A Longstaff and Schwartz Approach to the Early Election Problem

Table 1

Step

Summary

Description

0

Develop initial estimate for the value function (Detail Section 4.6)

Establish an estimate for the value function at time across all poll states .

1

Simulation of the poll process (Detail Section 4.2)

Generate trajectories of the SDE (4.6) by simulating solutions. Denote each of the N simulated poll process as .

2

Simulation of electoral outcome if calling an election (Detail Section 4.4)

At each decision point , and poll state , we establish the expectation of the value function from a decision to call an election. The electoral result arises from: (i) Continue the diffusion to time (ii) Simulate a sampling error for the poll state (iii) Simulate the randomness in relationship in Section 3.3 for the imperfect relationship between nationwide polling and the distribution of seats.

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 and the decision experiment is repeated.

Regress over all N simulations the value function conditioned upon the poll state under the scenario that an election IS called (step 2). Regress over all N simulations the value function conditioned upon the poll state under the scenario that an election IS NOT called (step 3).

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 , , from the maximum between the two regressions.

6

Repeat over all timesteps

Repeat over all time steps stepping backwards from to .

7

Update initial value estimate

When time zero is reached, the value function will disagree with the estimate from step 0. Update with the derived initial value and recommence from step 0 until convergence of the initial estimate is achieved.

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