chapter 7 monte carlo simulation - ------- what is the last thing you do before you climb on a...

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

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1,Monte Carlo Simulation Simulation is a procedure in which random nu

mbers 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 f(x)=1 ,x∈[0,1], and

f(x)=0 ,otherwise x=random(), or, x=random() ?How to generate an uniform

distribution between [0,2]

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2,Uniform distribution Example: Cal the mean of uniform distribution (1), (2) dime tem(1000)

a=0for I=1 to 1000tem(i)=random()a=a+tem(i)endforex=a/1000 (3) Excel: See 007simulation-uniform.xls

2/1)(1

0xdxXE 0833.012/1)2/1()(

1

0

2 dxxXD

5

2,Uniform distribution Strengths of Monte Carlo : can be used

to price complex securities for which no analytical method is available

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) 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 * X will follow Bernoulli Distribution, X~B(1,0,p)

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3,Bernoulli Distribution X~B(1,0,p) 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/sig

ma)/2 S(i-th month)=s(I-1 month)*u^Y*d^(1-Y)

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4,Normal Distribution by Monte Carlo (1) If Xi (I=1,2,…,12) is uniform distributed

between 0 and 1. then ∑Xi–6~approximately follow N(0,1).

Proof:

Since is Xi i.i.d. by Central Limitation Theorem That is ∑Xi–6~approximately follow N(0,1).

2/112/

12/1)()2/1()(

2/1)(

1

0

2

1

0

i

i

i

XEXE

dxxfxXVar

dxxfxEX

12

1

144/1144/)()(i

iXVarXVar

)1,0(~12/)2/1( NX

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4,Normal Distribution by Monte Carlo 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 when I is 1000, mean=0.0398, var=0.98 when tem=12*ran()-6, var=11 Excel : See 007simulation-uniform.xls (What’s analytical solution , and numeri

cal solution by Monte Carlo)

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5, Simulate normal related R.V. 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. independ

ent F(1,5): K-square (1)/ K-square (5), R.V. independ

ent See 007simulation.xls

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6 ， Simulate GBM (Stock Price ) by Monte Carlo

(1)Basic Principle:

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.

(without-log)

),)2/((lnln 220 ttNSSt

})2/{()0()(

})2/{(2

20

ttEXPStS

ttEXPSS t

})2/()(ln)(ln 21 ii SS

)1,0(~, NttS

s

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Function provided by Matlab D:\ 金融数学讲义 \2003\Brown.mfunction blssample(S0,T,n,mu,sigma)%This function produces a sample path of a diffus

ion process%given by the SDE, dS=mu*S*dt+sigma*S*dX,%in the interval [0,T] discretized with n division p

oints%variables T=length of the interval, n = number of

division points,%mu=drift, sigma=volatility

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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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(2) How to estimate the drift and volatility coefficient

See 007 estim drif volat

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7,Cal expected stock price . Suppose s0=RMB 5, mu=0.03, sigma=0.08.

1, suppose price is GBM, calculate the expected price 3 years later.

(1)Analytical Method

47.55

%8%3

),)2/((lnln

09.00

220

eeSES

ttNSS

tT

t

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(2)Numerical Method Step one: divide 3 years into 100 interval, and simulate the price 3 years late

r. 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

007poison.xls 007 jump [including mean reversion ,BM]

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Exercise:

Simulate Normal (5,1)and Poison(lamda=5).

Draw their histogram

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Exercise:

Use Excel to Solve 3.1(b). Two condition:

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)

%20%1

),(lnln,

),)2/((lnln2

0

220

ttNSSand

ttNSS

t

t

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