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Replication of financial derivatives under extreme market models given marginals

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  • Tongseok Lim

Abstract

The Black-Scholes-Merton model is a mathematical model for the dynamics of a financial market that includes derivative investment instruments, and its formula provides a theoretical price estimate of European-style options. The model's fundamental idea is to eliminate risk by hedging the option by purchasing and selling the underlying asset in a specific way, that is, to replicate the payoff of the option with a portfolio (which continuously trades the underlying) whose value at each time can be verified. One of the most crucial, yet restrictive, assumptions for this task is that the market follows a geometric Brownian motion, which has been relaxed and generalized in various ways. The concept of robust finance revolves around developing models that account for uncertainties and variations in financial markets. Martingale Optimal Transport, which is an adaptation of the Optimal Transport theory to the robust financial framework, is one of the most prominent directions. In this paper, we consider market models with arbitrarily many underlying assets whose values are observed over arbitrarily many time periods, and demonstrates the existence of a portfolio sub- or super-hedging a general path-dependent derivative security in terms of trading European options and underlyings, as well as the portfolio replicating the derivative payoff when the market model yields the extremal price of the derivative given marginal distributions of the underlyings. In mathematical terms, this paper resolves the question of dual attainment for the multi-period vectorial martingale optimal transport problem.

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  • Tongseok Lim, 2023. "Replication of financial derivatives under extreme market models given marginals," Papers 2307.00807, arXiv.org.
    • Handle: RePEc:arx:papers:2307.00807
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    References listed on IDEAS

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    1. Fouque,Jean-Pierre & Papanicolaou,George & Sircar,Ronnie & Sølna,Knut, 2011. "Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives," Cambridge Books, Cambridge University Press, number 9780521843584.
    2. Anna Aksamit & Ivan Guo & Shidan Liu & Zhou Zhou, 2021. "Superhedging duality for multi-action options under model uncertainty with information delay," Papers 2111.14502, arXiv.org, revised Nov 2023.
    3. Victor Chernozhukov & Iv·n Fern·ndez-Val & Alfred Galichon, 2010. "Quantile and Probability Curves Without Crossing," Econometrica, Econometric Society, vol. 78(3), pages 1093-1125, May.
    4. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    5. Laurent Carraro & Nicole El Karoui & Jan Ob{l}'oj, 2009. "On Az\'ema-Yor processes, their optimal properties and the Bachelier-drawdown equation," Papers 0902.1328, arXiv.org, revised Sep 2012.
    6. Guillaume Carlier & Victor Chernozhukov & Alfred Galichon, 2015. "Vector quantile regression: an optimal transport approach," CeMMAP working papers 58/15, Institute for Fiscal Studies.
    7. Victor Chernozhukov & Alfred Galichon & Marc Henry & Brendan Pass, 2021. "Identification of Hedonic Equilibrium and Nonseparable Simultaneous Equations," Journal of Political Economy, University of Chicago Press, vol. 129(3), pages 842-870.
    8. A. Galichon & P. Henry-Labord`ere & N. Touzi, 2014. "A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options," Papers 1401.3921, arXiv.org.
    9. Matteo Burzoni & Frank Riedel & H. Mete Soner, 2021. "Viability and Arbitrage Under Knightian Uncertainty," Econometrica, Econometric Society, vol. 89(3), pages 1207-1234, May.
    10. repec:hal:wpspec:info:hdl:2441/5rkqqmvrn4tl22s9mc4b6ga2g is not listed on IDEAS
    11. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    12. Mathias Beiglbock & Pierre Henry-Labord`ere & Friedrich Penkner, 2011. "Model-independent Bounds for Option Prices: A Mass Transport Approach," Papers 1106.5929, arXiv.org, revised Feb 2013.
    13. repec:hal:spmain:info:hdl:2441/5rkqqmvrn4tl22s9mc4b6ga2g is not listed on IDEAS
    14. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," Post-Print hal-03460952, HAL.
    15. Figalli, Alessio & Kim, Young-Heon & McCann, Robert J., 2011. "When is multidimensional screening a convex program?," Journal of Economic Theory, Elsevier, vol. 146(2), pages 454-478, March.
    16. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," SciencePo Working papers Main hal-03460952, HAL.
    17. Mathias Beiglböck & Pierre Henry-Labordère & Friedrich Penkner, 2013. "Model-independent bounds for option prices—a mass transport approach," Finance and Stochastics, Springer, vol. 17(3), pages 477-501, July.
    18. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," SciencePo Working papers hal-03460952, HAL.
    19. 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.
    20. repec:hal:spmain:info:hdl:2441/4c5431jp6o888pdrcs0fuirl40 is not listed on IDEAS
    21. Guillaume Carlier & Victor Chernozhukov & Alfred Galichon, 2016. "Vector Quantile Regression: An Optimal Transport Approach," SciencePo Working papers hal-03567920, HAL.
    22. Gaoyue Guo & Xiaolu Tan & Nizar Touzi, 2015. "Optimal Skorokhod embedding under finitely-many marginal constraints," Papers 1506.04063, arXiv.org, revised Aug 2016.
    23. Victor Chernozhukov & Ivan Fernandez-Val & Alfred Galichon, 2007. "Quantile and probability curves without crossing," CeMMAP working papers CWP10/07, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
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