Robust deep hedging

Abstract

We study pricing and hedging under parameter uncertainty for a class of Markov processes which we call generalized affine processes and which includes the Black–Scholes model as well as the constant elasticity of variance (CEV) model as special cases. Based on a general dynamic programming principle, we are able to link the associated nonlinear expectation to a variational form of the Kolmogorov equation which opens the door for fast numerical pricing in the robust framework. The main novelty of the paper is that we propose a deep hedging approach which efficiently solves the hedging problem under parameter uncertainty. We numerically evaluate this method on simulated and real data and show that the robust deep hedging outperforms existing hedging approaches in highly volatile periods.

Keywords:

• Affine processes
• Knightian uncertainty
• Kolmogorov equation
• Deep learning
• Robust hedging

JEL Classifications:

• C02
• C45
• G13

Open Scholarship

This article has earned the Center for Open Science badge for Open Data. The data are openly accessible at https://doi.org/10.34691/FK2/ULMNHV.

Acknowledgments

We thank the editor and two anonymous referees for several comments which significantly improved our paper. Moreover, we are thankful to David Criens for helpful remarks.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Note that α does not depend on P as the quadratic variation is a path property. We therefore write $C^P=C$ and ${\alpha }^P=\alpha$.

2 The name (nonlinear) generalized affine process and the setting of a NGA is inspired by the master thesis Denk (2021).

3 The considered constituents are: Apple Inc, Microsoft Corporation, Amazon.com Inc., Alphabet Inc. Class C, Berkshire Hathaway Inc. Class B, JPMorgan Chase $\mathrm{\&}$ Co., Johnson $\mathrm{\&}$ Johnson, Visa Inc. Class A, UnitedHealth Group Incorporated, NVIDIA Corporation, Procter $\mathrm{\&}$ Gamble Company, Home Depot Inc., Mastercard Incorporated Class A, Bank of America Corp, Walt Disney Company, Comcast Corporation Class A, Exxon Mobil Corporation, Adobe Inc., Verizon Communications Inc., Intel Corporation.

4 Note that the Black–Scholes model can be considered as a special case of an NGA process through setting $\mathrm{\Theta }=\left\{0\right\}\times \left\{\mu \right\}\times \left\{0\right\}\times \left\{\sigma \right\}\times \left\{1\right\}$ for some mean $\mu \in \mathbb{R}$ and some variance ${\sigma }^2>0$.

Additional information

Funding

The financial support of the NAP Grant Machine Learning based Algorithms in Finance and Insurance and from Deutsche Forschungsgemeinschaft (DFG) [grant number CHM 2160/13-1] is gratefully acknowledged.