COMPARISON OF THE EFFECTIVENESS OF OPTION PRICE FORECASTING:BLACK-SCHOLES vs. SIMPLE AND HYBRID NEURAL NETWORKS

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IntroductionIn general, two types of measurements have been employed - historical and implied. The output of the Black-Scholes model greatly depends on the estimate of the variance σ2, all other inputs being equal. The more accurate one's prediction is, the more accurate the result will be. Historical is calculated on an ex post basis, but used on an ex ante basis to provide an equilibrium model price. The process assumes the stability of the estimated parameter. The implied is derived through the inversion of the Black-Scholes formula by using the actual observed option prices but leaving the variance/standard deviation as the unknown. This algorithm was first utilized by Latane and Rendleman3. Several studies, as outlined by Rubinstein4, as well as several others, have shown that Black-Scholes formula consistently have two systematic biases - moneyness and time to maturity. It appeared that for the same underlying asset implied volatilities differ across strike prices and expiration dates. The bias is called a moneyness bias, well known as a volatility smile. Indeed, option prices tend to be higher when the option is deep in the money or deep out of the money.Neural Networks as a Forecasting ToolNeural Networks are an alternative to closed-form equilibrium models. Neural networks and genetic algorithms are largely non-parametric models. While a generally accepted definition of neural networks does not exist, consider the working definition of Haykin5:A neural network is a massively parallel distributed processor that has a natural propensity for storing experiential knowledge and making it available for use. It resembles the brain in two respects:* Knowledge is acquired by the network through a learning process.* Interneuron connection strengths known as synaptic weights are used to store the knowledge.The basic premise behind a neural network is to imitate the functioning of the human brain. While a plethora of neural networks exists, a generalized typology would be:1. Supervised:* Feedforward* Linear* Multilayer perceptron (MLP)* Radial Basis Function (RBF) Networks* Cerebellar Model Articulation Controller (CMAC)* Classification only* Regression only* Feedback* Bidirectional Associative Memory (BAM)* Boltzman Machine* Recurrent Time Series* Competitive* ARTMAP* Fuzzy ARTMAP* Gaussian ARTMAP* Counterpropagation* Neocognitron2. Unsupervised:* Competitive* Vector Quantization* Self-Organizing Map* Adaptive Resonance Theory* Differential Competitive Learning (DCL)* Dimension Reduction* Autoassociation* Linear Autoassociator* Brain State in The Box (BSFB)* Hopfield3. Nonlearning:* Hopfield* Various networks for optimizationHornik, et al.6 showed that artificial neural networks (ANN) are able to approximate an almost infinite array of functions and are therefore very powerful tools for function approximation and time series forecasting. This property allowed ANN to be widely used in financial areas (Cybenko7). The following is a schematic representation of a simple feedforward neural network:The schematic above employs four input independent variables (I^sub k^), then a certain number of nodes (neurons n^sub j^) in the hidden layer, and finally three output dependent variables (O^sub i^). Every neuron n^sub j^ represents the following operation:Each interconnection of nodes (neurons) has a corresponding weight (w^sub jk^). Table 1 summarizes implementation of the neural networks in the option pricing area.Malliaris and Salchenberger8, as well as Hutchison, Lo and Poggio9, were among the first to implement nonparametric option pricing models. Malliaris and Salchenberger used daily transactions of OEX options with the five traditional variables, S, X, T, r, and σ. …