Lecture 21
Monte Carlo Methods in Finance
21.1
What Monte Carlo for?
Monte Carlo simulation, considered as mathematical experimentation, can help in statistical mod
eling, optimization for a function of many variables, numerical integration over a highdimensional
space, and a lot more ...
Example 21.1: Statistical modeling
J
J
›
›
'
'
H
H
J
J
›
›
P
P
‡
‡
diagnostics
statistical inference
(bottomup)
descriptive
statistics
fitted model
(topdown)
Monte Carlo
simulation
proposed model
synthetic data
real data
See the diagram of “reconstruction cycle”. Consider the “usual” procedure: Given a data set,
summarize it, propose a model, fit the model (statistical inference), generate synthetic data from
the fitted model (this is the simulation step!), compare the simulated data with the real one. May
need to start over again, etc. Such cases are plenty: regression, time series, spatial stat, with many
applications. Why need a model? For prediction, better understanding, generalization, etc.
1
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Example 21.2: Optimization
Maybe the last resort when calculus, discrete search, etc.
cannot do it.
Optimization via
simulation is usually very timeconsuming.
Consider the 2D Ising model in statistical physics.
Minimizing the energy function
U
(
x
) is equivalent to finding the ground state ˆ
x
, i.e. the mode of
the Gibbs distribution. This can be done by simulated annealing (SA). See Liu’s book, sec. 1.3 for
the Ising model, sec. 10.2 for SA. Geman and Geman (1984) applies SA to image analysis.
Example 21.3: Numerical integration
Consider the integral
I
=
Z
D
g
(
x
)
π
(
x
)
dx
(21.1)
where the function
g
and probability density
π
are defined over a domain
D
⊂
IR
d
. How to compute
I
numerically?
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 Spring '08
 Staff
 Numerical Analysis, Monte Carlo method, Monte Carlo methods in finance

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