The History of Monte Carlo
Simulation Methods in
Financial Engineering
Paul Wilmott
Monte Carlo
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One of Monaco’s four quarters
Not its capital…Monaco doesn’t have one
Formula One Monaco Grand Prix
Casino (1873, Joseph Jagger ‘broke the
bank’)
Probabilistic Concepts
• Games of chance, dice games
• Formal probability theory 17th Century
(Pascal and Fermat)
Calculating p
• Buffon’s needle (1777)
– Parallel lines one inch apart, needle one inch long
2
– Probability of ‘hit’ is
p
Brownian Motion 1827
• The Scottish botanist, Robert Brown, gave
his name to the random motion of small
particles in a liquid. This idea of the
random walk has permeated many
scientific fields and is commonly used as
the model mechanism behind a variety of
unpredictable continuous-time processes.
The lognormal random walk based on
Brownian motion is the classical paradigm
for the stock market.
Louis Bachelier 1900
• Louis Bachelier was the first to quantify the
concept of Brownian motion. He developed a
mathematical theory for random walks, a theory
rediscovered later by Einstein. He proposed a
model for equity prices, a simple normal
distribution, and built on it a model for pricing the
almost unheard of options. His model contained
many of the seeds for later work, but lay
‘dormant’ for many, many years.
Fokker, Planck, Kolmogorov 1913
• Adriaan Fokker and Max Planck showed
how to relate transition probability density
functions for random variables to partial
differential equations, this is the ‘forward
equation.’ Later Andrey Kolmogorov
derived a related ‘backward equation.’
Manhattan Project
• Stanislaw Ulam invented the technique for
solving certain types of differential equation
using probabilistic methods. (Again inspired by a
card game.)
• The name “Monte Carlo” was given by Nicholas
Metropolis.
Both of these were working on the Manhattan
Project, towards the development of nuclear
weapons.
Quadrature
• Suppose you want to evaluate a complicated
integral

b
a
f ( x)dx
• This can be interpreted as (b – a) times
average of f(x).
• So…a relationship between probabilities and
integrals…just pick x at random, calculate f(x)
and then average!
Quadrature cont’d
• And this works in any number of dimensions!
• And speed of convergence is roughly the
same, independent of number of dimensions.
1
N
1
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Genuinely uniform random in two dimensions
1960s Sobol’, Faure, Hammersley,
Haselgrove, Halton…
• Many people were associated with the definition
and development of quasi random number
theory or low-discrepancy sequence theory. The
subject concerns the distribution of points in an
arbitrary number of dimensions so as to cover
the space as efficiently as possible, with as few
points as possible. The methodology is used in
the evaluation of multiple integrals among other
things. These ideas would find a use in finance
almost three decades later.
Low-discrepancy sequences
• Converges like 1
N
1
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Low-discrepancy sequences
0.9
1
Black-Scholes 1973
• Fischer Black, Myron Scholes and Robert
Merton derived the Black-Scholes
equation for options in the early seventies,
publishing it in two separate papers in
1973. The date corresponded almost
exactly with the trading of call options on
the Chicago Board Options Exchange.
Scholes and Merton won the Nobel Prize
for Economics in 1997. Black had died in
1995.
Option Prices as Expectations
• The famous Black-Scholes option-pricing
model bears close resemblance to the
Kolmogorov backward equation.
• A close examination of these two
equations leads to…
Option Prices as Expectations
cont’d
• The fair value of an option is
The present value of the expected payoff
Phelim Boyle 1977
• Phelim Boyle related the pricing of options
to the simulation of random asset paths.
He showed how to find the fair value of an
option by generating lots of possible future
paths for an asset and then looking at the
average that the option had paid off. The
future important role of Monte Carlo
simulations in finance was assured.
Stochastic Models in Finance
• The most successful models in financial
engineering use the mathematics of
stochastic calculus.
• Without a doubt, there would not have
been the enormous growth in quantity and
types of contracts traded if there wasn’t
such a solid theoretical foundation.
Equities, FX and Commodities
• The classic model starts with the
lognormal random walk.
• The asset model contains a deterministic
growth rate and a volatility, the latter
representing the amount of randomness in
the asset path.
Recent developments
• As profit margins shrink and markets
become more efficient, researchers
develop new (and better?) models to try to
capture reality:
– Stochastic volatility
– Jump diffusion
Fixed Income
• Similar models are used in the fixedincome world.
• Originally the random quantity was just the
spot interest rate (a very short-term rate),
in a ‘single-factor’ model.
Fixed Income cont’d
• Then there came the multi-factor models
• Now people model entire yield curves
moving randomly, not just single values.
So interest rates of different maturities are
modelled simultaneously.
The Market Price of Risk
• One of the inputs into fixed-income models
is the ‘market price of risk.’
• This can be interpreted as ‘how much the
market wants to be compensated for
taking risk.’
• But how rational is the market?
The Market Price of Risk cont’d
15
GREED
10
Lambda
5
Time
0
29/03/1986 11/08/1987 23/12/1988 07/05/1990 19/09/1991 31/01/1993 15/06/1994 28/10/1995
-5
-10
-15
-20
FEAR
-25
Maybe we should model this as
stochastic as well!
Credit Risk
• Very hot at the moment are credit derivatives.
• These financial instruments give a return to investors
that is contingent on default of a company, for example.
• These instruments can be valued using Monte Carlo
simulation via models for jump processes.
• Roll a dice, and you get a one, you default.
• And we are back with games of chance!