Statistics for Financial Engineering

Part1: ProbabilityInstructor: Youngju Lee

MFE, Haas Business SchoolUniversity of California, Berkeley

Overview of Class

Part1: Probability – March 23rd, 2006 Part2: Statistics – March 25th, 2006 Class will be organized as

Definitions Some comments about from definition Problems Applications in financial engineering – I will give short

examples how I apply these concepts in my real life and practice since I assume you do not have any idea about financial engineering as of now.

Probability

1. Probability

2. Random Variables – Discrete and Continuous

3. Distribution and Probability Density

4. Moments and Moments Generating Function

5. Stochastic Independence

6. Basic Limit Theorem

Probability

Definition1

A probability function (P) is a function which assigns to each event A a number denoted by P(A), called the probability of A and satisfies the following requirements. a) P is non-negative; P(A) 0 for every event A b) P us normed: that is P(S) = 1 c) P is additive: that is, for every collection of pairwise(or mutually) disjoint events, we

have P( jj

A ) = ( )jj

P A

c-1) P is finite additive

Probability

Some consequences of definition 1

a) P(0) = 0

b) P(n

jj

A ) = ( )n

jj

P A

c) P( cA ) = 1 – P(A) d) 1 2 1 2, ( ) ( )A A P A P A

e) 0 ( ) 1P A

f) 1 2 1 2 1 2( ) ( ) ( ) ( )P A A P A P A P A A

g) P is sub additive: 11

( ) ( )j j jj

P A P A

Probability

Try this

Twenty balls numbered from 1 to 20 are mixed in an urn and two balls are drawn successively and without replacement. If 1x and 2x are the numbers written on the first and second ball drawn,

respectively, what is the probability that

i. 1 2 8x x

ii. 1 2 5x x

Probability

In finance world?

This is the very basic concept of everything. – States, Monte-Carlo simulations and Binomial Trees, etc.

Conditional Probability

Definition 2

Let A be an event such that ( ) 0P A . Then the conditional probability, given A, is the function denoted by ( | )P A and defined for every event B as follows:

( )( | )

( )

P A BP B A

P A

( | )P B A is called the conditional probability of B given A.

Conditional Probability

Some consequences from definition 2

a) (Multiplicative Theorem) Let , 1, 2,3...,jA j n be events such that 11( ) 0n

j jP A then 11 1 2 1 1 1 2 2 2 1 1( ) ( | ) ( | ) ( | ) ( )n

j j n n n nP A P A A A A P A A A A P A A P A

b) (The total probability theorem) Let , 1, 2,3...,jA j n be a partition of S with

( ) 0jP A , all j. Then for B A , we have ( ) ( | ) ( )j jj

P B P B A P A

c) (Bayes Fomular) If , 1, 2,3...,jA j n is a partition of S and ( ) 0jP A and if

( ) 0P B then ( | ) ( )

( | ) 0( | ) ( )

j jj

i ii

P B A P AP A B

P B A P A

Conditional Probability

Try this.

Suppose that a test for diagnosing a certain heart disease is 95% accurate when applied to both those who have the disease and those who do not. If it is known that 5 of 1000 in a certain population have the disease in question, compute the probability that a patient actually has the disease if the test indicates that he does. (try to explain by intuitive reasoning.)

Conditional Probability

In finance world?

Fancy empirical model – Regime Switch Model

Independence

Definition 3

The events A and B are said to be independent if ( ) ( ) ( )P A B P A P B .

Definition4: The events , 1, 2,...,jA j n are said to be mutually or completely independent

if the following relationships hold. 1 2 1( ... ) ( )... ( )n nP A A A P A P A .

Independence

Some consequences from definition 3

a) If the events 1 2, ,... nA A A are independent, so are the events ' ' '1 2, ,... ,nA A A where '

jA is CjA or jA , j = 1,2,…n.

Independence

Try this. – Easy!

Six fair dice are tossed once. What is the probability that all six faces appear?

Seven fair dice are tossed once. What is the probability that every face appears at least once?

Independence

In finance world?

Is there any independent event in the financial world or at least in practice?

Random Variables

Definition 4

Random variable is a function which assigns to each sample point s S a real number, the value of the r.v. at s.

Discrete Random Variable

Definition 5: Binomial distribution is associated with binomial experiments – success or fail

( ) {0,1,2..., }

( ) ( )

0 1, 1

x n x

X S n

nP X x f x p q

x

p q p

Discrete Random Variables

Definition 6: Poisson distribution

( ) ( )!

x

P X x f x ex

Discrete Random Variables

Definition 7: Discrete uniform distribution

( ) {0,1,2..., 1}

1( ) ( )

0,1,2...

X S n

P X x f xn

x n

Discrete Random Variables

Definition 8: Hyper-geometric distribution

( ) {0,1,2..., }

( ) ( )

X S r

m n

x r xP X x f x

m n

r

Discrete Random Variables

Definition 9: Negative binomial distribution

( ) {0,1,2...}

1( ) ( )

0 1, 1 , 0,1,2....

r x

X S

r xP X x f x p q

x

p q p x

Discrete Random Variables

Definition 10: Multi-nominal distribution

'1 2

1

1 21 1 1

1 2

1

( ) { ( , ,...., ) : 0, 0,1, 2... , }

( ) ( ) ...! !... !

1

k

k j jj

r x x xk

k

k

jj

X S x x x x x j k x n

nP X x f x p p p p

x x x

p

Continuous Random Variables

Definition 11: Normal distribution

Rx

xxfRSX

,

2exp

2

1)(,)(

2

2

Continuous Random Variables

Some consequences from definition 11

Normal distribution is symmetric. Normal distribution has maximum value at mean.

Continuous Random Variables

Try this.

Let X to be distributed as N(0,1) and for a < b, let p P a X b . Then use the

symmetry of the p.d.f. f in order to show that:

1. For 0 ,a b p b a

2. For 0 , 1a b p b a

3. For 0 ,a b p a b

4. For 0, 2 1c P c X c c

Continuous Random Variables

In finance world?

Everything is assumed normal distribution in financial engineering. To check normality,

Use K-S test or Normal Probability Plot. I will cover this later.

Continuous Random Variables

Definition 12: Gamma distribution

0,0

0,0

0,1

)(

),0()(

1

x

xexxf

SXx

Where

0

1 dyey y . This integral is known as the Gamma function.

Continuous Random Variables

Definition 13: Chi-square distribution

0

0,22/1

1)( 212/

21

r

xexr

xfxr

Continuous Random Variables

Definition 14: Negative exponential distribution

0,0,0

0,)(

x

xexf

x

Continuous Random Variables

Definition 15: Continuous uniform distribution

else

xxf

RSX

,0

,1)(

)(

Continuous Random Variables

Definition 16: Beta distribution

0,0

,0

10,1)(

)(

11

else

xxxxf

RSX

Continuous Random Variables

Definition 17: Cauchy distribution

0,,

1)(

)(

22

RRx

xxf

RSX

Continuous Random Variables

Definition 18: Lognormal distribution

0,0

0,2

loglogexp

2

1)(

)(

2

2

x

xx

xxf

RSX

Continuous Random Variables

Definition 19: Bi-variate normal distribution

2

2

22

2

22

1

11

2

1

112

2

221

21

21

1

12

1,

xxxxq

exxfq

D.F. and P.D.F.

Definition 20: The distribution function

The distribution function F of a random variable X satisfies the following properties.

RxxF ,1)(0

F is non-decreasing F is continuous from the right

xxF

xxF

,1)(

,0)(

D.F. and P.D.F.

Some consequences from definition 20

Let X be an ),( 2N distributed r.v. and set

xY . Then Y is an r.v. and its

distribution is )1,0(N

Let X be an )1,0(N distributed r.v. then 2XY is distributed as .21

Let X be a ),( 2N distributed r.v. then the r.v. 2

x

is distributed as .21

D.F. and P.D.F.

Try this. – It is better to know what logistic distribution is.

Show that the following function F is a d.f. (Logistic distribution) and derive the corresponding p.d.f., f.

( )

1( ) , , 0,

1 xF x x R R

e

D.F. and P.D.F.

In finance world?

You probably want to remember some consequences from last slide. We use this all the time to make trading signals.

D.F. and P.D.F.

Definition 21: Joint distribution function

The joint distribution function of 1 2,X X X is 1 2 1 1 2 2, ,F X X P X x X x

D.F. and P.D.F.

Definition 22: Quantile of a distribution

Let X be an r.v. with d.f. F and consider a number p such that 0 < p < 1. A pth quantile of the r.v.X or of its d.f. F is a number denoted by px and having the following property:

pxXP p and pxXP p 1 .

D.F. and P.D.F.

Definition 23: Mode

Let X be an r.v. with p.d.f. f. Then a mode of f, if it exists, is any number which maximize f(x).

D.F. and P.D.F.

Try this.

Let X be an r.v. with p.d.f. f symmetric about a constant c then show c is a median of f.

D.F. and P.D.F.

In finance world?

Moments

Definition 24: Moments of random variables

For n=1,2,…, the nth moment of g(x) is denoted by nxgE )( and is defined by

nn

nn

x

n

n

dxdxdxxxxfxxxg

xfxg

xgE...)...,()...,(

)()(

)(212121

The first moment is called mean and the difference between the second moment and the square of the first moment is called variance.

Moments

Some consequences from definition 24The basic properties of mean

|)(||)]([|

)()(,

])([])([

)(

XgEXgE

YEXEYX

dxgcEdxcgE

ccE

The basic properties of variance

222

222

2

)]([])([)]([

)]([])([

0

XgEXgEXg

XgcdXcg

c

Moments

Try this.

A roulette wheel has 38 slots of which 18 are red, 18 black, and 2 green. Suppose a gambler is placing a bet of $M on red. What is the gambler’s expected gain or loss and what is the standard deviation?

Moments

Try this. But do not calculate!

Let X be an r.v. taking on the values -2,-1,1,2 each with probability 0.25. Set Y=X*X and compute the following quantities. EX, Var(X), EY and Var(Y).

Moments

In finance world?

I do not think you can be in finance industry without talking about Sharpe ratio a lot. (mean/sd)

We also need to look at skewness and kurtosis.

Stochastic Independence

Definition 25: Stochastic independence

The r.v.’s ,...2,1, jX j are said to be independent if, for sets kjRB j ,...2,1, , it holds

)(,...,1, 1 jjkjjj BXPkjBXP

Stochastic Independence

Some consequences from definition 25

Let 21 , XX have the bivariate normal distribution. Then 21 , XX are independent if and only if they are uncorrelated.

Let jX be ),( pnB j , j=1,2,…,k and independent. Then

k

jjXX

1

is B(n,p), where

k

jjnn

1

.

Let jX be )( jP , j=1,2,…,k and independent. Then

k

jjXX

1

is )(P , where

k

jj

1

Stochastic Independence

Some consequences from definition 25

Let jX be ),( 2jjN , j=1,2,…,k and independent. Then

k

jjXX

1

is ),( 2N , where

k

jj

1

and

k

j

j

1

22

Let jX be 2

jr , j=1,2,…,k and independent. Then

k

jjXX

1

is 2r where

k

jjrr

1

Let jX be ),( 2

jjN , j=1,2,…,k and independent. Then

22 /kS is 21 k where

k

jj XX

kS

1

22 1

The Central Limit Theorem

Definition 26: Central Limit Theorem

Let 1 2, ... nX X X be i.i.di r.v. with mean and variance 2 . Then

10,1

n

n

n XN

S