No Access

AN ANALYTIC RECURSIVE METHOD FOR OPTIMAL MULTIPLE STOPPING: CANADIZATION AND PHASE-TYPE FITTING

TIM LEUNG

IEOR Department, Columbia University, New York, NY 10027, USA

Corresponding author.

Search for more papers by this author

KAZUTOSHI YAMAZAKI

Department of Mathematics, Kansai University, Osaka, Japan

Search for more papers by this author

, and

HONGZHONG ZHANG

Statistics Department, Columbia University, New York, NY 10027, USA

Search for more papers by this author

https://doi.org/10.1142/S0219024915500326Cited by:16 (Source: Crossref)

Abstract

We study an optimal multiple stopping problem for call-type payoff driven by a spectrally negative Lévy process. The stopping times are separated by constant refraction times, and the discount rate can be positive or negative. The computation involves a distribution of the Lévy process at a constant horizon and hence the solutions in general cannot be attained analytically. Motivated by the maturity randomization (Canadization) technique by Carr (1998), we approximate the refraction times by independent, identically distributed Erlang random variables. In addition, fitting random jumps to phase-type distributions, our method involves repeated integrations with respect to the resolvent measure written in terms of the scale function of the underlying Lévy process. We derive a recursive algorithm to compute the value function in closed form, and sequentially determine the optimal exercise thresholds. A series of numerical examples are provided to compare our analytic formula to results from Monte Carlo simulation.

Keywords:

References

L. Alili and A. E. Kyprianou, Annals of Applied Probability 15(3), 2062 (2005). Web of Science, Google Scholar
S. Asmussen, F. Avram and M. R. Pistorius, Stochastic Processes and Their Applications 109(1), 79 (2004). Web of Science, Google Scholar
S. Asmussen, D. Madan and M. Pistorius, Journal of Computational Finance 11(2), 79 (2007). Google Scholar
S. Asmussen , O. Nerman and M. Olsson , Scandinavian Journal of Statistics 23 , 419 ( 1996 ) . Web of Science, Google Scholar
S. Asmussen and J. Rosiński, Journal of Applied Probability 38(2), 482 (2001). Web of Science, Google Scholar
F. Avram , A. E. Kyprianou and M. R. Pistorius , Annals of Applied Probability 14 , 215 ( 2004 ) . Web of Science, Google Scholar
F. Avram, Z. Palmowski and M. R. Pistorius, Annals of Applied Probability 17(1), 156 (2007). Web of Science, Google Scholar
C. Bender , Finance and Stochastics 15 , 1 ( 2011 ) . Web of Science, Google Scholar
F. Black, Journal of Finance 50(5), 1371 (1995). Web of Science, Google Scholar
B. Bouchard , N. El Karoui and N. Touzi , Annals of Applied Probability 15 , 2575 ( 2005 ) . Web of Science, Google Scholar
N. Cai and L. Sun, Journal of Economic Dynamics and Control 40(3), 213 (2014). Web of Science, Google Scholar
R. Carmona and S. Dayanik, Mathematics of Operations Research 33(2), 446 (2008). Web of Science, Google Scholar
R. Carmona and N. Touzi, Mathematical Finance 18(2), 239 (2008). Web of Science, Google Scholar
P. Carr , Review of Financial Studies 11 , 597 ( 1998 ) . Web of Science, Google Scholar
N. Chiara, M. Garvin and J. Vecer, Journal of Infrastructure Systems 13(2), 97 (2007). Web of Science, Google Scholar
S. Christensen and J. Lempa, Applied Mathematical and Optimization 71(1), 95 (2015). Web of Science, Google Scholar
E. Dahlgren and T. Leung , Journal of Economic Dynamics and Control 53 , 251 ( 2015 ) . Web of Science, Google Scholar
M. Dai and Y.-K. Kwok, Journal of Economic Dynamics and Control 32(7), 2269 (2008). Web of Science, Google Scholar
A. Dixit and R. Pindyck , Investment Under Uncertainty ( Princeton University Press , Princeton , 1994 ) . Google Scholar
M. Egami, T. Leung and K. Yamazaki, Stochastic Processes and Their Applications 123(2), 347 (2013). Web of Science, Google Scholar
M. Egami and K. Yamazaki, Stochastics 85(1), 111 (2013). Google Scholar
M. Egami and K. Yamazaki , Journal of Computational and Applied Mathematics 264 , 1 ( 2014 ) . Web of Science, Google Scholar
A. Ferreiro-Castilla et al. , Stochastic Processes and Their Applications 124 , 985 ( 2014 ) . Web of Science, Google Scholar
M. Grasselli and V. Henderson, Journal of Economic Dynamics and Control 33(1), 109 (2009). Web of Science, Google Scholar
V. Henderson and D. Hobson, Mathematical Finance 21(3), 365 (2011). Web of Science, Google Scholar
B. Hilberink and L. C. G. Rogers, Finance and Stochastics 6(2), 237 (2002). Web of Science, Google Scholar
J. Jacod and A. N. Shiryaev , Limit Theorems for Stochastic Processes 288 ( Springer-Verlag , Berlin , 2003 ) . Google Scholar
F. Kleinert & K. van Schaik (2013) A variation of the Canadisation algorithm for the pricing of American options driven by Lévy processes. Working Paper . Google Scholar
M. Kobylanski, M.-C. Quenez and E. Rouy-Mironescu, Annals of Applied Probability 21(4), 1365 (2011). Web of Science, Google Scholar
A. Kuznetsov , A. Kyprianou and V. Rivero , Springer Lecture Notes in Mathematics 2061 , 97 ( 2013 ) . Web of Science, Google Scholar
A. Kuznetsov et al. , Annals of Applied Probability 21 , 1749 ( 2011 ) . Web of Science, Google Scholar
A. E. Kyprianou , Introductory Lectures on Fluctuations of Lévy Processes with Applications ( Springer-Verlag , Berlin , 2006 ) . Google Scholar
A. E. Kyprianou and Z. Palmowski, Journal of Applied Probability 44(2), 428 (2007). Web of Science, Google Scholar
A. E. Kyprianou and M. R. Pistorius, Annals of Applied Probability 13(3), 1077 (2003). Web of Science, Google Scholar
A. E. Kyprianou and B. A. Surya, Finance and Stochastics 11(1), 131 (2007). Web of Science, Google Scholar
T. Leung and M. Ludkovski, SIAM Journal on Financial Mathematics 2(1), 768 (2011). Web of Science, Google Scholar
T. Leung and M. Ludkovski, Mathematics and Financial Economics 6(4), 363 (2012). Web of Science, Google Scholar
T. Leung and R. Sircar, Mathematical Finance 19(1), 99 (2009). Web of Science, Google Scholar
T. Leung and R. Sircar, SIAM Journal on Control and Optimization 48(3), 1422 (2009). Web of Science, Google Scholar
T. Leung and K. Yamazaki, Quantitative Finance 13(1), 137 (2013). Web of Science, Google Scholar
T. Leung , K. Yamazaki and H. Zhang , SIAM Journal on Control and Optimization ( 2015 ) . Google Scholar
S. Levendorskiı, SIAM Journal on Financial Mathematics 2(1), 79 (2011). Web of Science, Google Scholar
R. L. Loeffen, Annals of Applied Probability 18(5), 1669 (2008). Web of Science, Google Scholar
R. McDonald and D. Siegel, International Economic Review 26(2), 331 (1985). Web of Science, Google Scholar
N. Meinshausen and B. Hambly, Mathematical Finance 14(4), 557 (2004). Web of Science, Google Scholar
E. Mordecki , Finance and Stochastics 6 , 473 ( 2002 ) . Web of Science, Google Scholar
B. A. Surya and K. Yamazaki, International Journal of Theoretical and Applied Finance 17(2), 1450013 (2014). Link, Google Scholar
H. G. Tucker , Pacific Journal of Mathematics 12 , 1125 ( 1962 ) . Web of Science, Google Scholar
J. Xia and X. Zhou, Mathematical Finance 17(2), 307 (2007). Web of Science, Google Scholar
K. Yamazaki , Applied Mathematics and Optimization ( 2014 ) . Google Scholar
A. B. Zeghal and M. Mnif, International Journal of Theoretical and Applied Finance 9(8), 1267 (2006). Link, Google Scholar