Chapter 7 Monte Carlo Simulation
-------- What is the last thing you do before you
climb on a ladder?
You shake it.
“That is Monte Carlo Simulation”
Standford University Researcher Sam Savage
1,Monte Carlo Simulation
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Simulation is a procedure in which random
numbers are generated according to probabilities
assumed to be associated with a source of
uncertainty, such as, stock prices, interest rates..
Outcomes associated with these random drawings
are then analyzed to determine the likely related
results.
Oftentimes this technique is called Monte Carlo
simulation, being named for the city of Monte
Carlo, which is noted for its casinos. (卡西诺赌场.)
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2,Uniform distribution
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f(x)=1 ,x∈[0,1], and f(x)=0 ,otherwise
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x=random(), or, x=random()
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?How to generate an uniform distribution
between [0,2]
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2,Uniform distribution
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Example: Cal the mean of uniform distribution
(1), E( X )   xdx  1/ 2 D( X )   ( x  1/ 2) dx  1/ 12  0.0833
(2)
dime tem(1000)
1
1
0
0
2
a=0
for I=1 to 1000
tem(i)=random()
a=a+tem(i)
endfor
ex=a/1000
(3)
Excel: See 007simulation-uniform.xls
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2,Uniform distribution
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Strengths of Monte Carlo : can be used to
price complex securities for which no
analytical method is available
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Weakness:
provide only statistical
estimates, not exact results. Analytic
methods, where available, provides more
insight into cause-and-effect relationships.
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3,Bernoulli Distribution X~B(1,0,p)
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How to generate a Bernoulli Distribution ,X=1 (p)
X=0(1-p)?
Example : p=0.3
Z=random()
If z<=0.3
X=1
Else X=0
Endif
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* X will follow Bernoulli Distribution, X~B(1,0,p)
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3,Bernoulli Distribution X~B(1,0,p)
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Stimulate stock price by Bernoulli
Suppose annual mu=10%,sigma=8%
S0=10.
What would be the monthly price ?
U=exp{sigma*√1/12},
P=(1+mu√1/12/sigma)/2
S(i-th month)=s(I-1 month)*u^Y*d^(1-Y)
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4,Normal Distribution by Monte Carlo
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(1)
If Xi (I=1,2,…,12) is uniform
distributed between 0 and 1. then ∑Xi–
6~approximately follow N(0,1).
Proof: EX   x f ( x)dx  1 / 2
1
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i
0
1
Var ( X i )  
( x  1 / 2) 2 f ( x)dx  1 / 12
0
EX  E  X i / 12  1 / 2
Since is Xi i.i.d. Var( X )  Var( X ) /144  1/144
by Central Limitation Theorem ( X  1/ 2) / 12 ~ N (0,1)
That is ∑Xi–6~approximately follow N(0,1).
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i
i 1
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4,Normal Distribution by Monte Carlo
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2)Example : Use Monte Carlo to Cal mean and variance of standard
normal R.V.（Foxpro，D:\金数教学\2003\讲义\monte1.prg）
dime tem (10000)
for i= 1 to 10000
tem(i)=rand()+rand()+rand()+rand()+rand()+rand()+rand()+rand()
+rand()+rand()+rand()+rand()-6
endfor
mean=0
for i=1 to 10000
mean=mean+tem(i)
endfor
mean=mean/10000
?mean
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4,Normal Distribution by Monte Carlo
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when I is 1000, mean=0.0398, var=0.98
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when tem=12*ran()-6, var=11
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Excel : See 007simulation-uniform.xls
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(What’s analytical solution , and numerical
solution by Monte Carlo)
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5, Simulate normal related R.V.
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K-square (1): square of Standard Normal
K-square (5): sum the squares of five Standard
Normal
T-(1): X/sqrt{K-square (1)},R.V. independent
T-(5): X/sqrt{K-square (5)/5}, R.V. independent
F(1,1): K-square (1)/ K-square (1), R.V.
independent
F(1,5): K-square (1)/ K-square (5), R.V.
independent
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See 007simulation.xls
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6，Simulate GBM (Stock Price ) by Monte Carlo
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(1)Basic Principle:
2
2
ln
S

ln
S
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N
((



/
2
)
t
,

t)
t
0
2
S t  S 0 EXP{(    / 2)t   t  }
S (t )  S (0) EXP{(    2 / 2)t   t  }
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If stock moves t trading days, and we divide t into
n intervals(i=1,…t/n),the length of the interval is
Δ, the last formula is equal to
S(Δi) is stock’s price at the end of the i-th interval.
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ln S (i )  ln S (i 1 )  (   2 / 2)   }
s
 t   t  ,  ~ N (0,1)
S
(without-log)
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6，Simulate GBM (Stock Price ) by Monte Carlo
Function provided by Matlab
 D:\金融数学讲义\2003\Brown.m
function blssample(S0,T,n,mu,sigma)
%This function produces a sample path of a
diffusion process
%given by the SDE, dS=mu*S*dt+sigma*S*dX,
%in the interval [0,T] discretized with n division
points
%variables T=length of the interval, n = number of
division points,
%mu=drift, sigma=volatility
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6，Simulate GBM (Stock Price ) by Monte Carlo
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t=0:T/n:T-(1/n);
dt=T/n;
phi=random('Normal',0,1,n,1);
X=zeros(1,n); S=zeros(1,n);
M=zeros(1,n); M(1)=S0;
S(1)=S0;
for i = 1:1:n-1
dX(i)=sigma*phi(i)*sqrt(dt);
S(i+1)= S(i)+mu*S(i)*dt+sigma*S(i)*dX(i);
M(i+1)=M(i)+mu*M(i)*dt;
End
plot(t,S);
Hold
plot(t,M);
hold; end
Initial price is 5, annual drift and volatility are 3% and 8%, give an possible
path of the stock in future 3 years .
调用函数blssample(5,3,100,0.03,0.08)得到上图 007 brownian simaulation。
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6，Simulate GBM (Stock Price ) by Monte Carlo
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(2) How to estimate the drift and volatility
coefficient
See 007 estim drif volat
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7,Cal expected stock price .
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Suppose
s0=RMB
5,
mu=0.03,
sigma=0.08. 1, suppose price is GBM,
calculate the expected price 3 years later.
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(1)Analytical Method
ln S t  ln S 0  N ((   2 / 2)t ,  2 t )
  3%  8%
EST  S 0 e t  5e 0.09  5.47
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(2)Numerical Method
Step one: divide 3 years into 100 interval, and simulate the price 3 years later.
Step two : Repeat step one 1000 times, and calculate the average price(D:\金
数教学\2003\讲义\montebs.prg计算股票的期望值
dime E(1001),s(100)
s(1)=5
E(1)=0
mu=0.03
sigma=0.08
t=3
n=100
dt=t/n
for j=1 to 1000
for i= 1 to 99
normz=rand()+rand()+rand()+rand()+rand()+rand()+rand()+rand()+rand()
+rand()+rand()+rand()-6
s(i+1)=s(i)+mu*s(i)*dt+sigma*s(i)*sqrt(dt)*normz
endfor
e(j+1)=s(100)+e(j)
endfor
?e(j)/1000 =5。48
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8,Simulate Poison Distribution ,
Jump GBM and mean reversion
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007poison.xls
007 jump [including mean reversion ,BM]
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Exercise:
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Simulate Normal (5,1)and Poison(lamda=5).
Draw their histogram
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Exercise:
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Use Excel to Solve 3.1(b). Two condition:
ln S t  ln S 0  N ((   2 / 2)t ,  2 t )
and, ln S t  ln S 0  N ( t ,  2 t )
  1%  20%
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Divide 10 years into 100 interval, and cal P[S(10)>100]
Which condition is more closer to the analytical method?
{Hint: By Monte Carlo, repeat your simulation 50 times , then count
the times when S(10)>100}.
( by email)
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