We provide a framework for learning risk-neutral measures (Martingale measures) for pricing options. In a simple geometric Brownian motion model, the price volatility, fixed interest rate and a no-arbitrage condition suce to determine a unique risk-neutral measure. On the other hand, in our framework, we relax some of these assumptions to obtain a class of allowable risk-neutral measures. We then propose a framework for learning the appropriate risk-neural measure. Since the risk-neutral measure prices all options simultaneously, we can use all the option contracts on a particular stock for learning. We demonstrate the performance of these models on historical data. In particular, we show that both learning without a no-arbitrage condition and a no-arbitrage condition without learning are worse than our framework; however the combination of learning with a no-arbitrage condition has the best result. These results indicate the potential to learn Martingale measures with a no-arbitrage condition providing just the right constraint. We also compare our approach to standard Binomial models with volatility estimates (historical volatility and GARCH volatility predictors). Finally, we illustrate the power of such a framework by developing a real time trading system based upon these pricing methods.
[1]
F. Black,et al.
The Pricing of Options and Corporate Liabilities
,
1973,
Journal of Political Economy.
[2]
S. Ross,et al.
The valuation of options for alternative stochastic processes
,
1976
.
[3]
David M. Kreps,et al.
Martingales and arbitrage in multiperiod securities markets
,
1979
.
[4]
S. Ross,et al.
Option pricing: A simplified approach☆
,
1979
.
[5]
J. Harrison,et al.
Martingales and stochastic integrals in the theory of continuous trading
,
1981
.
[6]
J. Harrison,et al.
A stochastic calculus model of continuous trading: Complete markets
,
1983
.
[7]
S. Pliska,et al.
On the fundamental theorem of asset pricing with an infinite state space
,
1991
.
[8]
P. Wilmott,et al.
The Mathematics of Financial Derivatives: Contents
,
1995
.
[9]
B. E. Sorenson.
FINANCIAL CALCULUS: AN INTRODUCTION TO DERIVATIVE PRICING
,
1998,
Econometric Theory.
[10]
Malik Magdon-Ismail,et al.
The equivalent martingale measure: an introduction to pricing using expectations
,
2001,
IEEE Trans. Neural Networks.
[11]
Sheldon M. Ross,et al.
An elementary introduction to mathematical finance
,
2002
.