FE-Whttp://pluto.mscc.huji.ac.il/
~mswiener/zvi.htmlEMBAF
Zvi Wiener
02-588-3049
Financial Engineering
FE-Whttp://pluto.mscc.huji.ac.il/
~mswiener/zvi.htmlEMBAF
Following
Paul Wilmott, Introduces Quantitative Finance
Chapter 4, see www.wiley.co.uk/wilmott
Math
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 3
eNatural logarithm
2.718281828459045235360287471352662497757…
ex = Exp(x)
e0 = 1
e1 = e
0
5432
!...
!5!4!321
i
ix
i
xxxxxxe
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 4
-2 -1 1 2
1
2
3
4
5
6
7
x
Exp(x)
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 5
LnLogarithm with base e.
eln(x) = x, or ln(ex) = x
Determined for x>0 only!
...5432
)1(5432
yyyy
yyLn
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 6
Ln
1 2 3 4
-2
-1.5
-1
-0.5
0.5
1
xLn(x)
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 7
0.5 1 1.5 2
1
2
3
4
5
6
7
Differentiation and Taylor series
x
f(x)
1
)1('
xx
ff
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 8
Differentiation and Taylor series
x
xfxxf
dx
dfxf
x
)()(lim)('
0
xdx
dfxfxxf )()(
!3!2
)()(3
3
32
2
2 x
dx
fdx
dx
fdx
dx
dfxfxxf
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 9
Differentiation and Taylor series
x x+x
xxfxf )(')(
)(xf
)( xxf
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 10
Taylor seriesone variable
0 !)(
i
i
i
i
i
x
dx
fdxxf
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 11
Taylor seriestwo variable
2
22
2),(
),(
S
VS
S
VS
t
VttSV
ttSSV
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 12
Differential Equations
Ordinary
Partial
Boundary conditions
Initial Conditions
FE-Whttp://pluto.mscc.huji.ac.il/
~mswiener/zvi.htmlEMBAF
Chapter 2Quantitative Analysis
Fundamentals of ProbabilityFollowing P. Jorion 2001
Financial Risk Manager Handbook
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 14
Random Variables
Values, probabilities.
Distribution function, cumulative probability.
Example: a die with 6 faces.
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 15
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
xx
i
i
xfxF )()(
If X is continuous then
x
duufxF )()(
Note thatdx
xdFxf
)()(
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 16
Random Variables
Probability density function of a random
variable X has the following properties
0)( xf
duuf )(1
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 17
Independent variables
)()(),( 22112112 ufufuuf
)()(),( 22112112 uFuFuuF
Credit exposure in a swap depends on two randomvariables: default and exposure.If the two variables are independent one canconstruct the distribution of the credit loss easily.
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 18
MomentsMean = Average = Expected value
dxxxfXE )()(
Variance
dxxfXExXV )()()( 22
VarianceDeviationdardS tan
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 19
221121 ),( XEXXEXEXXCov
Its meaning ...
3
3
1XEXE
21
2121
),(),(
XXCov
XX
Skewness (non-symmetry)
4
4
1XEXE
Kurtosis (fat tails)
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 20
Main properties
)()( XbEabXaE
)()( XbbXa
)()()( 2121 XEXEXXE
),(2)()()( 2122
12
212 XXCovXXXX
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 21
Portfolio of Random Variables
XwXwY TN
iii
1
N
iiiX
TTp wwXEwYE
1
)()(
N
i
N
jjiji
T wwwwY1 1
2 )(
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 22
Portfolio of Random Variables
NNNNN
N
N
w
w
w
www
Y
2
1
211
11211
21
2
,,,
)(
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 23
Product of Random Variables
Credit loss derives from the product of the
probability of default and the loss given default.
),()()()( 212121 XXCovXEXEXXE
When X1 and X2 are independent
)()()( 2121 XEXEXXE
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 24
Transformation of Random Variables
Consider a zero coupon bond
TrV
)1(
100
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 25
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%
Example
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 26
Quantile
Quantile (loss/profit x with probability c)
cduufxFx
)()(
50% quantile is called median
Very useful in VaR definition.
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 27
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 28
FRM-99, Question 11
37254.0225 22
BABA 222
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 29
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 30
FRM-99, Question 21
BA
BACov
),(
89.2),(
AB
BACov
33.82 B
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 31
Uniform DistributionUniform distribution defined over a range of values axb.
bxaab
xf
,1
)(
12
)()(,
2)(
22 ab
Xba
XE
xb
bxaab
ax
ax
xF
,1
,
,0
)(
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 32
Uniform Distribution
a b
ab 1
1
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 33
Normal DistributionIs defined by its mean and variance.
2
2
2
)(
2
1)(
x
exf
22 )(,)( XXE
Cumulative is denoted by N(x).
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 34
-3 -2 -1 1 2 3
0.1
0.2
0.3
0.4
Normal Distribution66% of events liebetween -1 and 1
95% of events liebetween -2 and 2
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 35
Normal Distribution
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 36
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 50
3.715 3.09 2.326 2.000 1.96 1.645 1.282 1 0
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 37
Lognormal DistributionThe normal distribution is often used for rate of return.
Y is lognormally distributed if X=lnY is normally distributed. No negative values!
2
2
2
))(ln(
2
1)(
x
ex
xf
22
2
22222 )(,)(
eeXeXE
222 )(ln)(,)(ln)( XYXEYE
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 38
Lognormal DistributionIf r is the expected value of the lognormal variable X, the mean of the associated normal variable is r-0.52.
0.5 1 1.5 2 2.5 3
0.1
0.2
0.3
0.4
0.5
0.6
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 39
Student t DistributionArises in hypothesis testing, as it describes the distribution of the ratio of the estimated coefficient to its standard error. k - degrees of freedom.
2
12
1
11
2
2
1
)(
k
k
xkk
k
xf
0
1)( dxexk xk
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 40
Student t DistributionAs k increases t-distribution tends to the normal one.This distribution is symmetrical with mean zero and variance (k>2)
2)(2
k
kx
The t-distribution is fatter than the normal one.
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 41
Binomial DistributionDiscrete random variable with density function:
nxppx
nxf xnx ,,.1,0,)1()(
nppXpnXE )1()(,)( 2
For large n it can be approximated by a normal.
)1,0(~)1(
Nnpp
pnxz
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 42
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 43
FRM-99, Question 16If 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 44
FRM-99, Question 5Which 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 45
FRM-98, Question 10For 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 46
FRM-98, Question 10
02.1][ 2
2.00
2
22
eeXE
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 47
FRM-98, Question 16Which of the following statements are true?I. The sum of normal variables is also normalII. The product of normal variables is normalIII. The sum of lognormal variables is lognormalIV. The product of lognormal variables is lognormalA. I and IIB. II and IIIC. III and IVD. I and IV
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 48
FRM-99, Question 22Which of the following exhibits positively skewed distribution?I. Normal distributionII. Lognormal distributionIII. The returns of being short a put optionIV. The returns of being long a call optionA. II onlyB. III onlyC. II and IV onlyD. I, III and IV only
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 49
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 50
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 greaterB. The kurtosis of the leptokurtotic distribution is greaterC. The skew of the leptokurtotic distribution is smallerD. The kurtosis of the leptokurtotic distribution is smaller
Zvi Wiener FE-Wilmott-IntroQF Ch4 slide 51
Home AssignmentRead 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.