fe-w mswiener/zvi.html embaf zvi wiener mswiener@mscc.huji.ac.il 02-588-3049 financial engineering

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Zvi Wiener

mswiener@mscc.huji.ac.il

02-588-3049

Financial Engineering

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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

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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.