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^{1}Energy Edge Pty Ltd, 141 Queen Street, Brisbane, Queensland, 4000, Australia

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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 Longstaff and Schwartz in 2001 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.

This paper is concerned with a new approach for establishing the optimal decision criteria for calling an early election within an electoral environment which permits a government such an option. The problem is predicated on the assumption that a government endures a stochastic level of popularity, and that popularity can be translated into a probability distribution for the likelihood of being returned to government at a general election. This problem has been studied in [

Intuitively, as the government rises higher in the popular opinion polls, it should become more beneficial to call an early election, as a successful election outcome will yield the government another full term in power. However, the decision is not entirely trivial, because calling an election (even when high in the polls) is not entirely risk free, and the government puts in jeopardy their remaining (certain) term in power for an (uncertain) extended period in power. As the popularity polls become higher and the electoral term grows closer to an end, the decision becomes clear to call an early election; as the government’s popularity diminishes, it becomes clear to defer an election. The optimal boundary is the distinct transition point at which the decision is made to call or defer the election, depending on the state.

Approaches to establish the optimal boundary for the early election have been treated by various methods. Balke [

Optimal exercise and free boundary problems occur frequently in physical and decision sciences. Depending on the particular context, the problem can be phrased as a free boundary problem, a moving boundary problem, or an optimal-stopping problem. The Stefan problem in physical sciences [

In the decision sciences, the American option problem has been solved numerically with binary, tertiary, and multinodal trees [

However, all of the techniques are computationally intensive and are best suited for a problem structured with a single state variable (i.e., an option which is written on a single underlying).

Under the fundamental theorem of financial calculus, the problem of option pricing reverts to the calculation of conditional expectations for a payoff under particular probability measures. Monte Carlo techniques have been applied with great success for valuing standard and exotic financial options. Until recently, it was widely held that the Monte Carlo approach was not applicable for options with a control such as an exercise decision.

Longstaff and Schwartz [

The practical benefits become apparent as Monte Carlo simulations are only required to

Analytical studies [

The problem of establishing the optimal exercise boundary for political elections has been successfully attacked using several of the standard free boundary approaches from applied mathematics. Publications such as [

This paper applies the basic philosophy of the Monte Carlo method of Longstaff-Schwartz adapted to the particular subtleties of the election problem. It is our understanding that the approach in [

In the current paper, the novel extensions of the standard Longstaff-Schwartz method are summarized as follows.

The random process describing the stochastic behaviour is governed by a different SDE compared to the geometric Brownian motion typically applied for American option pricing on assets.

The payoff for the option is not a deterministic quantity. For the American option, the payoff upon exercise is known precisely according to the standard payoff formula for the put or call option. In the present context, if an election is called (i.e., the option is exercised), then government is still subject to the uncertainty of the electoral polls and a win is never guaranteed. The election outcome, and therefore the payoff are random variables because there remains a positive probability that the party will lose government, even from very high in the polls.

Once an election is called, there is a delay until the exercise date. This differs from the American option in which the call yields an immediate exercise. This element contributes a source of randomness described in the previous point. However, we must adapt the Longstaff-Schwartz method in order to accommodate the subtleties.

Most importantly, the recursive nature of the payoff makes the implementation of a MLS more complex than the original implementation in [

The structure of the remainder of this paper is as follows. In Section

Define a time-varying state variable

Constitutionally, there is also a positive period between announcing and holding the election, which we call the

Define a

The statement of the optimal election problem is

to determine an optimal strategy which maximizes the

to establish the expected remaining life.

The optimal election problem is couched in an environment which does not accommodate real or financial hedging as is conducted in derivative markets. While the analogous problem for American options is developed under a risk-neutral measure [

We model

We assume that opinion polls are driven by random processes and obey a Markov property, where the current state depends only upon the last observed polls. This underlying process consists of increments driven by the current state and a Gaussian process. The parametric formulation of the SDE for

The state of the polls,

The sampling error

The model (

Article [

Proportion of seats won depending on nationwide proportion of votes won. Resultant probability of winning majority of seats as a function of nationwide vote.

This section develops the algorithm for establishing the value function and the optimal exercise boundary by modifying the Longstaff-Schwartz method. We term it the Modified Longstaff-Schwartz (MLS) method, as adapted for the optimal election exercise problem. The main stages in the MLS algorithm are described here. The details on each individual component are explained in the next subsections. The main stages in the MLS algorithm are described in Table

Step | Summary | Description |
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0 | Develop initial estimate for the value function (Detail Section |
Establish an estimate for the value |

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1 | Simulation of the poll process |
Generate trajectories of the SDE ( |

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2 | Simulation of electoral outcome if calling an election (Detail Section |
At each decision point |

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3 | Simulation of electoral outcome if not calling an election | The alternative to the step 2 is to NOT call an election. In that case, the polls will diffuse to the next timestep |

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4 | Perform regression (Detail Section |
Regress over all |

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5 | Establish max and strategy | Establish the maximum arising from steps 3 or 4; if 3 then assign the optimal strategy as CALL, else CONTINUE. Assign the value function to each of the points |

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6 | Repeat over all timesteps | Repeat over all time steps stepping backwards from |

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7 | Update initial value estimate | When time zero is reached, the value function will disagree with the estimate from step 0. |

The solution space is discretised

At the heart of the Monte Carlo approach is the simulation of solutions to the SDE (

Sample trajectories of the poll process are modeled using the Euler Maruyama formulation expressed in (

The value function is defined in Section

Figure

Recursive nature of establishing the value function. The electoral cycle commences at time zero in government. As time progresses, the poll state (red curve) diffuses. The latest date for an election (compulsory election) is at time

In more formal terms, express the state of the system as the triple

In the case that the election has been already called and the system is in a campaign state, then there are no decisions to be made.

Let

The value function

The optimization problem is then a strategy selection problem:

According to the Bellman principle of optimality [

The terminal condition is a forced election, that is,

The first term in (

We initially assume a structure for

A stochastic dynamic program implements the principle of optimality by back stepping from the terminal condition, progressing through timesteps

The MLS method does not enumerate all states explicitly but creates a regression to approximate the two terms in (

The method requires an estimate for the initial value

In contrast to the method on American options [

Unlike the typical American Option problem in finance, when the option holder (the government) exercises the option (calls an early election), the payoff is not known with certainty. Instead, there is a lag of around 55 days from calling to holding the election, during which time the polls can diffuse. The measured poll state at the final date is distorted from the actual poll state owing to sampling and response errors. And finally, the mapping of the proportion of votes to the number of parliamentary seats won is not certain and can result in an exaggerated majority.

Consequently, at each decision point of call/continue, the MLS method requires to establish the expectation of the value function under the alternatives of calling an election or continuing to the next decision point. There following alternatives are available for the practical implementation to estimate those expectations:

In this paper, we will pursue the

As the MLS proceeds, at each time step a regression is performed between the poll state and the payoff or value function. In the seminal paper of Longstaff and Schwartz on the American Option problem [

Figure

Value function regressions on basis of call or continue. Figure contains remaining life on each simulation under assumption of calling the election (blue scatterplot) and its regression (blue curve), remaining life under assumption of continuing without calling (red scatterplot) and its regression (red curve), and maximum outcome of the two decisions (broken black curve).

If continuing, the value function for simulation

And if the process continues, the value function will be determined from the simulated election outcome:

The value

Figure

The figure illustrates the dichotomous nature of the electoral process: the outcome of the election is either a win (with a greatly extended life in power) or loss (with limited life in power remaining).

The figure is also overlaid with the call decision outcome and with the continue decision outcomes. Under the strategy to continue until the next decision time

Comparing the two regressions, it becomes an optimal strategy for the government to select the alternative which maximizes the value function, which is a call decision if the polls are sufficiently high and a continue decision otherwise. The ability to defer the election decision delivers asymmetric benefit to the government, and with some imagination, the “kink”, the black curve at

In the present problem, we apply a polynomial fit for the regression. Some motivation for the polynomial fit is provided by the rapid speed of fitting a large number of points to a smooth curve: a task which must be done many times in the algorithm. Alternative fits such as sigmoidal functions provide other avenues of research.

Intuitively, given the high mean reversion tendency of the process (

The value function near the terminal time (

In selecting the initial value, we choose a constant function. Motivation arrives from the following reasons.

The mean reversion of the SDE dictates that the poll state tends to a stationary distribution.

The relatively high mean reversion tendency

Under the assumption of a fast mean reversion rate and no systematic bias for one party to win an election, the unconditional probability at time zero of a party losing the next election without an ability for early exercise is 0.5 (i.e., life in power is

Figure

Sample trajectories (grey) and historical polls April 1993–March 2012 (blue).

The algorithm described in Section

10,000 simulations,

campaign time 55 days,

daily exercise decisions,

iterating the algorithm until a practical level of convergence of the initial condition

Owing to the random nature of the algorithm, perfect convergence of the initial condition is not expected, and we have set a threshold of achieving no more than 1% error (in norm) of

Figure

Progressive updates to estimate of initial value

The call boundary is generated by the algorithm to maximize the value function. In reality, because the approach is based on a statistical regression, the exercise boundary will not be a smooth contour.

Articles [

The algorithm generates a collection of sample points in the state space which are all distinguished as

Figure

Outcome of all call decisions.

Calculation for each trajectory and the position in

Figure

The minimum level

A fit over the minimum values with polynomial and square root.

The behavior of the boundary near the terminal time at

Figure

Intensity plot of exercise decisions.

The value function is calculated from 10,000 simulations with a 3-year term. There are 20 recursive iterations to converge on the starting value at time = 0. The “kink” in the value function is apparent in the surface plot in Figure

Value function surface plot.

The 2-d and surface plots actually represent the period 0 to

Timing performance of the algorithm was very good, with the algorithm as described executing in around 20 seconds on a standard 32 bit desktop PC, which represents an order of magnitude in improved speed over a SDP. However, the nature of the Monte Carlo yields nonsmoothness in the estimated boundary and requires a high volume of simulations to deliver the accuracy of a SDP.

In this paper we have developed adaptations of a technique which has proven to be successful in financial engineering and applied it to find the optimal exercise boundary in the early political election problem. Our technique was based on the Longstaff-Schwartz method used in estimating the exercise boundary for American options. The solution method is fast, and the results compare favourably with traditional methods such as SDP and PDE.

The solutions clearly display some inaccuracies in the fit for poll outcomes in the extreme. We expect (and can prove) that solutions

The use of a sigmoidal or logistic function for the regression is likely to introduce a superior fit, at the cost of additional computation time. This is another avenue to pursue for further research.

A case study is presented for the Australian commonwealth electoral system.