Frequently Asked Questions In Quantitative Finance

Chapter 10: Brainteasers 389

Now we settle down into a rhythm. Here’s the entire
allocation table.

Pirate 10 Pirate 9 Pirate 8 Pirate 7 Pirate 6 Pirate 5 Pirate 4 Pirate 3 Pirate 2 Pirate 1

1000 0

1000 0 0

998011

997 0 1 2 / 0 0 / 2

995 0 1 2 / 2 / 0 2 / 0 / 2 0 / 2 / 2

99501

99301

99301

991 0 1 Two doubloons to any four of these seven

Two doubloons to any two of these four

Two doubloons to any three of these five

Two doubloons to any three of these six

This Brainteaser is particularly relevant in quantitative
finance because of the backward induction nature of
the solution. This is highly reminiscent of the binomial
model in which you have to calculate today’s option
price by working backwards from expiration by consid-
ering option prices at different times.

Another of these backward induction types is the
famous Brainteaser, the unexpected hanging. In this
problem we have a prisoner who has been condemned
to be executed in ten days’ time and an unusually
considerate executioner. The executioner wants the
prisoner to suffer as little mental anguish as possible
during his last days and although the prisoner knows
that sometime in the next ten days he will be executed
he doesn’t know when. If the executioner can surprise
him then the prisoner will be able to enjoy his last few
days, at least relatively speaking. So, the executioner’s
task is to wake the prisoner up one morning and exe-
cute him but must choose the day so that his visit was
not expected by the prisoner.

Let’s see how to address this problem by induction
backwards from the last of the ten days. If the prisoner
has not been executed on one of the first nine days then
he goes to bed that night in no doubt that tomorrow he
will be woken by the executioner and hanged. So he
can’t be executed on the last day, because it clearly
wouldn’t be a surprise. Now, if he goes to bed on the
night of the eighth day, not having been executed during