Pricing and hedging derivative securities with neural networks and a homogeneity hint

We estimate a generalized option pricing formula that has a functional shape similar to the usual Black–Scholes formula by a feedforward neural network model. This functional shape is obtained when the option pricing function is homogeneous of degree one with respect to the underlying asset price (St) and the strike price (K). We show that pricing accuracy gains can be made by exploiting this generalized Black–Scholes shape. Instead of setting up a learning network mapping the ratio St/K and the time to maturity (τ) directly into the derivative price, we break down the pricing function into two parts, one controlled by the ratio St/K, the other one by a function of time to maturity. The results indicate that the homogeneity hint always reduces the out-of-sample mean squared prediction error compared with a feedforward neural network with no hint. Both feedforward network models, with and without the hint, provide similar delta-hedging errors that are small relative to the hedging performance of the Black–Scholes model. However, the model with hint produces a more stable hedging performance.