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Deep hedging is a framework for hedging derivatives in the presence of marketfrictions. In this study, we focus on the problem of hedging a given targetoption by using multiple options. To extend the deep hedging framework to thissetting, the options used as hedging instruments also have to be priced duringtraining. While one might use classical pricing model such as the Black-Scholesformula, ignoring frictions can offer arbitrage opportunities which areundesirable for deep hedging learning. The goal of this study is to develop anested deep hedging method. That is, we develop a fully-deep approach of deephedging in which the hedging instruments are also priced by deep neuralnetworks that are aware of frictions. However, since the prices of hedginginstruments have to be calculated under many different conditions, the entirelearning process can be computationally intractable. To overcome this problem,we propose an efficient learning method for nested deep hedging. Our methodconsists of three techniques to circumvent computational intractability, eachof which reduces redundant computations during training. We show throughexperiments that the Black-Scholes pricing of hedge instruments can admitsignificant arbitrage opportunities, which are not observed when the pricing isperformed by deep hedging. We also demonstrate that our proposed methodsuccessfully reduces the hedging risks compared to a baseline method that doesnot use options as hedging instruments.

Efficient Learning of Nested Deep Hedging using Multiple Options