Discrete-time pricing and optimal exercise of American perpetual warrants in the geometric random walk model. (English) Zbl 1260.93180
Summary: An American option (or, warrant) is the right, but not the obligation, to purchase or sell an underlying equity at any time up to a predetermined expiration date for a predetermined amount. A perpetual American option differs from a plain American option in that it does not expire. In this study, we solve the optimal stopping problem of a perpetual American option (both call and put) in discrete time using linear programming duality. Under the assumption that the underlying stock price follows a discrete time and discrete state Markov process, namely a geometric random walk, we formulate the pricing problem as an infinite dimensional Linear Programming (LP) problem using the excessive-majorant property of the value function. This formulation allows us to solve complementary slackness conditions in closed-form, revealing an optimal stopping strategy which highlights the set of stock-prices where the option should be exercised. The analysis for the call option reveals that such a critical value exists only in some cases, depending on a combination of state-transition probabilities and the economic discount factor (i.e., the prevailing interest rate) whereas it ceases to be an issue for the put.
MSC:
| 93E20 | Optimal stochastic control |
| 93C05 | Linear systems in control theory |
| 49N15 | Duality theory (optimization) |
| 91B24 | Microeconomic theory (price theory and economic markets) |
| 90C05 | Linear programming |
Keywords:
American perpetual warrants; optimal stopping; optimal exercise; linear programming; duality; combination of state-transition probabilities; geometric random walk model; discrete state Markov process; complementary slackness conditions; underlying stock price; economic discount factor; predetermined expiration date; prevailing interest rate; underlying equity; value function; critical value excessive-majorant property; discrete-time pricingReferences:
| [1] | Çınlar, E.: Introduction to Stochastic Processes. Prentice Hall, New York (1975) |
| [2] | Darling, D., Liggett, T., Taylor, H.: Optimal stopping for partial sums. Ann. Math. Stat. 43, 1363–1368 (1972) · Zbl 0244.60037 · doi:10.1214/aoms/1177692491 |
| [3] | Deligiannidis, G., Le, H., Utev, S.: Optimal stopping for processes with independent increments, and applications. J. Appl. Probab. 46, 1130–1145 (2009) · Zbl 1213.60082 · doi:10.1239/jap/1261670693 |
| [4] | Diermann, F.: Optimal stopping for processes under ambiguity via measure transformation. Master’s thesis, Bielefeld University, Germany. http://erasmus-mundus.univ-paris1.fr/fichiers_etudiants/3160_dissertation.pdf (2010) |
| [5] | Dynkin, E.B., Yushkevich, A.A.: Markov Processes: Theorems and Problems. Plenum, New York (1969). Translated from Russian by James S. Wood |
| [6] | Helmes, K., Stockbridge, R.: Construction of the value function and stopping rules in optimal stopping of one-dimensional diffusions. Adv. Appl. Probab. 42, 158–182 (2010) · Zbl 1195.60063 · doi:10.1239/aap/1269611148 |
| [7] | Hobson, D.: A survey of mathematical finance. Proc. R. Soc. A, Math. Phys. Eng. Sci. 460(2052), 3369–3401 (2004) · Zbl 1168.91386 · doi:10.1098/rspa.2004.1386 |
| [8] | Karatzas, I.: On the pricing of american options. Appl. Math. Optim. 17, 37–60 (1988) · Zbl 0699.90010 · doi:10.1007/BF01448358 |
| [9] | Myneni, R.: The pricing of the American option. Ann. Appl. Probab. 2(1), 1–23 (1992) · Zbl 0753.60040 · doi:10.1214/aoap/1177005768 |
| [10] | Samuelson, P.A.: Rational theory of warrant pricing. Ind. Manage. Rev. 6, 13–31 (1965). Appendix by H.P. McKean, pp. 32–39 |
| [11] | Shreve, S.E.: Stochastic Calculus fr Finance II Continuous-Time Models. Springer, New York (2004) · Zbl 1068.91041 |
| [12] | Sødal, S.: The stochastic rotation problem: a comment. J. Econ. Dyn. Control 26, 509–515 (2002) · Zbl 1007.91044 · doi:10.1016/S0165-1889(00)00076-2 |
| [13] | Sødal, S.: Entry and exit decisions based on a discount factor approach. J. Econ. Dyn. Control 30, 1963–1968 (2006) · Zbl 1162.91385 · doi:10.1016/j.jedc.2005.06.011 |
| [14] | Taksar, M.: Infinite dimensional linear programming approach to singular stochastic control. SIAM J. Control Optim. 35(2), 604–625 (1997) · Zbl 0880.93060 · doi:10.1137/S036301299528685X |
| [15] | van Moerbeke, P.: On optimal stopping and free boundary problems. Arch. Ration. Mech. Anal. 60, 101–148 (1976) · Zbl 0336.35047 |
| [16] | Vanderbei, R., Pınar, M.Ç.: Pricing american perpetual warrants by linear programming. SIAM Rev. 51(4), 767–782 (2009) · Zbl 1180.90191 · doi:10.1137/080728366 |
| [17] | Willassen, Y.: The stochastic rotation problem: A generalization of Faustmann’s formula to stochastic growth. J. Econ. Dyn. Control 22, 573–596 (1998) · Zbl 0899.90065 · doi:10.1016/S0165-1889(97)00071-7 |
| [18] | Wong, D.: Generalized Optimal Stopping Problems and Financial Markets. Addison-Wesley, Longman (1996) · Zbl 0898.90025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
