Methodology

Black-Scholes Dynamics

Under the risk-neutral Black-Scholes model, the underlying asset follows

\[dS_t = r S_t dt + \sigma S_t dW_t,\]

where \(S_t\) is the underlying price, \(r\) is the risk-free rate, \(\sigma\) is constant volatility, and \(W_t\) is a Brownian motion.

The terminal distribution is

\[S_T = S_0 \exp\left[ \left(r - \frac{1}{2}\sigma^2\right)T + \sigma \sqrt{T} Z \right], \qquad Z \sim \mathcal{N}(0,1).\]

European Call Pricing

The European call payoff is

\[\max(S_T - K, 0).\]

The analytical Black-Scholes price is used as the ground-truth label:

\[C_{BS} = S_0 N(d_1) - K e^{-rT} N(d_2),\]

with

\[d_1 = \frac{ \ln(S_0/K) + \left(r + \frac{1}{2}\sigma^2\right)T }{ \sigma\sqrt{T} }, \qquad d_2 = d_1 - \sigma\sqrt{T}.\]

Supervised Learning Task

The neural network is trained as a regression model:

\[f_\theta(S_0, K, T, r, \sigma) \approx C_{BS}.\]

Inputs are standardized and the target is scaled during training. Target scaling improves numerical conditioning because option prices may span a wider range than the normalized input features.

The final configuration uses the engineered moneyness feature \(S_0/K\), so the learned map becomes

\[f_\theta(S_0, K, T, r, \sigma, S_0/K) \approx C_{BS}.\]

Moneyness is included because it summarizes the relative position of the underlying price with respect to the strike, which is financially informative for option-pricing regression.

Monte Carlo Benchmark

Monte Carlo pricing estimates

\[\hat{C}_{MC} = e^{-rT} \frac{1}{M} \sum_{i=1}^{M} \max(S_T^{(i)} - K, 0).\]

Because the option is European and path-independent, the implementation samples the exact terminal distribution instead of simulating intermediate time steps.

Controlled Noisy Targets

The noisy-target experiments perturb only the training labels:

\[\widetilde{C}_{BS} = \max(C_{BS} + \varepsilon, 0), \qquad \varepsilon \sim \mathcal{N}(0, \tau^2).\]

The validation and test targets remain clean analytical Black-Scholes prices. This design measures robustness to imperfect training labels while preserving a known ground truth for evaluation.

SVR Baseline

Support Vector Regression is used as a classical machine learning reference. It is evaluated on reduced synthetic datasets because kernel SVR scales less favorably than the neural-network pipeline when the number of training samples becomes large. The purpose is therefore comparative and diagnostic, not to replace the final neural-network experiment.