Download Zvi Wiener slide 5

Financial Engineering
Zvi Wiener
[email protected]
02-588-3049
FE-W
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
EMBAF
Math
Following
Paul Wilmott, Introduces Quantitative Finance
Chapter 4, see www.wiley.co.uk/wilmott
FE-W
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
EMBAF
e
Natural logarithm
2.718281828459045235360287471352662497757…
ex = Exp(x)
e0 = 1
e1 = e
2
3
4
5

i
x
x
x
x
x
e  1  x      ...  
2 3! 4! 5!
i 0 i!
x
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 3
7
6
5
4
Exp(x)
3
2
1
-2
Zvi Wiener
-1
x
1
FE-Wilmott-IntroQF Ch4
2
slide 4
Ln
Logarithm with base e.
eln(x) = x, or ln(ex) = x
Determined for x>0 only!
2
3
4
5
y
y
y
y
Ln(1  y )  y 
    ...
2
3
4
5
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 5
Ln
1
0.5
Ln(x)
x
1
2
3
4
-0.5
-1
-1.5
-2
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 6
Differentiation and Taylor series
7
f(x)
6
5
4
3
f
f ' (1) 
x
2
1
0.5
Zvi Wiener
1
FE-Wilmott-IntroQF Ch4
1.5
x 1
x
2
slide 7
Differentiation and Taylor series
df
f ( x  x)  f ( x)
f ' ( x) 
 lim
dx x0
x
df
f ( x  x)  f ( x)  x
dx
df
d 2 f x 2 d 3 f x 3
f ( x  x)  f ( x)  x  2
 3

dx
dx 2!
dx 3!
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 8
Differentiation and Taylor series
f ( x)  f ' ( x)x
f ( x  x )
f (x )
x
Zvi Wiener
FE-Wilmott-IntroQF Ch4
x+x
slide 9
Taylor series
one variable
d f x
f ( x  x)   i
i!
i 0 dx

Zvi Wiener
FE-Wilmott-IntroQF Ch4
i
i
slide 10
Taylor series
two variable
V ( S  S , t  t ) 
V
V S  V
V ( S , t )  t
 S



2
t
S
2 S
2
Zvi Wiener
FE-Wilmott-IntroQF Ch4
2
slide 11
Differential Equations
Ordinary
Partial
Boundary conditions
Initial Conditions
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 12
Chapter 2
Quantitative Analysis
Fundamentals of Probability
Following P. Jorion 2001
Financial Risk Manager Handbook
FE-W
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
EMBAF
Random Variables
Values, probabilities.
Distribution function, cumulative probability.
Example: a die with 6 faces.
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 14
Random Variables
Distribution function of a random variable X
F(x) = P(X  x) - the probability of x or less.
If X is discrete then
F ( x )   f ( xi )
xi  x
x
If X is continuous then F ( x) 
dF ( x)
Note that f ( x) 
dx
Zvi Wiener
FE-Wilmott-IntroQF Ch4
 f (u)du

slide 15
Random Variables
Probability density function of a random
variable X has the following properties
f ( x)  0

1
 f (u )du

Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 16
Independent variables
f12 (u1 , u2 )  f1 (u1 )  f 2 (u2 )
F12 (u1 , u2 )  F1 (u1 )  F2 (u2 )
Credit exposure in a swap depends on two random
variables: default and exposure.
If the two variables are independent one can
construct the distribution of the credit loss easily.
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 17
Moments
Mean = Average = Expected value
  E( X ) 

xf
(
x
)
dx


Variance
  V (X ) 
2

 x  E ( X )
2
f ( x)dx

  S tan dard Deviation  Variance
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 18
Cov( X 1 , X 2 )  E X 1  EX 1  X 2  EX 2 
( X1, X 2 ) 
Cov( X 1 , X 2 )
 1 2
Skewness (non-symmetry)
Kurtosis (fat tails)
Zvi Wiener
Its meaning ...
 
 
1

3
1

FE-Wilmott-IntroQF Ch4
4

E  X  E X 

3
E  X  E  X 

4

slide 19
Main properties
E (a  bX )  a  bE ( X )
 (a  bX )  b ( X )
E( X 1  X 2 )  E( X 1 )  E( X 2 )
 ( X 1  X 2 )   ( X 1 )   ( X 2 )  2Cov( X 1 , X 2 )
2
Zvi Wiener
2
2
FE-Wilmott-IntroQF Ch4
slide 20
Portfolio of Random Variables
N
Y   wi X i  w X
T
i 1
N
E (Y )   p  w E ( X )  w  X   wi  i
T
T
i 1
N
N
 (Y )  w w   wi ij w j
2
T
i 1 j 1
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 21
Portfolio of Random Variables
 (Y ) 
2
  11  12

w1 , w2 ,, wN  
 N 11  N 2
Zvi Wiener
FE-Wilmott-IntroQF Ch4
 w1 
  1N   
  w2 
  
  NN   
 wN 
slide 22
Product of Random Variables
Credit loss derives from the product of the
probability of default and the loss given default.
E( X 1 X 2 )  E( X 1 ) E( X 2 )  Cov( X 1 , X 2 )
When X1 and X2 are independent
E( X 1 X 2 )  E( X 1 ) E( X 2 )
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 23
Transformation of Random Variables
Consider a zero coupon bond
100
V 
T
(1  r )
If r=6% and T=10 years, V = $55.84,
we wish to estimate the probability that the
bond price falls below $50.
This corresponds to the yield 7.178%.
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 24
Example
The probability of this event can be derived
from the distribution of yields.
Assume that yields change are normally
distributed with mean zero and volatility 0.8%.
Then the probability of this change is 7.06%
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 25
Quantile
Quantile (loss/profit x with probability c)
x
F ( x) 
 f (u)du  c

50% quantile is called median
Very useful in VaR definition.
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 26
FRM-99, Question 11
X and Y are random variables each of which
follows a standard normal distribution with
cov(X,Y)=0.4.
What is the variance of (5X+2Y)?
A. 11.0
B. 29.0
C. 29.4
D. 37.0
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 27
FRM-99, Question 11
    2  A B 
2
A
2
B
5  2  2  0.4  5  2  37
2
Zvi Wiener
2
FE-Wilmott-IntroQF Ch4
slide 28
FRM-99, Question 21
The covariance between A and B is 5. The
correlation between A and B is 0.5. If the
variance of A is 12, what is the variance of B?
A. 10.00
B. 2.89
C. 8.33
D. 14.40
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 29
FRM-99, Question 21

B 
Cov( A, B )
 A B
Cov( A, B)
 A
 2.89
  8.33
2
B
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 30
Uniform Distribution
Uniform distribution defined over a range of
2
values axb.
ab 2
(b  a)
E( X ) 
,  (X ) 
2
12
1
f ( x) 
, a xb
ba
xa
0,
x  a

F ( x)  
, a xb
b  a
bx
1,
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 31
Uniform Distribution
1
1
ba
a
Zvi Wiener
b
FE-Wilmott-IntroQF Ch4
slide 32
Normal Distribution
Is defined by its mean and variance.
f ( x) 

1
 2
e
( x )2
2 2
E( X )  ,  ( X )  
2
2
Cumulative is denoted by N(x).
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 33
Normal Distribution
66% of events lie
between -1 and 1
0.4
0.3
95% of events lie
between -2 and 2
0.2
0.1
-3
Zvi Wiener
-2
-1
1
FE-Wilmott-IntroQF Ch4
2
3
slide 34
Normal Distribution
1
0.8
0.6
0.4
0.2
-3
Zvi Wiener
-2
-1
1
FE-Wilmott-IntroQF Ch4
2
3
slide 35
Normal Distribution
symmetric around the mean
mean = median
skewness = 0
kurtosis = 3
linear combination of normal is normal
99.99 99.90 99
97.72 97.5 95
90
84.13
3.715 3.09 2.326 2.000 1.96 1.645 1.282 1
Zvi Wiener
FE-Wilmott-IntroQF Ch4
50
0
slide 36
Lognormal Distribution
The normal distribution is often used for rate
of return.
Y is lognormally distributed if X=lnY is
normally distributed. No negative values!
f ( x) 
E( X )  e

2
2
1
x 2

(ln(x )   ) 2
e
,  (X )  e
2
2 2
2   2 2
e
2   2
E (Y )  E (ln X )   ,  (Y )   (ln X )  
2
Zvi Wiener
FE-Wilmott-IntroQF Ch4
2
2
slide 37
Lognormal Distribution
If r is the expected value of the lognormal
variable X, the mean of the associated normal
variable is r-0.52.
0.6
0.5
0.4
0.3
0.2
0.1
0.5
Zvi Wiener
1
1.5
2
FE-Wilmott-IntroQF Ch4
2.5
3
slide 38
Student t Distribution
Arises in hypothesis testing, as it describes the
distribution of the ratio of the estimated
coefficient to its standard error. k - degrees of
freedom.
 k 1


1
2  1

f ( x) 
k 1
k
 
k
 

x2  2
1  
2

k 
k 1  x

(k )   x e dx
0
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 39
Student t Distribution
As k increases t-distribution tends to the
normal one.
This distribution is symmetrical with mean
zero and variance (k>2)
k
 ( x) 
k 2
2
The t-distribution is fatter than the normal one.
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 40
Binomial Distribution
Discrete random variable with density function:
n x
n x


f ( x)    p (1  p) , x  0,1,., n
 x
E ( X )  pn,  ( X )  p(1  p)n
2
For large n it can be approximated by a normal.
x  pn
z
~ N (0,1)
p(1  p)n
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 41
FRM-99, Question 13
What is the kurtosis of a normal distribution?
A. 0
B. can not be determined, since it depends on
the variance of the particular normal
distribution.
C. 2
D. 3
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 42
FRM-99, Question 16
If a distribution with the same variance as a
normal distribution has kurtosis greater than
3, which of the following is TRUE?
A. It has fatter tails than normal distribution
B. It has thinner tails than normal distribution
C. It has the same tail fatness as normal
D. can not be determined from the
information provided
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 43
FRM-99, Question 5
Which of the following statements best
characterizes the relationship between normal and
lognormal distributions?
A. The lognormal distribution is logarithm of the
normal distribution.
B. If ln(X) is lognormally distributed, then X is
normally distributed.
C. If X is lognormally distributed, then ln(X) is
normally distributed.
D. The two distributions have nothing in common
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 44
FRM-98, Question 10
For a lognormal variable x, we know that
ln(x) has a normal distribution with a mean of
zero and a standard deviation of 0.2, what is
the expected value of x?
A. 0.98
B. 1.00
C. 1.02
D. 1.20
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 45
FRM-98, Question 10
E[ X ]  e
Zvi Wiener

2
2
e
0.2 2
0
2
FE-Wilmott-IntroQF Ch4
 1.02
slide 46
FRM-98, Question 16
Which of the following statements are true?
I. The sum of normal variables is also normal
II. The product of normal variables is normal
III. The sum of lognormal variables is lognormal
IV. The product of lognormal variables is
lognormal
A. I and II
B. II and III
C. III and IV
D. I and IV
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 47
FRM-99, Question 22
Which of the following exhibits positively
skewed distribution?
I. Normal distribution
II. Lognormal distribution
III. The returns of being short a put option
IV. The returns of being long a call option
A. II only
B. III only
C. II and IV only
D. I, III and IV only
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 48
FRM-99, Question 22
C. The lognormal distribution has a long right
tail, since the left tail is cut off at zero. Long
positions in options have limited downsize,
but large potential upside, hence a positive
skewness.
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 49
FRM-99, Question 3
It is often said that distributions of returns from financial
instruments are leptokurtotic. For such distributions, which of
the following comparisons with a normal distribution of the
same mean and variance MUST hold?
A. The skew of the leptokurtotic distribution is greater
B. The kurtosis of the leptokurtotic distribution is greater
C. The skew of the leptokurtotic distribution is smaller
D. The kurtosis of the leptokurtotic distribution is smaller
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 50
Home Assignment
Read chapters 4, 5 in Wilmott.
Read and understand the xls files!!
Build a module for pricing of the Max, Min
and Mixture programs (BRIRA).
Analyze the program offered by BH.
Build a module for pricing of this program.
Describe in terms of options the client’s
position in the program offered by FIBI.
Zvi Wiener
FE-Wilmott-IntroQF Ch4
slide 51