A Longstaff and Schwartz approach to the early election problem.

*(English)*Zbl 1254.91129Summary: In many democratic parliamentary systems, election timing is an important decision availed to governments according to sovereign political systems. Prudent governments can take advantage of this constitutional option in order to maximize their expected remaining life in power. The problem of establishing the optimal time to call an election based on observed poll data has been well studied with several solution methods and various degrees of modeling complexity. The derivation of the optimal exercise boundary holds strong similarities with the American option valuation problem from mathematical finance. A seminal technique refined by F. A. Longstaff and E. S. Schwartz [“Valuing American options by simulation: a simple least-squares approach,” Rev. Fin. Stud. 14, No. 1, 113–147 (2001; doi:10.1093/rfs/14.1.113)] provided a method to estimate the exercise boundary of the American options using a Monte Carlo method and a least squares objective. In this paper, we modify the basic technique to establish the optimal exercise boundary for calling a political election. Several innovative adaptations are required to make the method work with the additional complexity in the electoral problem. The transfer of Monte Carlo methods from finance to determine the optimal exercise of real-options appears to be a new approach.

##### MSC:

91B12 | Voting theory |

91F10 | History, political science |

91G60 | Numerical methods (including Monte Carlo methods) |

91G20 | Derivative securities (option pricing, hedging, etc.) |

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\textit{E. Tonkes} and \textit{D. Lesmono}, Adv. Decis. Sci. 2012, Article ID 287579, 18 p. (2012; Zbl 1254.91129)

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##### References:

[1] | D. Lesmono, E. J. Tonkes, and K. Burrage, “An early political election problem,” ANZIAM Journal, vol. 45, pp. C16-C33, 2003. · Zbl 1097.91070 |

[2] | D. Lesmono and E. J. Tonkes, “Optimal strategies in political elections,” ANZIAM Journal, vol. 46, no. 5, pp. C764-C785, 2004. · Zbl 1074.91514 |

[3] | D. Lesmono, E. J. Tonkes, and K. Burrage, “A continuous time model for election timing,” Australian Mathematical Society Gazette, vol. 32, no. 5, pp. 329-338, 2005. · Zbl 1183.91044 |

[4] | D. Lesmono and E. Tonkes, “Stochastic dynamic programming for election timing: a game theory approach,” Asia-Pacific Journal of Operational Research, vol. 23, no. 3, pp. 287-309, 2006. · Zbl 1103.90070 |

[5] | D. Lesmono, E. Tonkes, and K. Burrage, “Opportunistic timing and manipulation in Australian federal elections,” European Journal of Operational Research, vol. 192, no. 2, pp. 677-691, 2009. · Zbl 1157.91331 |

[6] | N. S. Balke, “The rational timing of parliamentary elections,” Public Choice, vol. 65, no. 3, pp. 201-216, 1990. |

[7] | A. Smith, “Endogenous election timing in majoritarian parliamentary systems,” Economics and Politics, vol. 8, no. 2, pp. 85-110, 1996. |

[8] | A. Smith, “Election timing in majoritarian parliaments,” British Journal of Political Science, vol. 33, no. 3, pp. 397-418, 2003. |

[9] | A. Smith, Election Timing, Cambridge University Press, Cambridge, UK, 2004. |

[10] | R. Bonnerot and P. Jamet, “Numerical computation of the free boundary for the two-dimensional Stefan problem by space-time finite elements,” Journal of Computational Physics, vol. 25, no. 2, pp. 163-181, 1977. · Zbl 0364.65091 |

[11] | R. Carmona and N. Touzi, “Optimal multiple stopping and valuation of swing options,” Mathematical Finance, vol. 18, no. 2, pp. 239-268, 2008. · Zbl 1133.91499 |

[12] | C. Knessl, “A note on a moving boundary problem arising in the American put option,” Studies in Applied Mathematics, vol. 107, no. 2, pp. 157-183, 2001. · Zbl 1152.91522 |

[13] | R. A. Kuske and J. B. Keller, “Optimal exercise boundary for an American put option,” Applied Mathematical Finance, vol. 5, no. 2, pp. 107-116, 1998. · Zbl 1009.91025 |

[14] | H. Pham, “Optimal stopping, free boundary, and American option in a jump-diffusion model,” Applied Mathematics and Optimization, vol. 35, no. 2, pp. 145-164, 1997. · Zbl 0866.60038 |

[15] | V. Bally, G. Pagès, and J. Printems, “A quantization tree method for pricing and hedging multidimensional american options,” Mathematical Finance, vol. 15, no. 1, pp. 119-168, 2005. · Zbl 1127.91023 |

[16] | S. P. Zhu, “A new analytical approximation formula for the optimal exercise boundary of American put options,” International Journal of Theoretical and Applied Finance, vol. 9, no. 7, pp. 1141-1177, 2006. · Zbl 1140.91415 |

[17] | S. P. Zhu and W. T. Chen, “A predictor-corrector scheme based on the ADI method for pricing American puts with stochastic volatility,” Computers and Mathematics with Applications, vol. 62, no. 1, pp. 1-26, 2011. · Zbl 1228.91077 |

[18] | H. Ben-Ameur, M. Breton, and P. L’Ecuyer, “A dynamic programming procedure for pricing American-style Asian options,” Management Science, vol. 48, no. 5, pp. 625-643, 2002. · Zbl 1232.91645 |

[19] | S. P. Zhu, “An exact and explicit solution for the valuation of American put options,” Quantitative Finance, vol. 6, no. 3, pp. 229-242, 2006. · Zbl 1136.91468 |

[20] | F. A. Longstaff and E. S. Schwartz, “Valuing American options by simulation: a simple least-squares approach,” Review of Financial Studies, vol. 14, no. 1, pp. 113-147, 2001. · Zbl 1386.91144 |

[21] | E. Clement, D. Lamberton, and P. Protter, “An analysis of a least squares regression method for American option pricing,” Finance and Stochastics, vol. 6, no. 4, pp. 449-471, 2002. · Zbl 1039.91020 |

[22] | S. Jain and C. W. Oosterlee, “Pricing high-dimensional American options using the stochastic grid method,” 2010, http://ssrn.com/abstract=1723712. · Zbl 1255.91430 |

[23] | D. Lesmono, P. K. Pollett, E. J. Tonkes, and K. A. Burrage, “note on the existence and uniqueness of a bounded mean-reverting process,” Journal of the Indonesian Mathematical Society, vol. 14, no. 2, pp. 83-93, 2008. · Zbl 1214.60025 |

[24] | Enelow, J. M, and J. H. . Melvin, The Spatial Theory of Voting: An Introduction, Cambridge University Press, New York, NY, USA, 1984. · Zbl 1156.91303 |

[25] | J. A. Krosnick and M. K. . Berent, “Comparison of party identification and policy preferences: the impact of survey question format,” American Journal of Political Science, vol. 37, no. 3, pp. 941-964, 1993. |

[26] | J. Zaller and S. Feldman, “A simple theory of survey response: answering questions versus revealing preferences,” American Journal of Political Science, vol. 36, no. 3, pp. 579-616, 1992. |

[27] | Z. M. Kmietowicz, “Sampling errors in political polls,” Teaching Statistics, vol. 16, no. 3, pp. 70-74, 1994. |

[28] | P. E. Kloeden and E. Platen, Numerical Solutions of Stochastic Differential Equations, Springer, Berlin, Germany, 1999. · Zbl 0701.60054 |

[29] | R. E. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, USA, 1957. · Zbl 0077.13605 |

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