  Deep hedging is a framework for hedging derivatives in the presence of market
frictions. In this study, we focus on the problem of hedging a given target
option by using multiple options. To extend the deep hedging framework to this
setting, the options used as hedging instruments also have to be priced during
training. While one might use classical pricing model such as the Black-Scholes
formula, ignoring frictions can offer arbitrage opportunities which are
undesirable for deep hedging learning. The goal of this study is to develop a
nested deep hedging method. That is, we develop a fully-deep approach of deep
hedging in which the hedging instruments are also priced by deep neural
networks that are aware of frictions. However, since the prices of hedging
instruments have to be calculated under many different conditions, the entire
learning process can be computationally intractable. To overcome this problem,
we propose an efficient learning method for nested deep hedging. Our method
consists of three techniques to circumvent computational intractability, each
of which reduces redundant computations during training. We show through
experiments that the Black-Scholes pricing of hedge instruments can admit
significant arbitrage opportunities, which are not observed when the pricing is
performed by deep hedging. We also demonstrate that our proposed method
successfully reduces the hedging risks compared to a baseline method that does
not use options as hedging instruments.
