Learning The Pricing Kernel : Applications To Option Pricing

We seek to estimate a portfolio of option prices in an entirely data driven way, at any future time, for trading and risk management purposes in a model independent way. We do not know the model driving the dynamics of the actual stock prices, but only observe discretely their evolution in the historical measure. We must therefore map option prices from the historical measure to the risk-neutral measure. Fortunately, we can consider the no-arbitrage condition expressed in terms of conditional expectations and use it to derive the adjusted pricing kernel (APK), which allows for the mapping of options from the historical measure to the risk-neutral measure. We relate it to the market price of risk factor (MPRF), which we express in terms of a ratio of European options in the subjective measure over the observed forward price in the risk-neutral measure. We augment market data by using Machine Learning (ML) to generate consistent synthetic data closely related to the original time series that reproduces the well known stylised facts of financial assets. Thereafter, we apply Reinforcement Learning (RL) to estimate option prices on this dataset, and use these subjective prices with the observed forward price to compute the adjusted pricing kernel. We can then modify the discretely observed stock prices with the APK to recover the risk-adjusted process on average. Afterwards, we use this risk-adjusted process in a TD-model to compute the expected value of the contingent claims in the risk-neutral measure. We can also use this synthetic to disentangle the pricing kernel from the risk-neutral probability and the historical one. Finally, we illustrate our approach in a simple model where the market price of risk is driven by an Ornstein-Uhlenbeck process