# Free Boundary Problems in Asset Pricing with Transaction Costs

## 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## Preview

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## References

- [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]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]Bensaid, B., J. Lesne, H. Pages and J. Scheinkman. (1992). “Derivative asset pricing with transaction costs,” Mathematical Finance, 2, 63–86.zbMATHCrossRefGoogle Scholar
- [4]Black, F., and M. Scholes. (1973). “The pricing of options and corporate liabilities,” Journal of Political Economy, 81, 637–654.CrossRefGoogle Scholar
- [5]Boyle, P., and T. Vorst. (1992). “Option replication in discrete time with transaction costs,” Journal of Finance, 47, 271–293.CrossRefGoogle Scholar
- [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]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.MathSciNetzbMATHCrossRefGoogle Scholar
- [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]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.MathSciNetzbMATHCrossRefGoogle Scholar
- [10]Crandall, M. G., and P.-L. Lions. (1983). “Viscosity solutions of Hamilton-Jacobi equations,” Transactions of the American Mathematical Society, 277, 1–42.MathSciNetzbMATHCrossRefGoogle Scholar
- [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]Davis, M. H. A., and A. R. Norman. (1990). “Portfolio selection with transaction costs,” Mathematics of Operations Research. 15, 676–713.MathSciNetzbMATHCrossRefGoogle Scholar
- [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.MathSciNetzbMATHCrossRefGoogle Scholar
- [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]Duffie, D., and T. Zariphopoulou. (1993). “Optimal investment with undiversifiable income risk,” Mathematical Finance, 3, 135–148.zbMATHCrossRefGoogle Scholar
- [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]Figlewski, S. (1989). “Options arbitrage in imperfect markets,” Journal of Finance, 44, 1289–1311.CrossRefGoogle Scholar
- [18]Fleming, W. H., and H. M. Soner. (1993).
*Controlled Markov Processes and Viscosity Solutions*. New York, NY: Springer Verlag.zbMATHGoogle Scholar - [19]Fleming, W. H., and T. Zariphopoulou. (1991). “An optimal investment/consumption model with borrowing,” Mathematics of Operations Research, 16, 802–822.MathSciNetzbMATHCrossRefGoogle Scholar
- Grannan, E. R., and G. H. Swindle. “Minimizing transaction costs of option hedging strategies,” Mathematical Finance, to appear.Google Scholar
- [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]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]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.MathSciNetzbMATHCrossRefGoogle Scholar
- [24]Leland, H. E. (1985). “Option pricing and replication with transaction costs,” Journal of Finance, 40, 1283–1301.CrossRefGoogle Scholar
- [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.MathSciNetzbMATHCrossRefGoogle Scholar
- [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.MathSciNetzbMATHCrossRefGoogle Scholar
- [27]Magill, M. J. P., and G. Constantinides. (1976). “Portfolio selection with transaction costs,” Journal of Economic Theory, 13, 245–263.MathSciNetzbMATHCrossRefGoogle Scholar
- [28]Merton, R. C. (1969). “Lifetime portfolio selection under uncertainty: the continuous-time case,” Journal of Economic Theory, 3, 247–257.MathSciNetGoogle Scholar
- [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]Merton, R. C. (1973). “Theory of rational option pricing,” Bell Journal of Economics and Management Science, 4, 141–183.MathSciNetCrossRefGoogle Scholar
- [31]Merton, R. C. (1990).
*Continuous Time Finance*, Oxford, UK: Basil Blackwell.Google Scholar - [32]Pichler, A. (1996). “On transaction costs and HJB equations,” preprint.Google Scholar
- [33]Shen, Q. (1990). “Bid-ask prices for call options with transaction costs,” University of Pennsylvania Working Paper.Google Scholar
- [34]Shreve, S. E., and H. M. Soner. (1994). “Optimal investment and consumption with transaction costs,” Annals of Applied Probability, 4(3), 609–692.MathSciNetzbMATHCrossRefGoogle Scholar
- [35]Soner, H. M. (1986). “Optimal control with state space constraints,” SIAM Journal on Control and Optimization, 24, 552–562, 1110–1122.MathSciNetzbMATHCrossRefGoogle Scholar
- [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.MathSciNetzbMATHCrossRefGoogle Scholar
- [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]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.MathSciNetzbMATHCrossRefGoogle Scholar
- [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]Tourin, A., and T. Zariphopoulou. (1994). “Numerical schemes for investment models with singular transactions,” Computational Economics, 7, 287–307.MathSciNetzbMATHCrossRefGoogle Scholar
- [41]Tourin, A., and T. Zariphopoulou. (1995). “Portfolio selection with transactions costs,” Progress in Probability, 36, 385–391.MathSciNetGoogle Scholar
- [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]Tourin, A., and T. Zariphopoulou. (1998). “Super-replicating strategies with probability less than one in the presence of transaction costs,” preprint.Google Scholar
- [44]Zariphopoulou, T. (1989). “Investment-consumption models with constraints,” Providence, RI: Ph.D. Thesis, Brown University.Google Scholar
- [45]Zariphopoulou, T. (1992). “Investment/consumption model with transaction costs and markov-chains parameters,” SIAM Journal on Control and Optimization, 30, 613–636.MathSciNetzbMATHCrossRefGoogle Scholar
- [46]Zariphopoulou, T. (1994). “Investment and consumption models with constraints,” SIAM Journal on Control and Optimization, 32, 59–84.MathSciNetzbMATHCrossRefGoogle Scholar