 Hutchinson, Lo and Poggio raised the question that if learning works can learn the Black-Scholes formula, and they proposed the network mapping the ratio of underlying price to strike S_t/K and the\to  time to maturity \tau directly into the ratio of option priceto strike C_t/K . In this paper we propose a novel descision function and study the network mapping S_t/K and%DIFDELCMD < \tauinto %%% the ratio of time value to strike\to V_t/K . Time values' appearance in artificial intelligence fits into traders' natural intelligence. Empirical experiments will be carried out to demonstrate that it significantly improves Hutchinson-Lo-Poggio's original model by faster learning and better generalization performance. In order to take a conceptual viewpoint and to prove that V_t/K but not C_t/K can be approximated by superpositions of logistic functions on its domain of definition, we work on the theory of universal approximation on unbounded domains. We prove some general results which imply that an artificial neural network with a single hidden layer and sigmoid activation represents no function in L^{p(}%DIFDELCMD < \RR%%% ^2 \times 0, 1%DIFDELCMD < ]%%% ^{n neural network with a single hidden layer and logistic activation is a universal approximator of L^{2}( %DIFDELCMD < \RR %%%  \times [0, 1] ^{n^+ \times } [0, 1] ^{n .
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)\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 %DIFDELCMD < \tauinto %%% , \tau)\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. (}%DIFDELCMD < \RR%%% %DIFDELCMD < ]%%% We prove that a shallow neural network with the logistic activation is a universal approximator in L^{2}( %DIFDELCMD < \RR %%% \mathbb{R \times [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 } [0, 1] . This justifies the benefit of time value oriented decision functions in option pricing models .
A
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.
R
raised
asked
R
works
networks
R
proposed the network mapping the ratio of underlying price to strike
studied the network (
R
and the
, \tau)
A
C_t/K where S_t, K, \tau, C_t are the underlying price, strike,
R
\tau directly into the ratio of option priceto strike C_t/K
and option price
R
descision
decision
R
mapping
(
D
and
R
the ratio of time value to strike
, \tau)
R
. Time values' appearance in artificial intelligence fits into traders' natural intelligence.
where V_t is the time value.
R
it
this new decision function
D
original
D
In order to take a conceptual viewpoint and to prove that V_t/K but not C_t/K can be approximated by superpositions of logistic functions on its domain of definition, we work on the theory of universal approximation on unbounded domains. We prove some general results which imply that an artificial neural network with a single hidden layer and sigmoid activation represents no function in L^{p
D
^2 \times
D
0, 1
R
^{n
We prove that a shallow
R
a single hidden layer and
the
R
of
in
A
\mathbb{R
R
^{n
). As a corollary V_t/K but not C_t/K can be approximated by superpositions of logistic functions on \mathbb{R
R
^{n
. This justifies the benefit of time value oriented decision functions in option pricing models