By: Apoorva seth
SIMULATION: The Monte Carlo Method
What is a Monte Carlo
● The expression "Monte Carlo method" is actually very general.
● Monte Carlo methods are based on the use of random numbers and probability statistics to
● You can find MC methods used in everything from economics to nuclear physics to regulating
the flow of traffic.
● A Monte Carlo method is a way of solving complex problems through approximation using
many random numbers. They are very versatile, but are often slower and less accurate than
other available methods.
A little bit of History
The term "Monte Carlo method" was coined in the 1940s by
physicists working on nuclear weapon projects in the Los
Alamos National Laboratory.
The physicists were investigating radiation shielding and the
distance that neutrons would likely travel through various
materials. Despite having most of the necessary data, the
problem could not be solved with analytical calculations.
John von Neumann and Stanislaw Ulam suggested that the
problem be solved by modeling the experiment on a computer
The name is a reference to the Monte Carlo
Casino in Monaco where Stanislaw Ulam's uncle would borrow
money to gamble
There is no single Monte Carlo method.
Instead, the term describes a large and widely-used class of approaches.
Essentially, the Monte Carlo method solves a problem by directly simulating the
underlying (physical) process and then calculating the (average) result of the
Because of their reliance on repeated computation of random or “pseudo-
random” numbers, these methods are most suited to calculation by a
However, these approaches tend to follow a
1. Define a Domain of Possible inputs
2. Generate Inputs randomly from the domain
using a certain specified probability distribution
3. Perform a deterministic computation ( this
means that given a particular input it will always
produce the same output ) using inputs
4. Aggregate the results of the individual
computations into a final result
Why Use the Monte Carlo
They tend to be used when it is
unfeasible or impossible to compute
an exact result with a deterministic
More broadly, Monte Carlo
methods are useful for modeling
events with significant uncertainty
in inputs, such as the calculation of
risk in business.
The advantage of Monte Carlo
methods over other techniques
increases as the sources of
uncertainty of the problem
Monte Carlo Methods are
particularly useful in the valuation
of options with multiple sources of
uncertainty or with complicated
features which would make them
difficult to value through a
straightforward Black-Scholes style
The technique is thus widely used
in valuing Exotic options.
In Most Basic Terms
1. Draw a random number
2. Process this random number in some way, for
example plug it into an equation
3. Repeat steps 1 and 2 a large number of times
4. Analyze the cumulative results to find an estimation
for a non random value
A Simple Example of the Monte
● Monte Carlo
Calculation of Pi
● We will use the unit
by a square
● However, it is easier to
just use one quadrant
of the circle. Sooooo.
Monte Carlo Calculation of Pi
● So lets pretend you are a horrible dart
player. The worst. Every throw is
● Now, Imagine throwing darts at the unit
● Because your throws are completely
random, The number of darts that land
within the shaded unit circle is proportional
to the area of the circle
● In other words,
● If you remember your geometry, it is easy to show:
● If each dart thrown lands somewhere inside the square, the ratio of "hits" (in the
shaded area) to "throws" will be one-fourth the value of pi.
Last one about pi, I swear!
● If you actually tried this experiment, you
would soon realize that it takes a very
large number of throws to get a decent
value of pi...well over 1,000.
● To make things easy on ourselves, we
can have computers generate random
● So, How?
● If we say our circle's radius is 1.0, for each
throw we can generate two random
numbers, an x and a y coordinate
● we can then use (x,y) to calculate the
distance from the origin (0,0) using the
● If the distance from the origin is less than
or equal to 1.0, it is within the shaded area
and counts as a hit.
● Do this thousands (or millions) of times
then average, and you will wind up with an
estimate of the value of pi. How good it is
depends on how many iterations (throws)
Monte Carlo Methods for
● Mostly used to calculate the value of an
option with multiple sources of
uncertainty or with complicated features
● In terms of theory, Monte Carlo
valuation relies on risk neutral valuation.
This just means that the current value of
all financial assets is equal to
the expected future payoff of the
asset discounted at the risk-free rate.
● Here is the pattern that is used:
● 1. Generate several thousand possible
(but random) price paths for the
underlying (or underlyings) via
● 2. Then calculate the associated exercise
value (aka the "payoff") of the option for
● 3. These payoffs are then averaged
● 4. Discounted to today.
● This result is the value of the option
Monte Carlo methods can help solve problems
that are too complicated to solve using
equations, or problems for which no equations
They are useful for problems which have lots of
uncertainty in inputs
They can also be used as an alternate way to solve
problems that have equation solutions.
Drawbacks: Monte Carlo methods are often
slower and less accurate than solutions via