The mathematical theory of neural networks is ``meant to tell us what is possible and, sometimes equally importantly, what is not." This paper contributes a case that fits into this principle. We propose that "option price or time value " is a natural hyperparameter in the design of neural network option models. Hutchinson, Lo and Poggio asked the question that if learning networks can learn the Black-Scholes formula, and they studied the network (S_t/K, \tau)%DIFDELCMD < \to %%% C_t/K where S_t, K, \tau, C_t are the underlying price, strike, time to maturity and option price. In this paper we propose a novel decision function and study the network (S_t/K, \tau)%DIFDELCMD < \to %%% V_t/K where V_t is the time value. Empirical experiments will be carried out to demonstrate that this new decision function significantly improves  Hutchinson-Lo-Poggio 's model by faster learning and better generalization performance. We prove that a shallow neural network with the logistic activation is a universal approximator in L^{2(R \times }%DIFDELCMD < [%%% 0, 1 ). As a corollary V_t/K but not C_t/K can be approximated by superpositions of logistic functions on \mathbb{R^+ \times }%DIFDELCMD < [%%% 0, 1 . This justifies the benefit of time value oriented decision functions in option pricing models .
%DIFDELCMD < \to %%% %DIFDELCMD < \to %%% We propose a time value related decision function to treat a classical option pricing problem raised by Hutchinson-Lo-Poggio. In numerical experiments, the new decision function significantly improves the original model of Hutchinson-Lo-Poggio with faster convergence and better generalization performance. (R \times }%DIFDELCMD < [%%% By proving a novel universal approximation theorem, we show that our decision function rather than Hutchinson-Lo-Poggio's can be approximated ^+ \times }%DIFDELCMD < [%%% on the entire domain of definition by neural networks. Thus the experimental results are partially explained by the representation properties of networks .
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The mathematical theory of neural networks is ``meant to tell us what is possible and, sometimes equally importantly, what is not." This paper contributes a case that fits into this principle. We propose that "option price or time value " is a natural hyperparameter in the design of neural network option models. Hutchinson, Lo and Poggio asked the question that if learning networks can learn the Black-Scholes formula, and they studied the network (S_t/K, \tau)
D
C_t/K where S_t, K, \tau, C_t are the underlying price, strike, time to maturity and option price. In this paper we propose a novel decision function and study the network (S_t/K, \tau)
R
V_t/K where V_t is the time value. Empirical experiments will be carried out to demonstrate that this
We propose a time value related decision function to treat a classical option pricing problem raised by Hutchinson-Lo-Poggio. In numerical experiments, the
A
the original model of
R
's model by faster learning
with faster convergence
D
We prove that a shallow neural network with the logistic activation is a universal approximator in L^{2
D
0, 1
R
). As a corollary V_t/K but not C_t/K
By proving a novel universal approximation theorem, we show that our decision function rather than Hutchinson-Lo-Poggio's
D
by superpositions of logistic functions on \mathbb{R
D
0, 1
R
. This justifies the benefit of time value oriented decision functions in option pricing models
on the entire domain of definition by neural networks. Thus the experimental results are partially explained by the representation properties of networks