蒙特卡罗在金融中模拟方法实现

▪ We need to generate a sequence {Ui} of random samples, which are independent (iid)
▪ Then we can compute the empiric mean:
Îm

1 m
m i 1
g(Ui )
▪ From the law of Great Numbers, one gets :
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30 40
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▪ Results depends on :
▪ random number generation (Mersenne Twister)
▪ Drawbacks
▪ Dependency to random number generator ▪ Big variability (accuracy) ▪ Computation time
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Why MATLAB ?
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▪ Efficiency :
▪ 1 000 000 paths in less than 1 s ▪ 25 times fastest thanExcel
▪ Number of simulations
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Typical uses cases of Monte carlo in finance
▪ Derivatives pricing
▪ Risk
▪ Structurer
▪ Stochastic Asset Liability Management
▪…
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Typical uses cases of Monte carlo in finance
© The MathWorks, Inc.
Agenda
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▪ Principles and uses cases for Monte Carlo methods ▪ Using MATLAB toolbox for Monte Carlo simulations ▪ Develop you own Monte Carlo engine ▪ A quick overview of Variance reduction technics
▪ State of the Art Algorithms
▪ Mersenne Twister ▪ Linear algebra
▪ Lots of statistical distributions supported (+ than 20)
▪ Easy deployment
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Agenda
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▪ Principles and uses cases for Monte Carlo methods ▪ Using MATLAB toolbox for Monte Carlo simulations ▪ Develop you own Monte Carlo engine ▪ A quick overview of Variance reduction technics
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Which tools for Monte Carlo simulations ?
▪ MATLAB : Core linear algebra engine, matrix factorisation, … ▪ Statistics toolbox : Random numbers, copulas, … ▪ Financial toolbox :
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An simple example 100 90 80 70 60 50 40
▪ Compute the area of a lake 30
20 10
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▪ We shot N cannon balls
▪ n balls out of the “lake” ▪ N- n balls in the lake
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General Principles
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▪ Estimation technique based on the simulation of a great number of random variables
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▪ Let’s consider I g(x)dx
0
▪ This can be seen as the expectation g(U ), with U being
▪ Portsim :” Monte Carlo simulation of correlated asset returns”
▪ GARCH Toolbox
▪ garchsim
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Financial toolbox
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▪ On a time interval, performances are driven by the following equation : ds dt dt
a uniform random variable on (0,1), ie U~(0,1)
▪ We can estimate tis expectation using an empiric mean from random draws
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General Principles, continued
▪ One gets
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30 40
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AreaSquare AreaLake

N N n

AreaLake

N N
n
* AreaSquare
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Take outs
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▪ Variance :
lim
m
Îm

I
V
ar(
Îm
)

V
ar(
g (U m
i
))
▪ Confidence interval : using a Gaussian approximation
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Good points and drawbacks
▪ Good points
▪ Various application areas ▪ Few hypothesis ▪ Easy to develop