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On the possibility of hedging options in the presence of transaction costs. (English) Zbl 0883.90018
Summary: We study the continuous-time problem of hedging a generalized call option of the European and of the American type, in the presence of transaction costs. We show that if the price process of the relevant stock fluctuates with positive probability, then the only hedge that is possible for the American option is the trivial one. If the price of the stock, in addition to fluctuating with positive probability, is also stable with positive probability, then the same is true for the European option. We also show that in some sense, stable price with positive probability is a necessary condition for having only a trivial hedge for the European option. Our basic idea is to work with an appropriate discrete-time version of the problem which is transaction costs free. The mathematical tools that we use are elementary. A related result appears in Soner, Shreve and Cvitanic.

##### MSC:
 91G20 Derivative securities (option pricing, hedging, etc.) 60H30 Applications of stochastic analysis (to PDEs, etc.)
##### Keywords:
continuous-time; hedging a generalized call option
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##### References:
 [1] BENSAID, B., LESNE, J., PAGES, H. and SCHEINKMAN, J. 1992. Derivative asset pricing with transaction costs. Math. Finance 2 63 86. Z. · Zbl 0900.90100 [2] BLACK, F. and SCHOLES, M. 1973. The pricing of options and corporate liabilities. J. Polit. Economy 81 637 659. Z. · Zbl 1092.91524 [3] BOy LE, P. P. and VORST, T. 1992. Option replication in discrete time with transaction costs. J. Finance 47 272 293. Z. [4] COX, J. C. and RUBINSTEIN, M. 1985. Option Markets. Prentice-Hall, Englewood Cliffs, NJ. Z. [5] DAVIS, M. H. and CLARK, J. M. C. 1994. A note on super replicating strategies. Philos. Trans. Roy al Soc., London Ser. A 347 485 494. Z. · Zbl 0822.90020 [6] DUFFIE, D. 1992. Dy namic Asset Pricing Theory. Princeton Univ. Press. Z. [7] EDIRISINGHE, C., NAIK, V. and UPPAL, R. 1993. Optimal replication of options with transaction costs and trading restrictions. Journal of Financial and Quantitative Analy sis 28 117 138. Z. [8] HARRISON, J. M. and PLISKA, S. 1981. Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process. Appl. 11 215 260. Z. · Zbl 0482.60097 [9] KARATZAS, I. and SHREVE, S. 1987. Brownian Motion and Stochastic Calculus. Springer, New York.Z. [10] LELAND, H. E. 1985. Option pricing and replication with transaction costs. J. Finance 40 1283 1301. Z. [11] MERTON, R. C. 1992. Continuous-Time Finance. Blackwell, Cambridge, MA. Z. [12] PROTTER, P. 1990. Stochastic Integration and Differential Equations. Springer, New York. Z. · Zbl 0694.60047 [13] REVUZ, D. and YOR, M. 1991. Continuous Martingales and Brownian Motion. Springer, New York. Z. · Zbl 0731.60002 [14] Ry ZNAR, M. 1994. Private communication. Z. [15] SONER, H. M., SHREVE, S. E. and CVITANIC, J. 1995. There is no nontrivial hedging portfolio for option pricing with transaction costs. Ann. Appl. Probab. 5 327 355. · Zbl 0837.90012 [16] EAST LANSING, MICHIGAN 48824 E-MAIL: levental@stt.msu.edu
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