We propose a neural network approach to price EU call options that significantly outperforms some existing pricing models and comes with guarantees that its predictions are economically reasonable. To achieve this, we introduce a class of gated neural networks that automatically learn to divide-and-conquer the problem space for robust and accurate pricing. We then derive instantiations of these networks that are 'rational by design' in terms of naturally encoding a valid call option surface that enforces no arbitrage principles. This integration of human insight within data-driven learning provides significantly better generalisation in pricing performance due to the encoded inductive bias in the learning, guarantees sanity in the model's predictions, and provides econometrically useful byproduct such as risk neutral density.
There are many methods for implementing option pricing models including: Fourier-based @cite_0 , Tree-based @cite_13 , Finite difference @cite_26 and Monte Carlo methods @cite_11 . In this paper, we employ the fractional FFT method @cite_5 for our benchmark option pricing models as their characteristic functions are known.
@cite_26 @cite_0 @cite_5 @cite_13 @cite_11
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Abstract The option pricing model developed by Black and Scholes and extended by Merton gives rise to partial differential equations governing the value of an option. When the underlying stock pays no dividends – and in some very restrictive cases when it does – a closed form solution to the differential equation subject to the appropriate boundary conditions, has been obtained. But, in some relevant cases such as the one in which the stock pays discrete dividends, no closed form solution has been found. This paper shows how to solve these equations by numerical methods. In addition, the optimal strategy for exercising American options is derived. A numerical illustration of the procedure is also presented. This paper shows how the fast Fourier Transform may be used to value options when the characteristic function of the return is known analytically. An efficient method for the calculation of the interactions of a 2' factorial ex- periment was introduced by Yates and is widely known by his name. The generaliza- tion to 3' was given by (1). Good (2) generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series. In their full generality, Good's methods are applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices, where m is proportional to log N. This results inma procedure requiring a number of operations proportional to N log N rather than N2. These methods are applied here to the calculation of complex Fourier series. They are useful in situations where the number of data points is, or can be chosen to be, a highly composite number. The algorithm is here derived and presented in a rather different form. Attention is given to the choice of N. It is also shown how special advantage can be obtained in the use of a binary computer with N = 2' and how the entire calculation can be performed within the array of N data storage locations used for the given Fourier coefficients. Consider the problem of calculating the complex Fourier series N-1 (1) X(j) = EA(k)-Wjk, j = 0 1, * ,N- 1, k=0 This paper develops a Monte Carlo simulation method for solving option valuation problems. The method simulates the process generating the returns on the underlying asset and invokes the risk neutrality assumption to derive the value of the option. Techniques for improving the efficiency of the method are introduced. Some numerical examples are given to illustrate the procedure and additional applications are suggested.