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Deep Equal Risk Pricing of Financial Derivatives with Multiple Hedging Instruments

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  • Alexandre Carbonneau
  • Fr'ed'eric Godin

Abstract

This paper studies the equal risk pricing (ERP) framework for the valuation of European financial derivatives. This option pricing approach is consistent with global trading strategies by setting the premium as the value such that the residual hedging risk of the long and short positions in the option are equal under optimal hedging. The ERP setup of Marzban et al. (2020) is considered where residual hedging risk is quantified with convex risk measures. The main objective of this paper is to assess through extensive numerical experiments the impact of including options as hedging instruments within the ERP framework. The reinforcement learning procedure developed in Carbonneau and Godin (2020), which relies on the deep hedging algorithm of Buehler et al. (2019b), is applied to numerically solve the global hedging problems by representing trading policies with neural networks. Among other findings, numerical results indicate that in the presence of jump risk, hedging long-term puts with shorter-term options entails a significant decrease of both equal risk prices and market incompleteness as compared to trading only the stock. Monte Carlo experiments demonstrate the potential of ERP as a fair valuation approach providing prices consistent with observable market prices. Analyses exhibit the ability of ERP to span a large interval of prices through the choice of convex risk measures which is close to encompass the variance-optimal premium.

Suggested Citation

  • Alexandre Carbonneau & Fr'ed'eric Godin, 2021. "Deep Equal Risk Pricing of Financial Derivatives with Multiple Hedging Instruments," Papers 2102.12694, arXiv.org.
  • Handle: RePEc:arx:papers:2102.12694
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    References listed on IDEAS

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    1. Martin Schweizer, 1995. "Variance-Optimal Hedging in Discrete Time," Mathematics of Operations Research, INFORMS, vol. 20(1), pages 1-32, February.
    2. Alexandre Carbonneau, 2020. "Deep Hedging of Long-Term Financial Derivatives," Papers 2007.15128, arXiv.org.
    3. Saeed Marzban & Erick Delage & Jonathan Yumeng Li, 2020. "Equal Risk Pricing and Hedging of Financial Derivatives with Convex Risk Measures," Papers 2002.02876, arXiv.org, revised Sep 2020.
    4. Rama Cont & Jose da Fonseca, 2002. "Dynamics of implied volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 45-60.
    5. T. F. Coleman & Y. Kim & Y. Li & M. Patron, 2007. "Robustly Hedging Variable Annuities With Guarantees Under Jump and Volatility Risks," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 74(2), pages 347-376, June.
    6. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    7. Peter Carr & Liuren Wu, 2014. "Static Hedging of Standard Options," Journal of Financial Econometrics, Oxford University Press, vol. 12(1), pages 3-46.
    8. Suleyman Basak & Georgy Chabakauri, 2012. "Dynamic Hedging in Incomplete Markets: A Simple Solution," The Review of Financial Studies, Society for Financial Studies, vol. 25(6), pages 1845-1896.
    9. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    10. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    11. Blanka Horvath & Josef Teichmann & Zan Zuric, 2021. "Deep Hedging under Rough Volatility," Swiss Finance Institute Research Paper Series 21-88, Swiss Finance Institute.
    12. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    13. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. "On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
    14. Guo, Ivan & Zhu, Song-Ping, 2017. "Equal risk pricing under convex trading constraints," Journal of Economic Dynamics and Control, Elsevier, vol. 76(C), pages 136-151.
    15. Alexandre Carbonneau & Fr'ed'eric Godin, 2020. "Equal Risk Pricing of Derivatives with Deep Hedging," Papers 2002.08492, arXiv.org, revised Jun 2020.
    16. Blanka Horvath & Josef Teichmann & Žan Žurič, 2021. "Deep Hedging under Rough Volatility," Risks, MDPI, vol. 9(7), pages 1-20, July.
    17. Hans Buehler & Lukas Gonon & Josef Teichmann & Ben Wood & Baranidharan Mohan & Jonathan Kochems, 2019. "Deep Hedging: Hedging Derivatives Under Generic Market Frictions Using Reinforcement Learning," Swiss Finance Institute Research Paper Series 19-80, Swiss Finance Institute.
    18. Warren B. Powell, 2009. "What you should know about approximate dynamic programming," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(3), pages 239-249, April.
    19. Masaaki Fukasawa & Blanka Horvath & Peter Tankov, 2021. "Hedging under rough volatility," Papers 2105.04073, arXiv.org.
    20. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July.
    21. Guiyuan Ma & Song-Ping Zhu & Ivan Guo, 2019. "Valuation of contingent claims with short selling bans under an equal-risk pricing framework," Papers 1910.04960, arXiv.org, revised Aug 2021.
    22. Blanka Horvath & Josef Teichmann & Zan Zuric, 2021. "Deep Hedging under Rough Volatility," Papers 2102.01962, arXiv.org.
    23. Xin-Jiang He & Song-Ping Zhu, 2020. "A revised option pricing formula with the underlying being banned from short selling," Quantitative Finance, Taylor & Francis Journals, vol. 20(6), pages 935-948, June.
    24. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    25. Dimitris Bertsimas & Leonid Kogan & Andrew W. Lo, 2001. "Hedging Derivative Securities and Incomplete Markets: An (epsilon)-Arbitrage Approach," Operations Research, INFORMS, vol. 49(3), pages 372-397, June.
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    Citations

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    Cited by:

    1. 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.
    2. Nacira Agram & Bernt Øksendal & Jan Rems, 2024. "Deep learning for quadratic hedging in incomplete jump market," Digital Finance, Springer, vol. 6(3), pages 463-499, September.
    3. 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.
    4. Alexandre Carbonneau & Frédéric Godin, 2023. "Deep Equal Risk Pricing of Financial Derivatives with Non-Translation Invariant Risk Measures," Risks, MDPI, vol. 11(8), pages 1-27, August.
    5. 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.
    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. Horikawa, Hiroaki & Nakagawa, Kei, 2024. "Relationship between deep hedging and delta hedging: Leveraging a statistical arbitrage strategy," Finance Research Letters, Elsevier, vol. 62(PA).
    9. Ben Hambly & Renyuan Xu & Huining Yang, 2021. "Recent Advances in Reinforcement Learning in Finance," Papers 2112.04553, arXiv.org, revised Feb 2023.
    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. 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.
    12. Nacira Agram & Bernt {O}ksendal & Jan Rems, 2024. "Deep learning for quadratic hedging in incomplete jump market," Papers 2407.13688, arXiv.org.
    13. Chunhui Qiao & Xiangwei Wan, 2024. "Enhancing Black-Scholes Delta Hedging via Deep Learning," Papers 2407.19367, arXiv.org, revised Aug 2024.
    14. Carbonneau, Alexandre, 2021. "Deep hedging of long-term financial derivatives," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 327-340.
    15. 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.
    16. 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.
    17. 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.

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