Option Pricing using Neural Networks

The first attempt was made by Hutchinson, Lo and Poggio (1994) who used three different network architectures: Radial Basis Functions (RBF), Multi Layer Perceptron (MLP), Projection Pursuit Regression (PPR) to fit both Monte-Carlo simulated Brownian underlier and Black-Scholes option data, as well as S&P500 futures and options thereof. They used a minimalistic approach in their input selection, and limited the network inputs to time to maturity (T-t) and moneyness (ratio of underlying price to strike price, S/X) only, assuming interest rate and volatility to be constant. We also note that they used financial knowledge in this construction, namely the “homogeneity property” of the option pricing formula (Merton 1990), which justifies the use of the moneyness rather than the underlying price and strike price separately. Another important question this paper raises is that of measuring performance of the network. While the out-of-sample error, R is a reasonable measure, they also examine the discrete delta-hedging performance of the neural network methods, which is of greater practical relevance. The result of these empirical tests is that these non-linear neural networks are able to recover the BS formula with remarkable accuracy, and even outperform BS in some cases in the discrete delta-hedging of simulated data. On the actual S&P500 data, the results are even more promising: all three of the neural network methods studied outperform the BS model both in terms of R as well as delta-hedging performance, especially for longer term and out-ofthe money options. The authors note that increasing the number of inputs, investigating the network architecture, re-examining the performance measures and establishing statistical significance of results are directions of further research. We examine the work done in each of these directions.