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Equal risk pricing of derivatives with deep hedging

Author

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  • Alexandre Carbonneau
  • Frédéric Godin

Abstract

This article provides a universal and tractable methodology based on deep reinforcement learning to implement the equal risk pricing framework for financial derivatives pricing under very general conditions. The equal risk pricing framework entails solving for a derivative price which equates the optimally hedged residual risk exposure associated, respectively, with the long and short positions in the contingent claim. The solution to the hedging optimization problem considered, which is inspired from the [Marzban, S., Delage, E. and Li, J.Y., Equal risk pricing and hedging of financial derivatives with convex risk measures. arXiv preprint arXiv:2002.02876, 2020.] framework relying on convex risk measures, is obtained through the use of the deep hedging algorithm of [Buehler, H., Gonon, L., Teichmann, J. and Wood, B., Deep hedging. Q. Finance, 2019, 19, 1271–1291]. Consequently, the current paper's approach allows for the pricing and the hedging of a very large number of contingent claims (e.g. vanilla options, exotic options, options with multiple underlying assets) with multiple liquid hedging instruments under a wide variety of market dynamics (e.g. regime-switching, stochastic volatility, jumps). A novel ε-completeness measure allowing for the quantification of the residual hedging risk associated with a derivative is also proposed. The latter measure generalizes the one presented in [Bertsimas, D., Kogan, L. and Lo, A.W., Hedging derivative securities and incomplete markets: an ε-arbitrage approach. Oper. Res., 2001, 49, 372–397.] based on the quadratic penalty. Monte Carlo simulations are performed under a large variety of market dynamics to demonstrate the practicability of our approach, to perform benchmarking with respect to traditional methods and to conduct sensitivity analyses. Numerical results show, among others, that equal risk prices of out-of-the-money options are significantly higher than risk-neutral prices stemming from conventional changes of measure across all dynamics considered. This finding is shown to be shared by different option categories which include vanilla and exotic options.

Suggested Citation

  • Alexandre Carbonneau & Frédéric Godin, 2021. "Equal risk pricing of derivatives with deep hedging," Quantitative Finance, Taylor & Francis Journals, vol. 21(4), pages 593-608, April.
  • Handle: RePEc:taf:quantf:v:21:y:2021:i:4:p:593-608
    DOI: 10.1080/14697688.2020.1806343
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    Citations

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    Cited by:

    1. Karim Barigou & Valeria Bignozzi & Andreas Tsanakas, 2021. "Insurance valuation: A two-step generalised regression approach," Post-Print hal-03043244, HAL.
    2. Horikawa, Hiroaki & Nakagawa, Kei, 2024. "Relationship between deep hedging and delta hedging: Leveraging a statistical arbitrage strategy," Finance Research Letters, Elsevier, vol. 62(PA).
    3. Ben Hambly & Renyuan Xu & Huining Yang, 2021. "Recent Advances in Reinforcement Learning in Finance," Papers 2112.04553, arXiv.org, revised Feb 2023.
    4. Eva Lutkebohmert & Thorsten Schmidt & Julian Sester, 2021. "Robust deep hedging," Papers 2106.10024, arXiv.org, revised Nov 2021.
    5. Pascal Franc{c}ois & Genevi`eve Gauthier & Fr'ed'eric Godin & Carlos Octavio P'erez Mendoza, 2024. "Is the difference between deep hedging and delta hedging a statistical arbitrage?," Papers 2407.14736, arXiv.org, revised Oct 2024.
    6. Pascal Franc{c}ois & Genevi`eve Gauthier & Fr'ed'eric Godin & Carlos Octavio P'erez Mendoza, 2024. "Enhancing Deep Hedging of Options with Implied Volatility Surface Feedback Information," Papers 2407.21138, arXiv.org.
    7. Andrei Neagu & Fr'ed'eric Godin & Clarence Simard & Leila Kosseim, 2024. "Deep Hedging with Market Impact," Papers 2402.13326, arXiv.org, revised Feb 2024.
    8. Carbonneau, Alexandre, 2021. "Deep hedging of long-term financial derivatives," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 327-340.
    9. Alexandre Roch, 2023. "Optimal Liquidation Through a Limit Order Book: A Neural Network and Simulation Approach," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-29, March.
    10. Alexandre Carbonneau & Fr'ed'eric Godin, 2021. "Deep equal risk pricing of financial derivatives with non-translation invariant risk measures," Papers 2107.11340, arXiv.org.
    11. Saeed Marzban & Erick Delage & Jonathan Yumeng Li, 2021. "Deep Reinforcement Learning for Equal Risk Pricing and Hedging under Dynamic Expectile Risk Measures," Papers 2109.04001, arXiv.org.
    12. Jay Cao & Jacky Chen & Soroush Farghadani & John Hull & Zissis Poulos & Zeyu Wang & Jun Yuan, 2022. "Gamma and Vega Hedging Using Deep Distributional Reinforcement Learning," Papers 2205.05614, arXiv.org, revised Jan 2023.
    13. Nacira Agram & Bernt {O}ksendal & Jan Rems, 2024. "Deep learning for quadratic hedging in incomplete jump market," Papers 2407.13688, arXiv.org.
    14. Ajitha Kumari Vijayappan Nair Biju & Ann Susan Thomas & J Thasneem, 2024. "Examining the research taxonomy of artificial intelligence, deep learning & machine learning in the financial sphere—a bibliometric analysis," Quality & Quantity: International Journal of Methodology, Springer, vol. 58(1), pages 849-878, February.
    15. Ben Hambly & Renyuan Xu & Huining Yang, 2023. "Recent advances in reinforcement learning in finance," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 437-503, July.
    16. Chunhui Qiao & Xiangwei Wan, 2024. "Enhancing Black-Scholes Delta Hedging via Deep Learning," Papers 2407.19367, arXiv.org, revised Aug 2024.
    17. Parisa Davar & Fr'ed'eric Godin & Jose Garrido, 2024. "Catastrophic-risk-aware reinforcement learning with extreme-value-theory-based policy gradients," Papers 2406.15612, arXiv.org, revised Jun 2024.

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