Neural Networks for Option Pricing

Neural networks can learn to price options by training on simulated or historical data.

The Approach

Given market features $\mathbf{x} = (S, K, T, r, \sigma)$ (spot, strike, time to maturity, risk-free rate, volatility), a neural network $f_\theta$ learns to approximate the option price:

$C \approx f_\theta(\mathbf{x})$

Loss Function

The network is trained to minimize the pricing error:

$\mathcal{L}(\theta) = \frac{1}{N} \sum_{i=1}^{N} \left( f_\theta(\mathbf{x}_i) - C_{\text{true}}(\mathbf{x}_i) \right)^2$

Results

For European options, deep networks can achieve pricing errors below 0.1% of the option price, competitive with Black-Scholes when the model assumptions hold.