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Hydrodynamics of Markets:Hidden Links Between Physics and Finance

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  • Alexander Lipton

Abstract

An intriguing link between a wide range of problems occurring in physics and financial engineering is presented. These problems include the evolution of small perturbations of linear flows in hydrodynamics, the movements of particles in random fields described by the Kolmogorov and Klein-Kramers equations, the Ornstein-Uhlenbeck and Feller processes, and their generalizations. They are reduced to affine differential and pseudo-differential equations and solved in a unified way by using Kelvin waves and developing a comprehensive math framework for calculating transition probabilities and expectations. Kelvin waves are instrumental for studying the well-known Black-Scholes, Heston, and Stein-Stein models and more complex path-dependent volatility models, as well as the pricing of Asian options, volatility and variance swaps, bonds, and bond options. Kelvin waves help to solve several cutting-edge problems, including hedging the impermanent loss of Automated Market Makers for cryptocurrency trading. This title is also available as Open Access on Cambridge Core.

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  • Alexander Lipton, 2024. "Hydrodynamics of Markets:Hidden Links Between Physics and Finance," Papers 2403.09761, arXiv.org.
    • Handle: RePEc:arx:papers:2403.09761
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    1. Qiang Dai & Kenneth J. Singleton, 2000. "Specification Analysis of Affine Term Structure Models," Journal of Finance, American Finance Association, vol. 55(5), pages 1943-1978, October.
    2. Adrian Dragulescu & Victor Yakovenko, 2002. "Probability distribution of returns in the Heston model with stochastic volatility," Quantitative Finance, Taylor & Francis Journals, vol. 2(6), pages 443-453.
    3. Mark Craddock & Eckhard Platen, 2003. "Symmetry Group Methods for Fundamental Solutions and Characteristic Functions," Research Paper Series 90, Quantitative Finance Research Centre, University of Technology, Sydney.
    4. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(4), pages 627-627, November.
    5. Alexander Lipton & Andrey Gal & Andris Lasis, 2014. "Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: survey and new results," Quantitative Finance, Taylor & Francis Journals, vol. 14(11), pages 1899-1922, November.
    6. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    7. Pavel Cizek & Wolfgang Karl Härdle & Rafal Weron, 2011. "Statistical Tools for Finance and Insurance (2nd edition)," HSC Books, Hugo Steinhaus Center, Wroclaw University of Technology, number hsbook1101, December.
    8. Masaaki Fukasawa & Basile Maire & Marcus Wunsch, 2023. "Weighted variance swaps hedge against impermanent loss," Quantitative Finance, Taylor & Francis Journals, vol. 23(6), pages 901-911, June.
    9. Rainer Schöbel & Jianwei Zhu, 1999. "Stochastic Volatility With an Ornstein–Uhlenbeck Process: An Extension," Review of Finance, European Finance Association, vol. 3(1), pages 23-46.
    10. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    11. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    12. 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..
    13. Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," The Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 573-592.
    14. Alexander Lipton & Marcos López De Prado, 2020. "A Closed-Form Solution For Optimal Ornstein–Uhlenbeck Driven Trading Strategies," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(08), pages 1-34, December.
    15. 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.
    16. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    17. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48, January.
    18. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    19. Rubinstein, Mark, 1994. "Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    20. Virginia Giorno & Amelia G. Nobile, 2021. "Time-Inhomogeneous Feller-Type Diffusion Process in Population Dynamics," Mathematics, MDPI, vol. 9(16), pages 1-29, August.
    21. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
    22. Walter Schachermayer & Josef Teichmann, 2008. "How Close Are The Option Pricing Formulas Of Bachelier And Black–Merton–Scholes?," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 155-170, January.
    23. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    24. A. James Boness, 1964. "Elements of a Theory of Stock-Option Value," Journal of Political Economy, University of Chicago Press, vol. 72(2), pages 163-163.
    25. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179.
    26. Darrell Duffie & Rui Kan, 1996. "A Yield‐Factor Model Of Interest Rates," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 379-406, October.
    27. Alexander Lipton & Adrien Treccani, 2021. "Blockchain and Distributed Ledgers:Mathematics, Technology, and Economics," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 11857, August.
    28. Alexander Lipton, 2001. "Mathematical Methods for Foreign Exchange:A Financial Engineer's Approach," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4694, February.
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