Perpetual Options for L\'evy Processes in the Bachelier Model

É. Mordecki

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Stochastic financial mathematics, Tome 237 (2002), pp. 256-264.

Résumé

A solution to the optimal stopping problem $V(x)=\sup_\tau\mathsf Ee^{-\delta\tau}g(x+X_\tau)$ is given, where $X=\{X_t\}_{t\ge 0}$ is a Lévy process, $\tau$ is an arbitrary stopping time, $\delta\ge 0$ is a discount rate, and the reward function $g$ takes the form $g_c(x)=(x-K)^+$ or $g_p(x)=(K-x)^+$. The results interpreted as option prices of perpetual options in Bachelier's model are expressed in terms of the distribution of the overall supremum in the case $g=g_c$ and overall infimum in the case $g=g_p$ of the process $X$ killed at rate $\delta$. Closed-form solutions are obtained under mixed exponentially distributed positive jumps with arbitrary negative jumps for $g_c$ and under arbitrary positive jumps and mixed exponentially distributed negative jumps for $g_p$. In the case $g=g_c$, a prophet inequality comparing the prices of perpetual look-back call options and perpetual call options is obtained.

Bibliographie

[1900] Bachelier L., “Théorie de la spéculation”, Ann. Sci. Ecole Norm. Supér. Sér. 3, 17 (1900), 21–86 ; Reprint: The random character of stock market prices, ed. P. H. Cootner, MIT Press, Cambridge, MA, 1967, 17–78 | MR | Zbl

[1901] Bertoin J., Lévy processes, Cambridge Univ. Press, Cambridge, 1996 | MR

[1902] Darling D. A., Ligget T., Taylor H. M., “Optimal stopping for partial sums”, Ann. Math. Statist., 43 (1972), 1363–1368 | DOI | MR | Zbl

[1903] Hill T. P., Kertz R. P., “A survey of prophet inequalities in optimal stopping theory”, Strategies for sequential search and selection in real time, Proc. Conf. Amherst (MA), 1990, Contemp. Math., 125, Amer. Math. Soc., Providence, RI, 1992, 191–207 | MR

[1904] Jacod J., Shiryaev A. N., Limit theorems for stochastic processes, Springer, Berlin, Heidelberg, New York, 1987 | MR

[1905] Kertz R., “Pricing-differentials and bounds for look-back options, and prophet problems in probability”, Advances in stochastic inequalities (Atlanta, GA, 1997), Contemp. Math., 234, Amer. Math. Soc., Providence, RI, 1999, 97–120 | MR | Zbl

[1906] Mordecki E., “Ruin probabilities and optimal stopping for a diffusion with jumps”, Actas del IV Congreso de matemática "Dr. Antonio A.R. Monteiro." (Bahía Blanca, Argentina, 1997), eds. E.L. Fernández Stacco et al., Univ. Nac. Sur. Inst. Mat., Bahía Blanca, 1997, 39–48 | MR

[1907] Mordecki E., “Optimal stopping for a compound Poisson process with exponential jumps”, Publ. Mat. Uruguay, 7 (1998), 55–66 | MR

[1908] Mordecki E., Ruin probabilities for a Lévy process with mixed-exponential negative jumps, Prepubl. Mat. Uruguay. 99/28

[1909] Mordecki E., Optimal stopping and perpetual options for Lévy processes, Prepubl. Mat. Uruguay. 2000/36

[1910] Mordecki E., Optimal stopping, ruin probabilities and prophet inequalities for Lévy processes, Prepubl. Mat. Uruguay. 2000/39

[1911] Shiryaev A. N., Optimal stopping rules, Springer-Verl., New York, Heidelberg, 1978 | MR

[1912] Skorokhod A. V., Sluchainye protsessy s nezavisimymi prirascheniyami, Nauka, M., 1986 ; Skorokhod A. V., Kluwer Acad. Publ., Dordrecht, 1991 | MR | MR | Zbl

[1913] Taylor H. M., “Optimal stopping in a Markov process”, Ann. Math. Statist., 39 (1968), 1333–1344 | DOI | MR | Zbl

[1914] Sato K., Lévy processes and infinitely divisible distributions, Cambridge Stud. Adv. Math., 68, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[1915] Shiryaev A. N., “Essentials of stochastic finance: Facts, models, theory”, Adv. Ser. Statist. Sci. and Appl. Probab., 3, World Sci., Singapore, 1999 | MR | Zbl