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  3. Hydrodynamics of Markets
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  • Cited by 2
Publisher:
Cambridge University Press
Online publication date:
November 2024
Print publication year:
2025
Online ISBN:
9781009503129
DOI:
https://doi.org/10.1017/9781009503129
Creative Commons:
Creative Common License - CC Creative Common License - BY Creative Common License - NC
This content is Open Access and distributed under the terms of the Creative Commons Attribution licence CC-BY-NC 4.0 https://creativecommons.org/creativelicenses
Subjects:
Economics, Finance and Accountancy
Series:
Elements in Quantitative Finance
Information

Book description

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.

References

References

Abramowitz, M. & Stegun, I. A., eds. 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington, DC: US Government Printing Office.
Aksenov, A. V. 1995 Symmetries of linear partial differential equations and fundamental solutions. Doklady Mathematics 51(3), 329–331.
Andersen, L. B. G. & Piterbarg, V. 2007 Moment explosions in stochastic volatility models. Finance and Stochastics 11, 29–100.
Angeris, G., Kao, H. T., Chiang, R., Noyes, C. & Chitra, T. 2019 An analysis of Uniswap markets. Cryptoeconomic Systems Journal. 1(1). DOI: 10.21428/2F58320208.C9738E64
Arnold, L. 1974 Stochastic Equations: Theory and Applications. New York: John Wiley.
Bachelier, L. 1900 Théorie de la spéculation. Annales de l’Ecole Normale Supérieure 17, 21–86.
Barucci, E., Polidoro, S. & Vespri, V. 2001 Some results on partial differential equations and Asian options. Mathematical Models and Methods in Applied Sciences 11(3), 475–497.
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Physical Review Letters 57, 2160–2163.
Bayly, B. J., Holm, D. D. & Lifschitz, A., 1996. Three-dimensional stability of elliptical vortex columns in external strain flows. Philosophical Transactions of the Royal Society A 354(1709), 895–926.
Berest, Yu. Yu. 1993 Group analysis of linear differential equations in distributions and the construction of fundamental solutions. Differential Equations 29(11), 1700–1711.
Bergomi, L. 2015 Stochastic Volatility Modeling. Boca Raton, FL: CRC Press.
Bharucha-Reid, A. T. 1960 Elements of the Theory of Markov Processes and Their Applications. New York: McGraw-Hill.
Bick, A. & Reisman, H. 1993 Generalized implied volatility. Preprint. https://www.researchgate.net/publication/228601694_Generalized_implied_volatility
Black, F. 1976 The pricing of commodity contracts. Journal of Financial Economics 81(3), 167–179.
Black, F. & Scholes, M. 1973 The pricing of options and corporate liabilities. Journal of Political Economy 81(3), 637–659.
Bluman, G. & Kumei, S. 1989 Symmetries and Differential Equations. Berlin: Springer.
Boness, A. J. 1964 Elements of a theory of a stock option value. Journal of Political Economy 72(2), 163–175.
Boyarchenko, S. & Levendorskii, S. 2002 Non-Gaussian Merton–Black–Scholes Theory. River Edge, NJ: World Scientific.
Carr, P., Lipton, A. & Madan, D. 2002 The reduction method for valuing derivative securities. Working Paper, New York University. https://www.researchgate.net/publication/2860358_The_Reduction_Method_for_Valuing_Derivative_Securities
Cartea, Á. , Drissi, F. & Monga, M. 2023 Predictable losses of liquidity provision in constant function markets and concentrated liquidity markets (August 15). Applied Mathematical Finance. Available at SSRN 4541034: https://ssrn.com/abstract=4541034 or http://dx.doi.org/10.2139/ssrn.4541034
Chandrasekhar, S. 1943 Stochastic problems in physics and astronomy. Reviews of Modern Physics 15(1), 1–89.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon.
Chapman, S. 1928 On the Brownian displacements and thermal diffusion of grains suspended in a nonuniform fluid. Proceedings of the Royal Society of London. Series A 119(781), 34–54.
Cordes, H. O. 1995 The Technique of Pseudodifferential Operators. Cambridge: Cambridge University Press.
Cox, J. C., Ingersoll Jr., J. E. & Ross, S. A. 1985 A theory of the term structure of interest rates. Econometrica 53(2), 385–408.
Craddock, M. 2012 Lie symmetry methods for multi-dimensional parabolic PDEs and diffusions. Journal of Differential Equations 252(1), 56–90.
Craddock, M. & Platen, E. 2004 Symmetry group methods for fundamental solutions. Journal of Differential Equations 207(2), 285–302.
Craik, A. D. D. & Criminale, W. O. 1986 Evolution of wavelike disturbances in shear flows: A class of exact solutions of the Navier–Stokes equations. Proceedings of the Royal Society of London. Series A 406, 13–26.
Dai, Q. & Singleton, K. J. 2000 Specification analysis of affine term structure models. Journal of Finance 55(5), 1943–1978.
Davis, M. H. A. 2004 Complete-market models of stochastic volatility. Proceedings of the Royal Society of London. Series A 460, 11–26.
Derman, E. & Kani, I. 1994 Riding on a smile. Risk Magazine 7(2), 32–39.
Devreese, J. P. A., Lemmens, D. & Tempere, J. 2010 Path integral approach to Asian options in the Black–Scholes model. Physica A 389(4), 780–788.
Di Francesco, M. & Pascucci, A. 2004 On the complete model with stochastic volatility by Hobson and Rogers. Proceedings of the Royal Society of London. Series A 460(2051), 3327–3338.
Di Francesco, M. & Pascucci, A. 2005 On a class of degenerate parabolic equations of Kolmogorov type. Applied Mathematics Research eXpress 3, 77–116.
Dragulescu, A. A. & Yakovenko, V. M. 2002 Probability distribution of returns in the Heston model with stochastic volatility. Quantitative Finance 2(6), 443–453.
Duffie, D., Filipovic, D. & Schachermayer, W. 2003 Affine processes and applications in finance. Annals of Applied Probability 13(3), 984–10053.
Duffie, J. D. & Kan, R. 1996 A yield-factor model of interest rates. Mathematical Finance 6(4), 379–406.
Duffie, D., Pan, J. & Singleton, K. 2000 Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6), 1343–1376.
Duong, M. H. & Tran, H. M. 2018 On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems: Series A 38(7), 3407–3438.
Dupire, B. 1994 Pricing with a smile. Risk Magazine 7(1), 18–20.
Ebeling, W., Gudowska-Nowak, E. & Sokolov, I. M. 2008 On stochastic dynamics in physics: Remarks on history and terminology. Acta Physica Polonica B 39(5), 1003–10017.
Egorov, M. 2019 StableSwap: Efficient mechanism for Stablecoin liquidity. White paper.
Fabijonas, B., Holm, D. D. & Lifschitz, A. 1997 Secondary instabilities of flows with elliptic streamlines. Physical Review Letters 78(10), 1900–1903.
Feller, W. 1951 Two singular diffusion problems. Annals of Mathematics 54(1), 173–182.
Feller, W. 1952 The parabolic differential equations and the associated semi-groups of transformations. Annals of Mathematics 55, 468–518.
Feller, W. 1971 An Introduction to Probability Theory and Its Application, vol. 2, 2nd ed. New York: John Wiley.
Filipovic, D. 2009 Term-structure models. Berlin: Springer.
Fokker, A. D. 1914 Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld. Annalen der Physik 348(5), 810–820.
Fourier, J. B. 1822 Théorie Analytique de la Chaleur. Paris: Firmin Didot, Père et Fils.
Friedlander, S. & Lipton-Lifschitz, A. 2003 Localized instabilities in fluids. In Handbook of Mathematical Fluid Dynamics 2, 289–354. North-Holland: Amsterdam.
Friedlander, S. & Vishik, M. 1991 Instability criteria for the flow of an inviscid incompressible fluid. Physical Review Letters 66, 2204–2206.
Friz, P. & Keller-Ressel, M. 2010. Moment explosions in stochastic volatility models. Encyclopedia of Quantitative Finance, 1247–1253.
Fukasawa, M., Maire, B. & Wunsch, M. 2023 Weighted variance swaps hedge against impermanent loss. Quantitative Finance 23(6), 1–11.
Gatheral, J., Jaisson, T. & Rosenbaum, M. 2018 Volatility is rough. Quantitative Finance 18(6), 933–949.
Geman, H. & Eydeland, A. 1995 Asian options revisited: Inverting the Laplace transform. Risk Magazine 8(4), 65–67.
Gershon, D., Lipton, A., Rosenbaum, M. & Wiener, Z. eds. 2022 Options-45 Years Since the Publication of the Black–Scholes–Merton Model: The Gershon Fintech Center Conference. Singapore: World Scientific.
Gihman, I. I. & Skorohod, A. V. 1972 Stochastic Differential Equations. New York: Springer.
Giorno, V. & Nobile, A. G. 2021 Time-inhomogeneous Feller-type diffusion process in population dynamics. Mathematics 9(16), 1879.
Guyon, J. 2014 Path-dependent volatility. Risk Magazine 27(10).
Hagan, P., Kumar, D., Lesniewski, A. & Woodward, D. 2002 Managing smile risk. Wilmott Magazine 9, 84–1008.
Hänggi, P., Talkner, P. & Borkovec, M. 1990 Reaction-rate theory: Fifty years after Kramers. Reviews of Modern Physics 62(2), 251–341.
Hanson, F. B. 2007. Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis and Computation. Philadelphia, PA: Society for Industrial and Applied Mathematics.
He, C., Chen, J., Fang, H. & He, H. 2021 Fundamental solution of fractional Kolmogorov–Fokker–Planck equation. Examples and Counterexamples 1, 100031.
Heston, S. L. 1993 A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6, 327–343.
Hobson, D. G. & Rogers, L. C. G. 1998 Complete models with stochastic volatility. Mathematical Finance 8(1), 27–48.
Hörmander, L. 1967 Hypoelliptic second order differential equations. Acta Mathematica 119, 147–171.
Hull, J. & White, A. 1990 Pricing interest rate derivative securities. Review of Financial Studies 3, 573–592.
Ibragimov, N. H. 1985 Transformation Groups Applied to Mathematical Physics. Dordrecht: D. Reidel.
Ivasishen, S. D. & Medynsky, I. P. 2010 The Fokker–Planck–Kolmogorov equations for some degenerate diffusion processes. Theory of Stochastic Processes 16(1), 57–66.
Jacob, N. & Schilling, R. L. 2001 Lévy-type processes and pseudodifferential operators. In Lévy Processes: Theory and Applications, Barndorff-Nielsen, O. E. et al. (eds.), (pp. 139–168). Boston, MA: Birkhäuser.
Janek, A., Kluge, T., Weron, R. & Wystup, U. 2011 FX smile in the Heston model. In Statistical Tools for Finance and Insurance (pp. 133–162). Berlin, Heidelberg: Springer.
Jex, M., Henderson, R. & Wang, D. 1999 Pricing exotics under the smile. Risk Magazine 12(11), 72–75.
Kelvin, Lord 1887 Stability of fluid motion: Rectilinear motion of viscous fluid between two parallel planes. Philosophical Magazine 24, 188–196.
Klein, O. 1921 Zur statistischen Theorie der Suspension und Lösungen. Inaugural-Dissertation. Uppsala: Almqvist & Wiksells.
Kolmogoroff, A. 1931 Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Mathematische Annalen 104(1), 415–458.
Kolmogoroff, A. 1933 Zur Theorie der stetigen zufälligen Prozesse. Mathematische Annalen 108, 149–160
Kolmogoroff, A. 1934 Zufallige Bewegungen (Zur Theorie der Brownschen Bewegung), Annals of Mathematics 35(1), 116–117.
Kovalenko, S., Stogniy, V. & Tertychnyi, M. 2014 Lie symmetries of fundamental solutions of one (2+ 1)-dimensional ultra-parabolic Fokker–Planck–Kolmogorov equation. arXiv preprint arXiv:1408.0166.
Kramers, H. A. 1940 Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7(4), 284–304.
Kuptsov, L. P. 1972 The fundamental solutions of a certain class of elliptic-parabolic second order equations. Differential Equations 8, 1649–1660.
Lanconelli, E., Pascucci, A. & Polidoro, S. 2002 Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance. In Sh, Michael. Birman, Stefan Hildebrandt, Solonnikov, Vsevolod A., and Uraltseva, Nina N., eds., Nonlinear Problems in Mathematical Physics and Related Topics, vol. II, 243–265. New York: Kluwer/Plenum.
Langevin, P. 1908 Sur la théorie du mouvement brownien. Comptes rendus de l’Académie des Sciences 146, 530–533.
Lewis, A. 2000 Option Valuation under Stochastic Volatility with Mathematica Code. Newport Beach, CA: Finance Press.
Lewis, A. 2001 A simple option formula for general jump-diffusion and other exponential Lévy processes. Available at SSRN: https://ssrn.com/abstract=282110 or http://dx.doi.org/10.2139/ssrn.282110.
Lifschitz, A. 1991 Short wavelength instabilities of incompressible three-dimensional flows and generation of vorticity. Physics Letters A 157(8–9), 481–487.
Lifschitz, A. 1995 Exact description of the spectrum of elliptical vortices in hydrodynamics and magnetohydrodynamics. Physics of Fluids A 7, 1626–1636.
Lifschitz, A. & Hameiri, E. 1991a A universal instability in fluid dynamics. In 1991 International Sherwood Fusion Theory Conference.
Lifschitz, A. & Hameiri, E. 1991b Local stability conditions in fluid dynamics. Physics of Fluids A 3, 2644–2651.
Lipton, A. 1999 Similarities via self-similarities. Risk Magazine 12(9), 101–1005.
Lipton, A. 2000 Pricing and Risk-Managing Exotics on Assets with Stochastic Volatility. Presentation, Risk Minds, Geneva.
Lipton, A. 2001 Mathematical Methods for Foreign Exchange: A Financial Engineer’s Approach. Singapore: World Scientific.
Lipton, A. 2002 The volatility smile problem. Risk Magazine 15(2), 61–65.
Lipton, A. 2018 Financial Engineering: Selected Works of Alexander Lipton. Singapore: World Scientific.
Lipton, A. 2023 Kelvin Waves, Klein–Kramers and Kolmogorov Equations, Path-Dependent Financial Instruments: Survey and New Results. arXiv preprint arXiv:2309.04547.
Lipton, A., Gal, A. & Lasis, A. 2014 Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: Survey and new results. Quantitative Finance 14(11), 1899–1922.
Lipton, A. & Hardjono, T. 2021 Blockchain intra- and interoperability. In Innovative Technology at the Interface of Finance and Operations: Volume II (pp. 1–30). Cham: Springer International Publishing.
Lipton, A. & Lopez de Prado, M. 2020 A closed-form solution for optimal Ornstein–Uhlenbeck driven trading strategies. International Journal of Theoretical and Applied Finance 23(08), 2050056.
Lipton, A. & Reghai, A. 2023 SPX, VIX and scale-invariant LSV. Wilmott Magazine 2023(126), 78–84.
Lipton, A. & Sepp, A. 2008 Stochastic volatility models and Kelvin waves. Journal of Physics A: Mathematical and Theoretical 41(34), 344012 (23pp).
Lipton, A. & Sepp, A. 2022 Automated market-making for fiat currencies. Risk Magazine 35(5). https://www.risk.net/cutting-edge/investments/7948106/automated-market-making-for-fiat-currencies.
Lipton, A. & Shelton, D. 2012 Credit default swaps with and without counterparty and collateral adjustments. Stochastics: An International Journal of Probability and Stochastic Processes 84(5–6), 603–624.
Lipton, A. & Treccani, A. 2021 Blockchain and Distributed Ledgers: Mathematics, Technology, and Economics. Singapore: World Scientific.
Lipton-Lifschitz, A. 1999 Predictability and unpredictability in financial markets. Physica D: Nonlinear Phenomena 133(1–4), 321–347.
Masoliver, J. 2016 Nonstationary Feller process with time-varying coefficients. Physical Review E 93(1), 012122.
Merton, R. C. 1973 Theory of rational option pricing. Bell Journal of Economics and Management Science 4(1), 141–183.
Merton, R. C. 1976 Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3(1–2), 125–144.
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics, Part I. New York: McGraw-Hill.
Olver, P. J. 1986 Applications of Lie Groups to Differential Equations, 1st ed. New York: Springer.
Orr, W. McF. 1907 The stability or instability of the steady motions of a perfect fluid. Proceedings of the Royal Irish Academy A 27, 9–69.
Ovsiannikov, L. V. 1982 Group Analysis of Differential Equations. New York: Academic Press.
Pascucci, A. 2005 Kolmogorov equations in physics and in finance. In , Elliptic and Parabolic Problems, C. Bandle, et al., eds. Progress in Nonlinear Differential Equations and Their Applications, 63, 313–324. Basel: Birkhäuser.
Piessens, R. 2000 The Hankel Transform. In Poularikas, A. D., ed., The Transforms and Applications Handbook: Second Edition. Boca Raton, FL: CRC Press.
Planck, M. 1917 Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 324–341.
Reghai, A. 2015 Quantitative Finance: Back to Basics. New York: Palgrave MacMillan.
Risken, H. 1989 The Fokker–Planck Equation: Method of Solution and Applications. New York: Springer.
Rogers, L. C. G. & Shi, Z. 1995 The value of an Asian option. Journal of Applied Probability 32(4), 1077–10088.
Rubinstein, M. 1994 Implied binomial trees. Journal of Finance 49(3), 771–818.
Samuelson, P. A. 1965 Rational theory of warrant pricing. Industrial Management Review 6, 13–32.
Schachermayer, W. & Teichmann, J. 2008 How close are the option pricing formulas of Bachelier and Black–Merton–Scholes? Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics 18(1), 155–170.
Schmelzle, M. 2010 Option pricing formulae using Fourier transform: Theory and application. Preprint, https://pfadintegral.com/articles/option-pricing-formulae-using-fourier-transform.
Schöbel, R. & Zhu, J. 1999 Stochastic volatility with an Ornstein–Uhlenbeck process: An extension. Review of Finance 3(1), 23–46.
Sepp, A., 2007. Affine Models in Mathematical Finance: An Analytical Approach. Tartu: University of Tartu Press.
Stein, E. M. & Stein, J. C. 1991 Stock price distributions with stochastic volatility: An analytic approach. Review of Financial Studies 4, 727–752.
Terakado, S., 2019 On the option pricing formula based on the Bachelier model. Available at SSRN 3428994.
Uhlenbeck, G. E. & Ornstein, L. S. 1930 On the theory of Brownian motion. Physical Review 36, 823–841.
Vasicek, O. A. 1977 An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177-188.
Weber, M. 1951 The fundamental solution of a degenerate partial differential equation of parabolic type. Transactions of the American Mathematical Society 71, 24–37.
Wong, M. W. 2014 Introduction to Pseudo-Differential Operators. Singapore: World Scientific Publishing Company.
Zhang, Y., Chen, X. & Park, D. 2018 Formal specification of constant product (x × y = k) market maker model and implementation. White paper. https://pdfcoffee.com/uniswap-formulas-pdf-free.html

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