Pricing and hedging derivative securities with neural networks: Bayesian regularization, early stopping, and bagging

We study the effectiveness of cross validation, Bayesian regularization, early stopping, and bagging to mitigate overfitting and improving generalization for pricing and hedging derivative securities with daily S&P 500 index daily call options from January 1988 to December 1993. Our results indicate that Bayesian regularization can generate significantly smaller pricing and delta-hedging errors than the baseline neural-network (NN) model and the Black-Scholes model for some years. While early stopping does not affect the pricing errors, it significantly reduces the hedging error (HE) in four of the six years we investigated. Although computationally most demanding, bagging seems to provide the most accurate pricing and delta hedging. Furthermore, the standard deviation of the MSPE of bagging is far less than that of the baseline model in all six years, and the standard deviation of the average HE of bagging is far less than that of the baseline model in five out of six years. We conclude that they be used at least in cases when no appropriate hints are available.

I. Introduction

Reference [18] demonstrate that learning networks can be used successfully to estimate a pricing formula for options, with good out-of-sample pricing and delta-hedging performance. This nonparametric pricing method has the distinct advantage of not relying on specific assumptions about the underlying asset price dynamics and is therefore robust to specification errors which might adversely affect parametric models. Reference [18] assume that their option pricing network formula is homogeneous of degree one in the underlying stock price and in the strike price which enables them to use a smaller number of inputs in learning the nonparametric pricing function. This parsimony is an advantage since the rate of convergence of nonparametric estimators slows down considerably as the number of input increases.