Free Boundary Problems in Asset Pricing with Transaction Costs

  • Thaleia Zariphopoulou
Part of the Applied Optimization book series (APOP, volume 50)

Abstract

This presentation provides an overview of a class of free boundary problems that arise in valuation models in markets with transaction costs. Transaction costs are a realistic feature in numerous financial transactions and their presence affects considerably the theoretical asset and derivative prices.

In the area of optimal portfolio management, the valuation models give rise to singular stochastic control problems and the goal is to characterize the value function (maximal utility) and to specify the optimal control policies.

In the area of derivative pricing, the classical Black and Scholes valuation theory, based on exact replication, breaks down completely when transaction costs are present. Various approaches have been developed which lead to free boundary problems for the derivative prices. These methods include among others, the method of super-replicating strategies and the utility maximization theory.

Keywords

Transaction Cost Asset Price Viscosity Solution Portfolio Selection Free Boundary Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Akian, M., J. L. Menaldi, and A. Sulem. (1992). “Multi-asset portfolio selection problem with transaction costs. Probabilités numériques,” Mathematics and Computers in Simulation, 38, 163–172.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Avellaneda, M., and A. Paras. (1994). “Optimal hedging portfolios for derivative securities in the presence of large transaction costs,” Applied Mathematical Finance, 1, 165–193.CrossRefGoogle Scholar
  3. [3]
    Bensaid, B., J. Lesne, H. Pages and J. Scheinkman. (1992). “Derivative asset pricing with transaction costs,” Mathematical Finance, 2, 63–86.MATHCrossRefGoogle Scholar
  4. [4]
    Black, F., and M. Scholes. (1973). “The pricing of options and corporate liabilities,” Journal of Political Economy, 81, 637–654.CrossRefGoogle Scholar
  5. [5]
    Boyle, P., and T. Vorst. (1992). “Option replication in discrete time with transaction costs,” Journal of Finance, 47, 271–293.CrossRefGoogle Scholar
  6. [6]
    Capuzzo-Dolcetta, I., and P.-L. Lions. (1990). “Hamilton-Jacobi equations with state constraints,” Transactions of the American Mathematical Society, 318, 543–583.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Constantinides, G. M., and T. Zariphopoulou. (1999). “Bounds on prices of contingent claims in an intertemporal economy with proportional transaction costs and general preferences,” Finance and Stochastics, 3(3), 345–369.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Constantinides, G. M. and T. Zariphopoulou. (2000). “Price bounds on derivative prices in an intertemporal setting with proportional costs and multiple securities,” preprint.Google Scholar
  9. [9]
    Crandall, M. G., H. Ishii, and P.-L. Lions. (1992). “User’s guide to viscosity solutions of second order partial differential equations,” Bulletin of the American Mathematical Society, 27, 1–67.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Crandall, M. G., and P.-L. Lions. (1983). “Viscosity solutions of Hamilton-Jacobi equations,” Transactions of the American Mathematical Society, 277, 1–42.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Davis, M. H. A., and J. M. C. Clark. (1994). “A note on super-replicating strategies,” Philosophical Transactions of the Royal Society of London A, 485–494.Google Scholar
  12. [12]
    Davis, M. H. A., and A. R. Norman. (1990). “Portfolio selection with transaction costs,” Mathematics of Operations Research. 15, 676–713.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Davis, M. H. A., V. Panas, and T. Zariphopoulou. (1993). “European option pricing with transaction costs,” SIAM Journal on Control and Optimization, 31, 470–493.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Davis, M. H. A., and T. Zariphopoulou. (1995). “American options and transaction fees,” Mathematical Finance, IMA Volumes in Mathematics and its Applications, New York, NY: Springer-Verlag.Google Scholar
  15. [15]
    Duffie, D., and T. Zariphopoulou. (1993). “Optimal investment with undiversifiable income risk,” Mathematical Finance, 3, 135–148.MATHCrossRefGoogle Scholar
  16. [16]
    Edirisinghe, C., V. Naik and R. Uppal. (1993). “Optimal replication of options with transaction costs and trading restrictions,” Journal of Finance, 28, 117–138.Google Scholar
  17. [17]
    Figlewski, S. (1989). “Options arbitrage in imperfect markets,” Journal of Finance, 44, 1289–1311.CrossRefGoogle Scholar
  18. [18]
    Fleming, W. H., and H. M. Soner. (1993). Controlled Markov Processes and Viscosity Solutions. New York, NY: Springer Verlag.MATHGoogle Scholar
  19. [19]
    Fleming, W. H., and T. Zariphopoulou. (1991). “An optimal investment/consumption model with borrowing,” Mathematics of Operations Research, 16, 802–822.MathSciNetMATHCrossRefGoogle Scholar
  20. Grannan, E. R., and G. H. Swindle. “Minimizing transaction costs of option hedging strategies,” Mathematical Finance, to appear.Google Scholar
  21. [21]
    Hodges, S. D., and A. Neuberger. (1989). “Optimal replication of contingent claims under transactions costs,” The Review of Futures Markets, 8(2), 222–239.Google Scholar
  22. [22]
    Hoggard, T., E. Whalley and P. Wilmott. (1994). “Hedging option portfolios in the presence of transaction costs,” Advances in Futures and Options Research, 7, 21–35.Google Scholar
  23. [23]
    Ishii, H., and R-L. Lions. (1990). “Viscosity solutions of fully nonlinear second-order elliptic partial differential equations,” Journal of Differential Equations, 83, 26–78.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Leland, H. E. (1985). “Option pricing and replication with transaction costs,” Journal of Finance, 40, 1283–1301.CrossRefGoogle Scholar
  25. [25]
    Levental, S., and A. Skorohod. (1997). “On the possibility of hedging options in the presence of transaction costs,” Annals of Applied Probability, 7, 410–443.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    Lions, P.-L. (1983). “Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations 1: The dynamic programming principle and applications; 2: Viscosity solutions and uniqueness,” Communications in Partial Differential Equations, 8, 1101–1174; 1229–1276.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    Magill, M. J. P., and G. Constantinides. (1976). “Portfolio selection with transaction costs,” Journal of Economic Theory, 13, 245–263.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Merton, R. C. (1969). “Lifetime portfolio selection under uncertainty: the continuous-time case,” Journal of Economic Theory, 3, 247–257.MathSciNetGoogle Scholar
  29. [29]
    Merton, R. C. (1971). “Optimum consumption and portfolio rules in a continuous-time model,” Journal of Economic Theory, 3, 373–413.MathSciNetCrossRefGoogle Scholar
  30. [30]
    Merton, R. C. (1973). “Theory of rational option pricing,” Bell Journal of Economics and Management Science, 4, 141–183.MathSciNetCrossRefGoogle Scholar
  31. [31]
    Merton, R. C. (1990). Continuous Time Finance, Oxford, UK: Basil Blackwell.Google Scholar
  32. [32]
    Pichler, A. (1996). “On transaction costs and HJB equations,” preprint.Google Scholar
  33. [33]
    Shen, Q. (1990). “Bid-ask prices for call options with transaction costs,” University of Pennsylvania Working Paper.Google Scholar
  34. [34]
    Shreve, S. E., and H. M. Soner. (1994). “Optimal investment and consumption with transaction costs,” Annals of Applied Probability, 4(3), 609–692.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    Soner, H. M. (1986). “Optimal control with state space constraints,” SIAM Journal on Control and Optimization, 24, 552–562, 1110–1122.MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    Soner, H. M., S. Shreve and J. Cvitanic. (1995). “There is no non-trivial hedging portfolio for option pricing with transaction costs,” Annals of Applied Probability, 5(2), 327–355.MathSciNetMATHCrossRefGoogle Scholar
  37. [37]
    Sulem, A. (1997). “Dynamic optimization for a mixed portfolio with transaction costs,” Numerical Methods in Finance, Newton Institute, Cambridge, UK: Cambridge University Press.Google Scholar
  38. [38]
    Taksar, M., M. J. Klass and D. Assaf. (1988). “A diffusion model for optimal portfolio selection in the presence of brokerage fees,” Mathematics of Operations Research, 13, 277–294.MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    Toft, K. B. (1996). “On the mean-variance tradeoff in option replication with transactions costs,” Journal of Financial and Quantitative Analysis, 31, 233–263.CrossRefGoogle Scholar
  40. [40]
    Tourin, A., and T. Zariphopoulou. (1994). “Numerical schemes for investment models with singular transactions,” Computational Economics, 7, 287–307.MathSciNetMATHCrossRefGoogle Scholar
  41. [41]
    Tourin, A., and T. Zariphopoulou. (1995). “Portfolio selection with transactions costs,” Progress in Probability, 36, 385–391.MathSciNetGoogle Scholar
  42. [42]
    Tourin, A., and T. Zariphopoulou. (1997). Viscosity solutions and numerical schemes for investment/consumption models with transaction costs, Numerical Methods in Finance, Newton Institute, Cambridge, England: Cambridge University Press, 245–269Google Scholar
  43. [43]
    Tourin, A., and T. Zariphopoulou. (1998). “Super-replicating strategies with probability less than one in the presence of transaction costs,” preprint.Google Scholar
  44. [44]
    Zariphopoulou, T. (1989). “Investment-consumption models with constraints,” Providence, RI: Ph.D. Thesis, Brown University.Google Scholar
  45. [45]
    Zariphopoulou, T. (1992). “Investment/consumption model with transaction costs and markov-chains parameters,” SIAM Journal on Control and Optimization, 30, 613–636.MathSciNetMATHCrossRefGoogle Scholar
  46. [46]
    Zariphopoulou, T. (1994). “Investment and consumption models with constraints,” SIAM Journal on Control and Optimization, 32, 59–84.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Thaleia Zariphopoulou
    • 1
  1. 1.Department of Mathematics and School of BusinessUniversity of WisconsinMadisonUSA

Personalised recommendations