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The Path Integral Approach to Financial Modeling and Options Pricing

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Abstract

In this paper we review some applications of the path integral methodology of quantum mechanics to financial modeling and options pricing. A path integral is defined as a limit of the sequence of finite-dimensional integrals, in a much the same way as the Riemannian integral is defined as a limit of the sequence of finite sums. The risk-neutral valuation formula for path-dependent options contingent upon multiple underlying assets admits an elegant representation in terms of path integrals (Feynman–Kac formula). The path integral representation of transition probability density (Green's function) explicitly satisfies the diffusion PDE. Gaussian path integrals admit a closed-form solution given by the Van Vleck formula. Analytical approximations are obtained by means of the semiclassical (moments) expansion. Difficult path integrals are computed by numerical procedures, such as Monte Carlo simulation or deterministic discretization schemes. Several examples of path-dependent options are treated to illustrate the theory (weighted Asian options, floating barrier options, and barrier options with ladder-like barriers).

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References

  • Babel, D.F. and Eisenberg, L.K. (1993). Quantity adjusting options and forward contracts, Journal of Financial Engineering 2(2), 89-126.

    Google Scholar 

  • Beaglehole, D. and Tenney, M. (1991). General solutions of some interest rate contingent claim pricing equations, Journal of Fixed Income 1, 69-84.

    Google Scholar 

  • Beilis, A. and Dash, J. (1989a, b). A Multivariate Yield-Curve LognormalModel, CNRS PreprintCPT-89/PE.2334; A Strongly Mean-Reverting Yield Curve Model, CNRS Preprint CPT-89/PE.2337.

  • Birge, J. (1995). Quasi-Monte Carlo Approaches to Options Pricing, University of Michigan Technical Report.

  • Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities, Journal of Political Economy 81, 637-659.

    Google Scholar 

  • Chance, D. and Rich, D. (1995). Asset swaps with Asian-style payoffs, Journal of Derivatives, Summer, 64-77.

  • Cox, J. and Ross, S. (1976). The valuation of options for alternative stochastic processes, Journal of Financial Economics 3, 145-166.

    Google Scholar 

  • Creutz, M., Jacobs, L. and Rebbi, C. (1983). ‘Monte Carlo computations in lattice gauge theories’, Physics Reports 95, 203-282.

    Google Scholar 

  • Dash, J. (1988). Path Integrals and Options, Part I, CNRS Preprint CPT-88/PE.2206.

  • Dash, J. (1989). Path Integrals and Options, Part II, CNRS Preprint CPT-89/PE.2333.

  • Dash, J. (1993). Path Integrals and Options, Invited Talk, SIAM Annual Conference, July.

  • Derman, E., Karasinski, P. and Wecker, J.S. (1990). Understanding Guaranteed Exchange Rate Contracts in Foreign Stock Investments, Goldman Sachs Report, June.

  • Dittrich, W. and Reuter, M. (1994). Classical and Quantum Dynamics: From Classical Paths to Path Integrals, Springer-Verlag, Berlin.

    Google Scholar 

  • Duffie, D. (1996). Dynamic Asset Pricing, 2nd ed., Princeton University Press, Princeton, New Jersey.

    Google Scholar 

  • Durrett, R. (1984). Brownian Motion and Martingales in Analysis, Wadsworth Publishing Co., Belmont, California.

    Google Scholar 

  • Esmailzadeh, R. (1995). Path-Dependent Options, Morgan Stanley Report.

  • Eydeland, A. (1994). A fast algorithm for computing integrals in function spaces: financial applications, Computational Economics 7, 277-285.

    Google Scholar 

  • Feynman, R.P. (1942). The Principle of Least Action in Quantum Mechanics, Ph.D. thesis, Princeton, May 1942.

  • Feynman, R.P. (1948). Space-time approach to non-relativistic quantum mechanics, Review of Modern Physics 20, 367-287.

    Google Scholar 

  • Feynman, R.P. and Hibbs, A. (1965). Quantum Mechanics and Path Integrals, McGraw-Hill, New Jersey.

    Google Scholar 

  • Fradkin, E.S. (1965). The Green's functions method in quantum field theory and quantum statistics, in: Quantum Field Theory and Hydrodynamics, Consultants Bureau, New York.

    Google Scholar 

  • Freidlin, M. (1985). Functional Integration and Partial Differential Equations, Princeton University Press, Princeton, New Jersey.

    Google Scholar 

  • Garman, M. (1985). Towards a semigroup pricing theory, Journal of Finance 40, 847-861.

    Google Scholar 

  • Geman, H. and Eydeland, A. (1995). Domino effect, RISK 8(April).

  • Geman, H. and Yor, M. (1993). Bessel processes, asian options, and perpetuities, Mathematical Finance 3(October) 349-375.

    Google Scholar 

  • Geske, R. and Johnson, H.E. (1984). The American put option valued analytically, Journal of Finance 39, 1511-1524.

    Google Scholar 

  • Glimm, J. and Jaffe, A. (1981). Quantum Physics: A Functional Point of View, Springer-Verlag, Berlin.

    Google Scholar 

  • Ito, K. and McKean, H.P. (1974). Diffusion Processes and Their Sample Paths, Springer-Verlag, Berlin.

    Google Scholar 

  • Jamshidian, F. (1991). Forward induction and construction of yield curve diffusion models, Journal of Fixed Income 1(June), 62-74.

    Google Scholar 

  • Joy, C., Boyle, P.P. and Tan, K.S. (1995). Quasi-Monte Carlo Methods in Numerical FinanceWorking Paper.

  • Harrison, J.M. and Kreps, D. (1979). Martingales and arbitrage in multiperiod security markets, Journal of Economic Theory 20(July) 381-408.

    Google Scholar 

  • Harrison, J.M. and Pliska, S.R. (1981).Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and Applications 11, 215-260.

    Google Scholar 

  • Heynen, R.C. and Kat, G. (1994). Crossing barriers, RISK 7(June) 46-51.

  • Heynen, R.C. and Kat, G. (1994). Partial barrier options, Journal of Financial Engineering 3(3/4), 253-274.

    Google Scholar 

  • Hull, J. (1996). Options, Futures and Other Derivatives, 3rd ed, Prentice Hall, New Jersey.

    Google Scholar 

  • Kac, M. (1949). On distributions of certain wiener functionals, Transactions of American Mathematical Society 65, 1-13.

    Google Scholar 

  • Kac, M. (1951). On some connections between probability and differential and integral equations, Proceedings of the 2nd Berkeley symposium on mathematical statistics and probability, University of California Press, pp. 189-215.

  • Kac, M. (1980). Integration in Function Spaces and Some of its Applications, Academia Nazionale Dei Lincei, Pisa.

    Google Scholar 

  • Karatzas, I. and Shreve, S. (1992). Brownian Motion and Stochastic Calculus, Springer-Verlag, New York.

    Google Scholar 

  • Karlin, S. and Taylor, H.M. (1981). A Second Course in Stochastic Processes, Academic Press.

  • Kemna, A.G.Z. and Vorst, A.C.F. (1990). A pricing method for options based on average asset values, Journal of Banking and Finance 14, 113-124.

    Google Scholar 

  • Kunitomo, N. and Ikeda, M. (1992). Pricing options with curved boundaries, Mathematical Finance 2(October) 275-298.

    Google Scholar 

  • Langouche, F., Roekaerts, D. and Tirapegui, E. (1980). Short derivation of Feynman Lagrangian for general diffusion processes, Journal of Physics A 13, 449-452.

    Google Scholar 

  • Langouche, F., Roekaerts, D. and Tirapegui, E. (1982). Functional Integration and Semiclassical Expansion, Reidel, Dordrecht.

  • Levy, E. (1992). Pricing European average rate currency options, Journal of International Money and Finance, 11, 474-491.

    Google Scholar 

  • Levy, E. and Turnbull, S. (1992). Asian intelligence, RISK, (February) 53-59.

  • Linetsky, V. (1996). Step options and forward contracts, University of Michigan, IOE Technical Report 96-18.

  • Merton, R.C. (1973). Theory of rational options pricing, Bell Journal of Economics and Management Finance, 2, 275-298.

    Google Scholar 

  • Merton, R.C. (1990). Continuous-Time Finance, Blackwell, Cambridge, MA.

    Google Scholar 

  • Metropolis, N, Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E. (1953). Equation of state calculations by fast computing machines, Journal of Chemical Physics 21, 1087-1092.

    Google Scholar 

  • Paskov, S.H. and Traub, J.F. (1995). Faster valuation of financial derivatives, Journal of Portfolio Management, Fall, 113-120.

  • Reiner, E. (1992). Quanto mechanics, RISK, (March), 59-63.

  • Rich, D. (1996). The valuation and behavior of Black-Scholes options subject to intertemporal default risk', Review of Derivatives Research 1, 25-61.

    Google Scholar 

  • Ross, S.A. (1976). The arbitrage theory of capital asset pricing, Journal of Economic Theory 13(December) 341-360.

    Google Scholar 

  • Rubinstein, M. and Reiner, E. (1991). Breaking down the barriers, RISK 4(September), 28-35.

    Google Scholar 

  • Schulman, L.S. (1981). Techniques and Applications of Path Integration, Wiley, New York.

    Google Scholar 

  • Simon, B. (1979). Functional Integration and Quantum Physics, Academic Press, New York.

    Google Scholar 

  • Turnbull, S. and Wakeman, L. (1991). A quick algorithm for pricing european average options, Journal of Financial and Quantitative Analysis 26(September), 377-89.

    Google Scholar 

  • Wilmott, P., Dewynne, J.N. and Howison, S.D. (1993). Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford.

    Google Scholar 

  • Zhang, P. (1994). Flexible Asian options, Journal of Financial Engineering 3(1), 65-83.

    Google Scholar 

  • Zhang, P. (1995a). Flexible arithmetic Asian options, Journal of Derivatives, Spring, 53-63.

  • Zhang, P. (1995b). A unified formula for outside barrier options, Journal of Financial Engineering 4(4), 335-349.

    Google Scholar 

Download references

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  1. Department of Industrial and Operations Engineering, University of Michigan, 272 IOE Building, 1205 Beal Avenue, Ann Arbor, MI, 48109-2117, U.S.A.

    Vadim Linetsky

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  1. Vadim Linetsky
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Linetsky, V. The Path Integral Approach to Financial Modeling and Options Pricing. Computational Economics 11, 129–163 (1997). https://doi.org/10.1023/A:1008658226761

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