Deep Hedging of Long-Term Financial Derivatives

Abstract

This study presents a deep reinforcement learning approach for global hedging of longterm financial derivatives. A similar setup as in Coleman et al. ( 2007) is considered with the risk management of lookback options embedded in guarantees of variable annuities with ratchet features. The deep hedging algorithm of Buehler et al. (2019a) is applied to optimize neural networks representing global hedging policies with both quadratic and non-quadratic penalties. To the best of the author's knowledge, this is the first paper that presents an extensive benchmarking of global policies for long-term contingent claims with the use of various hedging instruments (e.g. underlying and standard options) and with the presence of jump risk for equity. Monte Carlo experiments demonstrate the vast superiority of non-quadratic global hedging as it results simultaneously in downside risk metrics two to three times smaller than best benchmarks and in significant hedging gains. Analyses show that the neural networks are able to effectively adapt their hedging decisions to different penalties and stylized facts of risky asset dynamics only by experiencing simulations of the financial market exhibiting these features. Numerical results also indicate that non-quadratic global policies are significantly more geared towards being long equity risk which entails earning the equity risk premium.

Deep Equal Risk Pricing of Financial Derivatives with Multiple Hedging Instruments

“…Note thatCarbonneau (2020) demonstrates the potential of the deep hedging algorithm for global hedging long-term lookback options embedded in segregated funds guarantees with multiple hedging instruments. It is also worth highlighting thatBarigou et al (2020) developed a pricing scheme consistent with local non-quadratic hedging procedures for insurance liabilities which relies on neural networks.28 In the context of segregated funds, the short position of the embedded option is assumed to be held by an insurance company who has to provide a quote and mitigate its risk exposure.…”

“…The periodic computation of long short-term memory neural networks is done with so-called LSTM cells, which are similar to but more complex than the typical hidden layer of RNNs. LSTMs have recently been applied with success to approximate global hedging policies in several studies: Buehler et al (2019a), Cao et al (2020) and Carbonneau (2020). Additional remarks are made in subsequent sections to motivate this choice of neural networks for the specific setup of this paper.…”

“…time taken to find a good set of trainable parameters). Note that log transformation for risky asset prices was also considered in Carbonneau (2020), Buehler et al (2019b) and Buehler et al (2019a).…”

“…Indeed, some categories of instruments are more effective to mitigate certain risk factors than others. For instance, it is well-known that in the presence of random jumps, option hedges are much more effective than trading exclusively the underlying stock due to the convex property of derivatives prices (see, for instance, Coleman et al (2007) and Carbonneau (2020)). More generally, the use of option hedges dampens tail risk stemming from different risk factors (e.g.…”

“…Carbonneau and Godin (2020) also introduce novel -completeness metrics to quantify the level of market incompleteness which will be used throughout this current study. Several papers have studied different aspects of the class of deep hedging algorithms: Buehler et al (2019a) extend upon the work of Buehler et al (2019b) by hedging path-dependent contingent claims with neural networks, Carbonneau (2020) presents an extensive benchmarking of global policies parameterized with neural networks to mitigate the risk exposure of very long-term contingent claims, Cao et al (2020) show that the deep hedging algorithm provides good approximations of optimal initial capital investments for variance-optimal hedging problems and Horvath et al (2021) deep hedge in a non-Markovian framework with rough volatility models for risky assets.…”