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We consider two data driven approaches, Reinforcement Learning (RL) and DeepTrajectory-based Stochastic Optimal Control (DTSOC) for hedging a European calloption without and with transaction cost according to a quadratic hedging P&Lobjective at maturity ("variance-optimal hedging" or "final quadratichedging"). We study the performance of the two approaches under various marketenvironments (modeled via the Black-Scholes and/or the log-normal SABR model)to understand their advantages and limitations. Without transaction costs andin the Black-Scholes model, both approaches match the performance of thevariance-optimal Delta hedge. In the log-normal SABR model without transactioncosts, they match the performance of the variance-optimal Barlett's Deltahedge. Agents trained on Black-Scholes trajectories with matching initialvolatility but used on SABR trajectories match the performance of Bartlett'sDelta hedge in average cost, but show substantially wider variance. To apply RLapproaches to these problems, P&L at maturity is written as sum of step-wisecontributions and variants of RL algorithms are implemented and used thatminimize expectation of second moments of such sums.

Reinforcement Learning and Deep Stochastic Optimal Control for Final Quadratic Hedging