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Kelvin Waves, Klein-Kramers and Kolmogorov Equations, Path-Dependent Financial Instruments: Survey and New Results

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  • Alexander Lipton

Abstract

We discover several surprising relationships between large classes of seemingly unrelated foundational problems of financial engineering and fundamental problems of hydrodynamics and molecular physics. Solutions in all these domains can be reduced to solving affine differential equations commonly used in various mathematical and scientific disciplines to model dynamic systems. We have identified connections in these seemingly disparate areas as we link together small wave-like perturbations of linear flows in ideal and viscous fluids described in hydrodynamics by Kevin waves to motions of free and harmonically bound particles described in molecular physics by Klein-Kramers and Kolmogorov equations to Gaussian and non-Gaussian affine processes, e.g., Ornstein-Uhlenbeck and Feller, arising in financial engineering. To further emphasize the parallels between these diverse fields, we build a coherent mathematical framework using Kevin waves to construct transition probability density functions for problems in hydrodynamics, molecular physics, and financial engineering. As one of the outcomes of our analysis, we discover that the original solution of the Kolmogorov equation contains an error, which we subsequently correct. We apply our interdisciplinary approach to advance the understanding of various financial engineering topics, such as pricing of Asian options, volatility and variance swaps, options on stocks with path-dependent volatility, bonds, and bond options. We also discuss further applications to other exciting problems of financial engineering.

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  • Alexander Lipton, 2023. "Kelvin Waves, Klein-Kramers and Kolmogorov Equations, Path-Dependent Financial Instruments: Survey and New Results," Papers 2309.04547, arXiv.org.
    • Handle: RePEc:arx:papers:2309.04547
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Devreese, J.P.A. & Lemmens, D. & Tempere, J., 2010. "Path integral approach to Asian options in the Black–Scholes model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(4), pages 780-788.
    4. 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..
    5. 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.
    6. 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.
    7. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    8. 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.
    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. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48, January.
    11. Virginia Giorno & Amelia G. Nobile, 2021. "Time-Inhomogeneous Feller-Type Diffusion Process in Population Dynamics," Mathematics, MDPI, vol. 9(16), pages 1-29, August.
    12. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    13. 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.
    14. Darrell Duffie & Rui Kan, 1996. "A Yield‐Factor Model Of Interest Rates," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 379-406, October.
    15. 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.
    16. 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.
    17. 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.
    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. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
    20. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    21. 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.
    22. 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.
    23. 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.
    24. 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.
    25. 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.
    26. Rubinstein, Mark, 1994. "Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
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