Approximation using Radial Basis Function Networks - Application to Pricing Derivative Securities

Lendasse, Amaury [UCL] Lee, John Aldo [UCL] de Bodt, Eric [UCL] Wertz, Vincent [UCL] Verleysen, Michel [UCL]

The general scheme of approximation supposes the existence of a relationship between several variables (the inputs) and a dependent variable (the output). This relationship is unknown; we try to build an approximator between the inputs and the output. In that purpose, we need a set of inputs-output couples that form the learning data of the approximator. In this paper, we will use a neural network approximator: the radial basis function network (or RBF network). Many techniques have been developed for the learning phase of RBF; we use the technique described in [5]. We will show that the results obtained with these RBF can be improved by a pre-processing technique. This pre-processing is based on linear models. It does not increase the complexity of the RBF learning phase but improves the accuracy of the approximator. These techniques are applied to pricing derivative securities. Hutchinson, Andrew and Poggio [6] have treated this problem. They used simulated data and showed that RBF are adequate models to pricing derivative securities and also to hedging them. The results we obtained are comparable to Hutchinson's results but the learning scheme is simplified in our case. We will use this example to illustrate the advantages of our pre-processing technique for RBF. BF are non-linear parametric approximation models based on combinations of Gaussian functions. In most cases, these Gaussian functions are radial (their output value depends only on the Euclidean distance between the input vector and a centre). As a consequence, input variables are not scaled in RBF, while a proper scaling could be more adequate. Our purpose is to eliminate this limitation, without penalising the simplicity of the RBF learning phase. If we build a linear model between the inputs and the output, this output will be approximated by a weighted sum of the inputs. The weight associated to each input determines its importance on the approximation of the output. This gives a very simple technique to determine the importance of all the inputs on the output. Therefore we are going to scale the inputs by the corresponding weights. A new so-called “Weighted RBF” is built, giving adequate importance to each of the inputs. We test our weighted RBF on the example of pricing derivative securities. This example has been used by par Hutchinson, Lo and Poggio, we apply the same method to generate the data. In this paper, the authors use the Black-Scholes formula to generate the data and to simulate the price of derivative securities (named C). The purpose of the RBF model is to give an approximation Ĉ of C. Three measures of performance are used. The first one is the determination coefficient R2 between C and Ĉ. The two other measures are the tracking and prediction errors. To measure the quality of the results obtained by the traditional RBF and the weighted RBF, we simulated a six-month price sample. In both cases, the number of Gaussian functions is chosen equal to 6. Then, 500 different six-months test sets are generated; the three performance measures are evaluated on both RBF. The results obtained for R2 are presented in the next Figure. The X-coordinate is a design coefficient k (used in the RBF approximation scheme) and the Y-coordinate is the value of R2. The dotted line shows the results obtained with the classical RBF and the solid line shows the results of the weighted RBF. The improvement obtained thanks to weighting is undeniable. In this paper, we used a traditional method to parameterise a RBF. We then suggested an improvement, consisting in weighting the inputs by the coefficients obtained by a linear model. This approximation scheme is tested to pricing derivative securities. The results obtained show an undeniable advantage for the weighted RBF. Moreover, the results are comparable to the best RBF and multi-layer perceptron built in the Hutchinson's paper. The advantage of the weighted RBF is the simplicity of the learbing phase, while keeping the quality of the approximation.

(fre) Nous proposons une méthode d’approximation de fonction par réseaux à fonctions radiales de base. Nous montrerons que cette méthode d’approximation peut être améliorée par un prétraitement des entrées basé sur un modèle linéaire. Cette méthode d’approximation sera appliquée à la détermination du prix d’achat d’une option.