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This paper shows how reinforcement learning can be used to derive optimal hedging strategies for derivatives when there are transaction costs. The paper illustrates the approach by showing the dierence between using delta hedging and optimal hedging for a short position in a call option when the objective is to minimize a function equal to the mean hedging cost plus a constant times the standard deviation of the hedging cost. Two situations are considered. In the rst, the asset price follows geometric Brownian motion. In the second, the asset price follows a stochastic volatility process. The paper extends the basic reinforcement learning approach in a number of ways. First, it uses two dierent Q-functions so that both the expected value of the cost and the expected value of the square of the cost are tracked for dierent state/action combinations. This approach increases the range of objective functions that can be used. Second, it uses a learning algorithm that allows for continuous state and action space. Third, it compares the accounting P&L approach (where the hedged position is valued at each step) and the cash ow approach (where cash inows and outows are used). We nd that a hybrid approach involving the use of an accounting P&L approach that incorporates a relatively simple valuation model works well. The valuation model does not have to correspond to the process assumed for the underlying asset price.

Deep Hedging of Derivatives Using Reinforcement Learning