_0 = \sigma $

$\text{d}W_t\text{d}B_t = \rho \text{d}t$

with following parameters:

• $f = 1$ (initial forward price)
• $\sigma = 0.3$ (initial volatility of the underlying)
• $\alpha = 0.2$ (volatility of forward price volatility)
• $\beta = 0.6$
• $\rho = 0.2$

Simulation parameters:

• $N_{simulations} = 10^5 $
• $N_{steps} = 1.2 * 10^3$

Parameters

'market' parameters are:

• $T = 2$ (Time horizon)
• $r = 0.05$ (Risk free rate)
• $s = e^{- r T}f $
• $v$ (if applicable, initial volatility is initiated randomly and optimised during training)
• $\rho$ (if applicable, correlation between Wiener processes is initiated randomly and optimised during training)

Simulation parameters for neuralSDE:

• $N_{simulations} = 4*10^5$
• $N_{steps} = 96 $
• $N_{options} = 4 $
• $N_{maturities} = 21 $

Our model takes one of three possible parametrizations:

• NeuralLV (neural Local Volatility)

  $\text{d}S_t = r S_t \text{d}t + \sigma(t, S_t) S_t \text{d}W_t, S_0 = s$

• NeuralLSV (neural Local Stochastic Volatility)

  $\text{d}S_t = r S_t \text{d}t + \sigma_S(t, S_t, V_t) S_t \text{d}W_t, S_0 = s$

  $\text{d}V_t = b_V(V_t) \text{d}t + \sigma_V(V_t) \text{d}B_t, V_0 = v$

  $\text{d}W_t\text{d}B_t=\rho\text{d}t$

• NeuralSDE (neural Stochastic Volatility)

  $\text{d}S_t = r S_t \text{d}t + \sigma_S(t, S_t, V_t) S_t \text{d}W_t, S_0 = s$

  $\text{d}V_t = b_V(t, S_t, V_t) \text{d}t + \sigma_V(t, S_t, V_t) \text{d}B_t, V_0 = v$

  $\text{d}W_t\text{d}B_t=\rho\text{d}t$,

where all functions are given by feedforward neural networks with optional batch normalization layers or residual connections. Then our model is used in tamed Euler algorithm.

Standard approach

In the standard approach we train models based on Mean Square Error between option prices from the model and from Monte Carlo simulations. File 'StandardApproach/Data/Options_results.csv' contains option prices for chosen maturities.

Wasserstein approach

In Wasserstein approach we train models based on similarity of distributions between paths from SABR model and paths from our model at various timesteps. File 'Wasserstein/Data/Wasserstein_target.pth.tar' contains simulated paths from SABR model. Our model simulates paths iteratively during training procedure.

Wasserstein GAN approach

In Wasserstein GAN approach we train two models: generator (based on above models) and discriminator (standard MLP). Our goal is to train generator to succesfully trick the trained discriminator. We make use of Wasserstein GAN with gradient penalty.