Simulation - Monte Carlo Methods - Introduction

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• Simulation - Monte Carlo Methods - Introduction

In practice many of the issues that we have to
deal with are too complex to enable them to be
easily distilled into a simple mathematical
equation or even for them to be governed by an
underlying probabilistic structure. In such
cases the approach that most will seek to adopt
is Monte Carlo simulation. Monty Hall
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• Simulation - Monte Carlo Methods - What is Monte
  Carlo Simulation?

Any variable that can be measured and that can
take on different values, such as the value of an
equity portfolio over a given number of years, is
commonly referred to as a random variable. The
first type of probability structure that is
normally created is called the distribution
function, because it shows the frequency with
which the random variable actually takes specific
values within a certain range.
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• Simulation - Monte Carlo Methods - What is Monte
  Carlo Simulation?

The second type of probability structure that
might be created is called the cumulative
distribution function. This displays the
accumulated probability that the random variable
falls below a certain value, effectively
representing the total of the number of events
that are below a value to the total number of
observations.
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• Simulation - Monte Carlo Methods - What is Monte
  Carlo Simulation?

Monte Carlo simulation takes the idea of using
statistical trials to get an approximate solution
that has been considered previously and enables
this to be applied to a complex problem. There
is a normally a process (such as the generation
of an equity portfolio return) where some
information is known, but other information that
is required is not known with certainty. The
expected return on an equity portfolio in 12
months time is clearly not known with certainty,
although we will know the historic returns that
have been achieved.
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• Simulation - Monte Carlo Methods - What is Monte
  Carlo Simulation?

Since these future values are not known exactly,
many observations need to be made so that the
uncertain values of the process can be estimated
with increasing accuracy.
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• Simulation - Monte Carlo Methods - Monte Carlo
  for the Monty Hall Problem

This problem was introduced at the conclusion of
the probability section. The set of Monty
Halls game show Lets Make a Deal has three
closed doors (A, B and C). Behind one of these
doors is a car behind the other two are goats.
The contestant does not know where the car is,
but Monty Hall does. The contestant picks a
door and Monty opens one of the remaining doors,
one he knows does not hide the car. If the
contestant has already chosen the correct door,
Monty is equally likely to open either of the two
remaining doors, since each would have a goat
behind it.
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• Simulation - Monte Carlo Methods - Monte Carlo
  for the Monty Hall Problem

After Monty has shown a goat behind the door that
he opens, the contestant is always given the
option to switch doors. What is the probability
of winning the car if the contestant stays with
their first choice? What if they decide to
switch? The winning letter in the Monty Hall
problem, representing a door is always known to
Monty.
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• Simulation - Monte Carlo Methods - Monte Carlo
  for the Monty Hall Problem

Let us assume it is A. The participants initial
choice is either A, B or C with equal
probabilities. The cumulative probabilities are
shown in the table and these can be coupled with
a random number to indicate the participants
first choice.
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• Simulation - Monte Carlo Methods - Cumulative
  Initial Probabilities for the Monty Hall Problem

The next stage is to decide which letter the
participant is shown. If the initial choice
were A, then Monty is able to reveal B or C with
equal likelihoods. However if B or C is
chosen, then C or B, respectively, must be
revealed by Monty. These options are set out in
the table.
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• Simulation - Monte Carlo Methods - Second Event
  Probabilities for the Monty Hall Problem

The next stage is to decide which letter the
participant is shown. If the initial choice were
A, then Monty is able to reveal B or C with equal
likelihoods. However if B or C is chosen, then
C or B, respectively, must be revealed by Monty.
These options are set out in the table.
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• Simulation - Monte Carlo Methods - Second Event
  Probabilities for the Monty Hall Problem

The player may either stick with their original
choice or switch. This is equivalent to choosing
the third letter option. A few simulations are
displayed in the table.
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• Simulation - Monte Carlo Methods - Simulation of
  the Monty Hall Problem

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• Simulation - Monte Carlo Methods - Simulation of
  the Monty Hall Problem

After 2,000 cycles the results are summarised in
the table.
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• Simulation - Monte Carlo Methods - Summary of a
  Monte-Carlo Simulation of the Monty Hall Problem

The probabilities of success agree reasonably
with the exact values calculated previously.
The best strategy is to switch, with a winning
probability that is approximately ?. The
convergence of successive estimates to this value
is shown in the figure.
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• Simulation - Monte Carlo Methods - The Simulation
  of 2000 Cycles of the Switching Probability

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• Simulation - Monte Carlo Methods - Conclusion

Simulation is a very good technique when you need
to bring together a lot of data and
probabilities. Normally software solutions are
used to show this. Indeed with both the
simulators themselves and computer capacity
having grown it is now possible to undertake very
large simulations comparatively easily. However
even in large populations you will normally see
that the simulation really has converged by the
time you have done 250 simulations, although more
can be conducted if required.
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• Simulation - Monte Carlo Methods - Conclusion

Whilst we have shown a few examples, which you
can calculate manually, more complex ones are not
so easy to demonstrate. So in brief all Monte
Carlo achieves is to throw a series of random
numbers at a series of probabilities, often shown
as distributions, to come up with a single
answer.
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• Next Week

Behavioural Traps
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