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Perpetual options and Canadization through fluctuation theory. (English) Zbl 1039.60044

Summary: It is shown that one is able to evaluate the price of perpetual calls, puts, Russian and integral options directly as the Laplace transform of a stopping time of an appropriate diffusion using standard fluctuation theory. This approach is offered in contrast to the approach of optimal stopping through free boundary problems. Following ideas of P. Carr [Rev. Fin. Studies 11, 597–626 (1998)], we discuss the Canadization of these options as a method of approximation to their finite time counterparts. Fluctuation theory is again used in this case.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60J65 Brownian motion
91B26 Auctions, bargaining, bidding and selling, and other market models
91B28 Finance etc. (MSC2000)
91B70 Stochastic models in economics
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[1] AVRAM, F., Ky PRIANOU, A. E. and PISTORIUS, M. R. (2001). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab. · Zbl 1042.60023
[2] BERTOIN, J. (1996). Lévy Processes. Cambridge Univ. Press. · Zbl 0861.60003
[3] BLUMENTHAL, R. M. (1992). Excursions of Markov Processes. Birkhäuser, Basel. · Zbl 0983.60504
[4] BORODIN, A. N. and SALMINEN, P. (1996). Handbook of Brownian Motion-Facts and Formulae. Birkhäuser, Basel. · Zbl 0859.60001
[5] BOy KOV, Y. and CARR, P. (2001). Analy tic approximation of some options.
[6] CARR, P. (1998). Randomization and the American put. Review of Financial Studies 11 597-626. · Zbl 1386.91134
[7] DUFFIE, J. D. and HARRISON, J. M. (1993). Arbitrage pricing of Russian options and perpetual lookback options. Ann. Appl. Probab. 3 641-651. · Zbl 0783.90009
[8] GERMAN, H. and YOR, M. (1993). Asian options and perpetualities. Math. Finance 3 349-375. · Zbl 0884.90029
[9] GRAVERSEN, S. E. and PESKIR, G. (1998). On the Russian option: The expected waiting time. Theory Probab. Appl. 42 416-425. · Zbl 0924.60012
[10] KARATZAS, I. and SHREVE, S. (1998). Methods of Mathematical Finance. Springer, New York. · Zbl 0941.91032
[11] KRAMKOV, D. O. and MORDECKI, E. (1995). Integral option. Theory Probab. Appl. 39 162-172. · Zbl 0836.90012
[12] LAMBERTON, D. and LAPEy ER, B. (1996). Introduction to Stochastic Calculus Applied to Finance. Chapman and Hall, London.
[13] LAMPERTI, J. W. (1972). Semistable Markov processes. Z. Wahrsch. Verw. Gebiete 22 205-225. · Zbl 0274.60052
[14] LEBEDEV, N. N. (1972). Special Functions and Their Applications. Dover, New York. · Zbl 0271.33001
[15] MCKEAN, H. (1965). Appendix: A free boundary problem for the heat equation arising from a problem of mathematical economics. Industrial Management Review 6 32-39.
[16] My NENI, R. (1992). The pricing of the American option. Ann. Appl. Probab. 2 1-23. · Zbl 0753.60040
[17] ØKSENDAL, B. (1999). Stochastic Differential Equations, 5th ed. Springer, New York. · Zbl 0945.60051
[18] REVUZ, D. and YOR, M. (1994). Continuous Martingales and Brownian Motion, 2nd ed. Springer, New York. · Zbl 0804.60001
[19] SHEPP, L. A. and SHIRy AEV, A. N. (1993). The Russian option: reduced regret. Ann. Appl. Probab. 3 603-631. · Zbl 0783.90011
[20] SHEPP, L. A. and SHIRy AEV, A. N. (1995). A new look at pricing of the ”Russian option.” Theory Probab. Appl. 39 103-119.
[21] SHIRy AEV, A. N. (1978). Optimal Stopping Rules. Springer, Berlin.
[22] SHIRy AEV, A. N., KABANOV, YU., KRAMKOV, D. O. and MELNIKOV, A. V. (1994). Toward the theory of options of both European and American ty pes. II. Continuous time. Theory Probab. Appl. 39 61-102. · Zbl 0833.60065
[23] WILLIAMS, D. (1979). Diffusions, Markov Processes and Martingales I: Foundations. Wiley, New York.
[24] YOR, M. (1984). On square root boundaries for Bessel processes and pole seeking Brownian motion. Stochastic Analy sis and Applications. Lecture Notes in Math. 1095. Springer, New York. · Zbl 0598.60086
[25] YOR, M. (1992). Some Aspects of Brownian Motion I. Birkhäuser, Basel. · Zbl 0779.60070
[26] YOR, M. (1997). Some Aspects of Brownian Motion II. Birkhäuser, Basel. · Zbl 0880.60082
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