QLBS: Q-Learner in the Black-Scholes(-Merton) Worlds

This paper presents a discrete-time option pricing model that is rooted in Reinforcement Learning (RL), and more specifically in the famous Q-Learning method of RL. We construct a riskadjusted Markov Decision Process for a discrete-time version of the classical Black-ScholesMerton (BSM) model, where the option price is an optimal Q-function, while the optimal hedge is a second argument of this optimal Q-function, so that both the price and hedge are parts of the same formula. Pricing is done by learning to dynamically optimize risk-adjusted returns for an option replicating portfolio, as in the Markowitz portfolio theory. Using Q-Learning and related methods, once created in a parametric setting, the model is able to go model-free and learn to price and hedge an option directly from data generated from a dynamic replicating portfolio which is rebalanced at discrete times. If the world is according to BSM, our risk-averse Q-Learner converges, given enough training data, to the true BSM price and hedge ratio of the option in the continuous time limit ∆t → 0, even if hedges applied at the stage of data generation are completely random (i.e. it can learn the BSM model itself, too!), because QLearning is an off-policy algorithm. If the world is different from a BSM world, the Q-Learner will find it out as well, because Q-Learning is a model-free algorithm. For finite time steps ∆t, the Q-Learner is able to efficiently calculate both the optimal hedge and optimal price for the option directly from trading data, and without an explicit model of the world. This suggests that RL may provide efficient data-driven and model-free methods for optimal pricing and hedging of options, once we depart from the academic continuous-time limit ∆t → 0, and vice versa, option pricing methods developed in Mathematical Finance may be viewed as special cases of model-based Reinforcement Learning. Further, due to simplicity and tractability of our model which only needs basic linear algebra (plus Monte Carlo simulation, if we work with synthetic data), and its close relation to the original BSM model, we suggest that our model could be used for benchmarking of different RL algorithms for financial trading applications. I would like to thank my students for their interest in this work and stimulating discussions that challenged me to look for simple explanations of complex topics. I thank Tom N.L. for an initial implementation of a timediscretized BSM model. This work is dedicated to my wife Lola on the occasion of her birthday and receiving a doctoral degree.