Download Ch. 6 Random Behavior

Financial Engineering
Zvi Wiener
[email protected]
02-588-3049
FE-W
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
EMBAF
Random Behavior of Assets
Following
Paul Wilmott, Introduces Quantitative Finance
Chapter 6
FE-W
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
EMBAF
Returns
S i 1  Si
Ri 
Si
ri  LnSi 1   LnSi 
 Si 1 
 Si  Si 1  Si 
  Ln
 
ri  Ln
Si
 Si 


 Si 1  Si  Si 1  Si
 
Ln1 

Si 
Si

Zvi Wiener
FE-Wilmott-IntroQF Ch6
slide 3
Returns
1
R
M
M
R
i 1
i
1
2
Ri  R 
 ( R) 

M  1 i 1
M
See file 6.Random Behavior of Assets.XLS
Zvi Wiener
FE-Wilmott-IntroQF Ch6
slide 4
Normal Distribution N(, )
  0,
 1
1
PDF ( x) 
e
2
Zvi Wiener
FE-Wilmott-IntroQF Ch6
x2

2
slide 5
Normal Distribution N(, )


Zvi Wiener
FE-Wilmott-IntroQF Ch6
slide 6
Normal Distribution
1%
quantile
Zvi Wiener

FE-Wilmott-IntroQF Ch6
slide 7
Lognormal Distribution
0.6
0.5
0.4
0.3
0.2
0.1
1
Zvi Wiener
2
FE-Wilmott-IntroQF Ch6
3
4
slide 8
Covariance
Shows how two random variables are connected
For example:
independent
move together
move in opposite directions
covariance(X,Y) =
Zvi Wiener
E X  X Y  Y 
FE-Wilmott-IntroQF Ch6
slide 9
Correlation
 XY
E X  X Y  Y 

 ( X )  (Y )
-1    1
=0
independent
=1
perfectly positively correlated
 = -1
perfectly negatively correlated
Zvi Wiener
FE-Wilmott-IntroQF Ch6
slide 10
Properties
E (A  B)  E ( A)  E ( B)
 (A  B) 
2
  ( A)    ( B)  2Cov( A, B)
2
2
2
2
 (A  B) 
2
  ( A)    ( B)  2( A) ( B) 
2
Zvi Wiener
2
2
2
FE-Wilmott-IntroQF Ch6
slide 11
Time Aggregation
T   annualT
 T   annual T
Assuming normality
Zvi Wiener
FE-Wilmott-IntroQF Ch6
slide 12
Time Aggregation
Assume that yearly parameters of CPI are:
mean = 5%, standard deviation (SD) = 2%.
Then daily mean and SD of CPI changes are:
1
d   y
 0.02%
250
1
d  y
 0.1265%
250
Zvi Wiener
FE-Wilmott-IntroQF Ch6
slide 13
Volatility
1
2
Ri  R 


( M  1) t i 1
M
Zvi Wiener
FE-Wilmott-IntroQF Ch6
slide 14
Simulation of a Random Walk

Si 1  Si 1    t    t  z


  RAND ()   6
 1


12
A general formula
Zvi Wiener
See spreadsheet
N

12 
 N
   RAND()   
N  1
 2
FE-Wilmott-IntroQF Ch6
slide 15
Arithmetical Brownian Motion
dS    dt    dW
Geometrical Brownian Motion
dS    S  dt    S  dW
Zvi Wiener
FE-Wilmott-IntroQF Ch6
slide 16
Central Limit Theorem
The mean of n independent and identically
distributed variables converges to a normal
distribution as n increases.
1 n
X   Xi
n i 1
 2 

X  N   ,
n 

Zvi Wiener
FE-Wilmott-IntroQF Ch6
slide 17
Home Assignment
Read chapter 6 in Wilmott.
Follow Excel files coming with the book.
Zvi Wiener
FE-Wilmott-IntroQF Ch6
slide 18