Learning Martingale Measures From High Frequency Financial Data to Help Option Pricing

We provide a framework for learning risk-neutral mea-sures (Martingale measures) for pricing options from high frequency financial data. In a simple geometric Brownian motion model, a price volatility, a fixed in-terest rate and a no-arbitrage condition suffice to de-termine a unique risk-neutral measure. On the other hand, in our framework, we relax some of these as-sumptions 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 under-lying stock for learning. We demonstrate the perfor-mance of these models on historical data. In particu-lar, we show that both learning without a no-arbitrage condition and a no-arbitrage condition without learn-ing are worse than our framework; however the com-bination of learning with a no-arbitrage condition has the best result. These results indicate the potential to learn Martingale measures with a no-arbitrage condi-tion providing just the right constraint. We also com-pare 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