Summary
We develop a method of assigning unique prices to derivative securities, including options, in the continuous-time finance model developed in Raimondo [47]. In contrast with the martingale method of valuing options, which cannot distinguish among infinitely many possible option pricing processes for a given underlying securities price process when markets are dynamically incomplete, our option prices are uniquely determined in equilibrium in closed form as a function of the underlying economic data.
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This paper is dedicated to the memory of Birgit Grodal, whose strength and character we greatly admired. We are very grateful to Theo Diasakos, Darrell Duffie, Steve Evans, Botond Koszegi, Roger Purves, Jacob Sagi, Chris Shannon, Bill Zame and an anonymous refereee for very helpful discussions and comments. The work of both authors was supported by U.S. National Science Foundation Grant SES-9710424, Anderson’s work was also supported by U.S. National Science Foundation Grant SES-0214164, while Raimondo’s work was also supported by Australian Research Council grant DP0558187. Anderson is also grateful for the gracious hospitality of the Economic Theory Center at the University of Melbourne. Some of these results appeared previously in Anderson and Raimondo (2005).
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References
Anderson, Robert M., “A Nonstandard Representation of Brownian Motion and Itô Integration,” Israel Journal of Mathematics 25(1976), 15–46.
Anderson, Robert M., “Star-finite Representations of Measure Spaces,” Transactions of the American Mathematical Society 271(1982), 667–687.
Anderson, Robert M., Infinitesimal Methods in Mathematical Economics, Preprint (2000).
Anderson, Robert M., Theodoros Diasakos and Roberto C. Raimondo, “Qualitative Properties of Equilibrium Option Pricing,” in preparation (2004).
Anderson, Robert M. and Roberto C. Raimondo, “Market Clearing and Derivative Pricing,” Economic Theory 25(2005), 21–34.
Anderson, Robert M. and Salim Rashid, “A Nonstandard Characterization of Weak Convergence,” Proceedings of the American Mathematical Society 69(1978), 327–332.
Anderson, Robert M. and Roberto C. Raimondo, “An Equilibrium Existence Theorem in Continuous-Time Finance,” preprint (2003).
Back, Kerry, “Asset Pricing for General Processes,” Journal of Mathematical Economics 20(1991), 371–395.
Bick, Avi, “On the Consistency of the Black-Scholes MOdel with a General Equilibrium Framework,” The Journal of Financial and Quantitative Analysis 22(1987), 259–275.
Bick, Avi, “On Viable Diffusion Price Processes of the Market Portfolio,” Journal of Finance 15(1990), 673–689.
Black, Fischer and Myron Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy 81(1973), 673–654.
Campbell, John Y. and Lo, Andrew W. and MacKinley, A. Craig, The Econometrics of Financial Markets, Princeton University Press, Princeton, New Jersey, 1997.
Cox, John C., Jonathan E. Ingersoll, Jr., and Stephen A. Ross, “An Intertemporal General Equilibrium Model of Asset Prices,” Econometrica 53(1985), 363–384.
Cutland, Nigel J., “Infinitesimal Methods in Control Theory: Deterministic and Stochastic,” Acta Applicandae Mathematicae 5(1986), 105–135.
Cutland, Nigel J., P. Ekkehard Kopp, and Walter Willinger, “A Nonstandard Approach to Option Pricing,” Mathematical Finance 1(1991a), 1–38.
Cutland, Nigel J., P. Ekkehard Kopp, and Walter Willinger, “Nonstandard Methods in Option Pricing,” Proceedings of the 30th I.E.E.E. Conference in Decision and Control (1991b), 1293–1298.
Cutland, Nigel J., P. Ekkehard Kopp, and Walter Willinger, “Convergence of Cox-Ross-Rubinstein to Black-Scholes,” manuscript (1991c).
Cutland, Nigel J., P. Ekkehard Kopp, and Walter Willinger, “From Discrete to Continuous Financial Models: New Convergence Results for Option Pricing,” Mathematical Finance 3(1993), 101–123.
Cutland, Nigel J., P. Ekkehard Kopp, and Walter Willinger, “Stock Price Returns and the Joseph Effect: A Fractional Version of the Black-Scholes Model,” Progress in Probability 36(1995a), 327–351.
Cutland, Nigel J., P. Ekkehard. Kopp, and Walter Willinger, “From Discrete to Continuous Stochastic Calculus,” Stochastics and Stochastics Reports 52(1995b), 173–192.
Cutland, Nigel J., P. Ekkehard Kopp, Walter Willinger and M. C. Wyman, “Convergence of Snell Envelopes and Critical Prices in the American Put,” in M. H. A. Demmpster and S. R. Pliska, eds., Mathematics for Derivative Securities, Isaac Newton Institute Proceedings Volume, Cambridge University Press, Cambridge (forthcoming).
Duffie, Darrell, Security Markets, Stochastic Models, Academic Press, Boston, MA, 1995.
Duffie, Darrell and Wayne Shafer, “Equilibrium in Incomplete Markets I: Basic Model of Generic Existence,” Journal of Mathematical Economics 14(1985), 285–300.
Duffie, Darrell and Wayne Shafer, “Equilibrium in Incomplete Markets II: Generic Existence in Stochastic Economies,” Journal of Mathematical Economics 15(1986), 199–216.
Duffie, Darrell and Costis Skiadas, “Continuous-time security pricing: A Utility Gradient Approach,” Journal of Mathematical Economics 23(1994), 107–131.
Harrison, J. Michael and David M. Kreps, “Martingales and Arbitrage in Multiperiod Securities Markets,” Journal of Economic Theory 20 (1979), 381–408.
Heston, Steven L., “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies 6(1993), 327–343.
He, Hua and Hayne Leland, “On Equilibrium Asset Price Processes,” Review of Financial Studies 6(1995), 593–617.
Hindy, Ayman and Chi-Fu Huang, “On Intertemporal Preferences for Uncertain Consumption: A Continuous Time Approach,” Econometrica 60(1992), 781–801.
Hindy, Ayman and Chi-Fu Huang, “Optimal Consumption and Portfolio Rules with Durability and Local Substitution,” Econometrica 61(1993), 85–121.
Hindy, Ayman, Chi-Fu Huang and David M. Kreps, “On Intertemporal Preferences with a Continuous Time Dimension: The Case of Certainty,” Journal of Mathematical Economics 21(1992), 401–420.
Hoover, Douglas N. and Edwin A. Perkins, “Nonstandard Construction of the Stochastic Integral and Applications to Stochastic Differential Equations,” preprint, 1980.
Hurd, Albert E. and Peter A. Loeb, An Introduction to Nonstandard Real Analysis, Academic Press, Orlando, 1985.
Keisler, H. Jerome, An Infinitesimal Approach to Stochastic Analysis, Memoirs of the American Mathematical Society 48(1984), Number 297.
Lindström, T. L., “Hyperfinite Stochastic Integration I: the Nonstandard Theory,” Mathematica Scandinavica 46(1980), 265–292.
Lindström, T. L., “Hyperfinite Stochastic Integration II: Comparison with the Standard Theory,” Mathematica Scandinavica 46(1980), 293–314.
Lindström, T. L., “Hyperfinite Stochastic Integration III: Hyperfinite Representations of Standard Martingales,” Mathematica Scandinavica 46(1980), 315–331.
Lindström, T. L., “Addendum to Hyperfinite Stochastic Integration III,” Mathematica Scandinavica 46(1980), 332–333.
Loeb, Peter A., “Conversion from Nonstandard to Standard Measure Spaces and Applications in Potential Theory”, Transactions of the American Mathematical Society, 211(1975), 113–122.
Lucas, Robert E. Jr., “Asset Prices in an Exchange Economy,” Econometrica 46(1978), 1426–1446.
Magill, Michael and Martine Quinzii, Theory of Incomplete Markets, Volume 1, MIT Press, Cambridge, MA 1996.
Merton, Robert C., Continuous-Time Finance, Blackwell, Cambridge (MA), 1990.
Métivier, M., Reelle und Vektorwertige Quasimartingele und die Theorie der Stochastischen Integration, Lecture Notes in Mathematics 607, Springer-Verlag, Berlin-Heidelberg-New York, 1977.
Nielsen, Lars Tyge, Pricing and Hedging of Derivative Securities, Oxford University Press, Oxford-New York, 1999.
Radner, Roy, “Existence of Equilibrium of Planes, Prices and Price Expectations in a Sequence of Markets,” Econometrica 40(1972), 289–303.
Raimondo, Roberto C., “Incomplete Markets with a Continuum of States,” preprint, 2001.
Raimondo, Roberto C., “Market Clearing, Utility Functions, and Securities Prices,” Economic Theory 25(2005), 265–285.
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Anderson, R.M., Raimondo, R.C. (2006). Equilibrium Pricing of Derivative Securities in Dynamically Incomplete Markets. In: Schultz, C., Vind, K. (eds) Institutions, Equilibria and Efficiency. Studies in Economic Theory, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-28161-4_3
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