financial mathematics class1

8/7/2019 Financial Mathematics Class1

1/50

Mike KokalariEmail: [email protected]

Email Work:[email protected]

Cell: 090-797-4408


2/50

Ton ti chnh Kha hc da trn 15 kinh nghim thc tin trong lnhvc ti chnh ca ti

Tp trung ch yu vo cch thc p dng, s dngtrong lnh vc ti chnh

Nhn s trong ngnh hc ny ang c tuyn dng

bi cng ty VFM


3/50

Ton ti chnh

V tr lm vic bn thi gian ang cn gp nh sau:

Matlab (hoc C++, bao gm STL)

Chun b ti liu hc bng file PowerPoint

Cc v tr lm vic ton thi gian v bn thi gian khc: Time series Factor analysis (PCA/SVD) Cc ti liu hc khc t kha hc ny


4/50

Ton ti chnh

2 ti chnh:

- Portfolio Optimization

- Derivatives Pricing(Options Pricing)


5/50

Portfolio Optimization:

- Modern Portfolio Theory(MPT) v cc l thuyt/m hnh m

rng khc

- Cc l thuyt/m hnh ton thng k

(Statistical Methods)Factor AnalysisTime Series


6/50

Option Pricing

- Black Scholes Formula

- Cc l thuyt/m hnh m rng(chng ta s cp n mt vi l

thuyt/m hnh ny trong cc bui sau)


7/50

Mathematical Methods:

2 m hnh ton hc mi:

- Risk Neutral Valuation

- Portfolio Variance as Measure of Risk

(chng ta s hc v nhng m hnh ny)


8/50

Phn ln nhng m hnh ton ti chnh u xut pht t

nhng lnh vc ca ton ng dng

PDE: Heat Equation

Measure Theory: Girsanov Theory

Functional Analysis: Riesz Representation

Stochastic Differential Equations:Itos Lemma


9/50

Phn ln nhng m hnh ton ti chnh khng kh

PDE n gin nht

Kt qu d nht v measury theory Kt qu d nht v SDEs


10/50

p dng nhng cng c ny vo thc t nh th no mi liu kh

im nhn ca kha hc ny l:Hiu bit nhng cng c ny qua trc gicKh nng ng dngV nhiu ti khng mang tnh thun ton hc

Gi nh tt c hm u mang tnh lin tc, L2,


11/50

Future Value, Present Value

r = Interest Rate

PV= Present Value (money you have today)

FV= Future Value (money you will have in T years)


12/50

Lets say:

r = 10%PV = $1T = 1 year

Deposit your $1 (PV) in the bank for 1 year (T=1).

In 1 year, you will have:

$1 + $.1 = $1.1

FV = $1.1

FV = PV(1+r)


13/50

Now deposit $1 in the bank and reinvest the moneyat the end of year 1

1(1+r)=1.1

1.1 1.1 (1+r) = 1.21

Year 1 Year 2

PV=1, T=2, FV=1.21

(1 ) (1 )FV PV r r !


14/50

Some

Formulas:

(1 )TFV PV r ! 1

(1 )T

PV FV r

!

Let: FV=1, r=.10, T=2

2

( )

1.8264 .8264

(1 1) DiscountFactor DFPV FV FV ! !

E55F


15/50

Discount Factors

Having $.75 today is same as getting paid $1 in 3 years

( , ) (1 ) TDF T r r !

(1,.10) .9091DF

!

(2,.10) .8264DF !

(3,.10) .7513DF !

( , )PV FV DF T r!


16/50

Interest Rates

Interest Rates Change Every Day

Inflation, Rates

Recession, Rates

o

q


17/50

( , ) (1 ) TDF T r r ! (2, .10) .8264

.8116

(2,.11) .0148

DF

DF

!

! ( 1)(1 ) TDF

T rr

x ! x

32 (1 1)!

1.50!

0148

01

DF

r

!

V

V

Sensitivity to Interest Rates:

1 48!


18/50

Bond Valuation

1

( ; , ) ( , ) 1 ( , )T

t

B r c T c DF t r DF T r!

!

1

( ; , ) (1 ) 1 (1 )T

t T

t

B r c T c r r

!

!

.10r !

(different notation then assigned readings)

.08c ! 5T !


19/50

Bond Price Sensitivity to Interest Rates

1

'( ) (1 ) (1 )T

t T

t

r c r r

r

!

x !

x

( 1) ( 1)

1

'( ) (1 ) (1 )T

t T

t

r t c r T r

!

!


20/50

Terminology

This is called Modified Duration

There are several different Duration

calculations

All Basically B(r)

'( )%

B rB

B! (


21/50

( 1) ( 1)

1

"( ) (1 ) (1 )T

t T

t

r t c r T rr

!

x ! x

( 1) ( 1)

1

"( ) ( 1) (1 ) ( 1) (1 )T

t T

t

r t t c r T T r

!

!


22/50

Terminology

"( )B r

B

"Bincluding

B

is called convexity

In finance f(x) is called Gamma or Convexity


23/50

Important !

Homework Question

Why is it good for the owners of a Bondthat B(r) > 0?

Write a small paragraph in your own words.

This is not a math question, this is a thinkingquestion.

Think about what happens to a bond price

when interest rates go up. Or go down

"( ) 0B r "


24/50

Compound Interest

r is an Annual Interest Rate

Banks pay you an Annual interest rate, but allow you to re-invest themoney after 6 months or daily, etc.

4(1 ) 1.1038

4

rFV PV ! !

2(1 ) 1.10752

rFV PV ! !

365(1 ) 1.1057365

rFV PV ! !

1.0 1.05 1.05 1.05

6 Months


25/50

Compound Interest

365(1 ) 1.1057365

rFV PV ! !

Daily Compounding, for 5 Years

365 5(1 ) 1.6486365

rFV PV

! !

Daily Compounding


26/50

.(1 )n Tr

FV PV n

!

.lim(1 )n T rTn

re

npg !

365.10(1 ) 1.105156365

!

.101 1.105171e !

Continous Compunding.pdf


27/50

rTFV PV e!

rTPV FV e

!

( , ) rTDF T r e!

rTDF Ter

x ! x


28/50

Link to ODEClass

dA rAdt!

1dA rdt

A!

( )Ln r t const!

( ) (0) rtt e!

Versus:

rt

FV PV e!

rtFVr P V e

t

x!

x

( )A t AmmountOfMoney!

FVr FV

t

x!

x


29/50

Summery of Money Growth Model

95.00

100.00

105.00

110.00

115.00

1

0.00

1

5.00

130.00

0.00 0.1

0.

4 0.36 0.48 0.60 0.71 0.83 0.95

A(t)

.20r !

.05r !

dA rAdt! ( ) (0)r t

A t A e

! ( )A t FV|


30/50

From Money Growth ODE

dA rAdt!

dS Sd t SdZQ W! %

Notes:1) r is a certain return, is expected return

2) > r because stocks are risky ( r + 6%)

3) dZ is random

to Stock Price SDE(Stochastic Differential Equations)


31/50

About dZ

( ) ~ (0, )Z t N t

(0) 0Z !

( ) ( ) (0, )Z t t Z t t ( (

? A0lim ( ) ( )t Z t t Z t dZ( p ( !


32/50

95

100

105

110

115

1 0

1 5

0 0.

0.4 0.6 0.8 1 1.

S(t)

dS Sdt SdZQ W! 21

( ) ( )2( ) (0)

T Z T

S T S eQ W W

!

Stock Simulation.xls


33/50

W

hy is this here ?

dA rAdt!

( ) (0)rT

A T A e!

dS Sd t SdZQ W!

What would happen if it were not there

21( ) ( )2

( ) (0)

T Z T

ST

Se

Q W W

!

Money Growth Stock Price


34/50

Suppose:

( )( ) (0) T Z TS T S eQ W!

Also suppose =0 (this will be easier to work with)

dS SdZ W!

( )( ) (0) Z TS T S eW!


35/50

Note: S is a random variable, so its not obvious that:

? A ? AE dS S E dZW!

? A ? AE dS E SdZW!

? A ? A 0E dS S E dZ SW W! !

In probability language, S is adapted

In real world, S is yesterday's stock price,end-of-day (see Stock Simulation.xls)


36/50

? A( ) (0)E S T S!

? A 0E dS !

Because

Then


37/50

If: ( )

( ) (0)

Z TS T S

e

W

!

? A ( )( ) (0) Z TE S T E S eW !

? A ( )( ) (0) Z TE S T S E eW !

Then:

To understand ( )Z TE eW we need

Jensens Inequality

Next


38/50

Jensens Inequality

If

x%is any random variable

Example:

"( ) 0f x " ? A ? A( ) ( )E f x f E xu% %

( )x

f x e!

5( ) 1.649f x e! !

? A1

1 0

0

( ) 1 1.718xE f x e dx e e! ! !%

then

~ [0,1]x U%


39/50

Back to our problem: ? A ( )( ) (0) Z TE S T S E eW !

? A( )( ) 1E Z TZ T

E e eWW u !

? A ( )( ) (0) (0) 1Z TE S T S E e SW ! u

? A( ) (0)E S T Su

? A 0 [ ( )] (0)E

dS E S

TS

!p

!

By Jensens inequality

( )1Z TE e

W uSincethen

Or

But


40/50

Clearly something is wrong with

( )( ) (0) T Z TS T S eQ W!

21( ) ( )2( ) (0)

T Z T

dS Sd t Sdz S T S eQ W W

Q W

! p !

In the next class we will use Itos Lemma to show:

Today lets check E[S(T)] when = 0

Using Moment Generating Function (MGF)


41/50

Moment Generating Function (MGF)

( ) kxm k E e ! % x%

kx kx

dE e E xe

dk ! % %%

? A0

kx

k

dE e E x

dk ! !

% %

is any random variable


42/50

Moment Generating Functions

? A'(0)m E x

!%

2"(0)m E x ! %

3"'(0)m E x ! %

( ) (0) [ ]k km E x! %


43/50

~ ( , )x N Q W%

21

( ) var( )2( ) mean x k x kkxm k E e e ! !% %%

xE e

%

y ax b! % %

( ) ( )kby x

m k e m ka!

If then

Homework Exercise, prove this.

1) Find

2) Show if

~ (0,1)x N%


44/50

21( ) ( )2

( ) (0)

T Z T

ST

Se

Q W W

! 0Q !

? A21

( )2( ) (0)T

Z TE S T s e E e

W W

!

21( ) var( )2

mean x k x kkx

E e e

!% %

%

k W!? A( ) 0mean Z T ! ? Avar ( )Z T T!

210( ) 2

k TZ T

E e eW

W

!

MGF

Plug-in:

Back to our Problem:


45/50

Put everything together:

? A

21

( )2

( ) (0)

TZ T

E S T S e E e

WW

!

? A2 21 1

2 2( ) (0)T T

E S T S e eW W

!

? A( ) (0)E S T S!

? A 0E dS ! 0Q !

Which agrees with:


46/50

Homework

1) Why are bond owners happy B(r)>0

2 21

2k k

kxE e e

Q W !

%2

~ ( , )x N Q W%

xE e

%

2~ (0, )x N W%2 .05W !

2) Show

3) Use Excel to simulate

To generate normal random variables in excel: normsinv(rand())

If you are stuck in these problems, make an appointment to visit me at

my office at VFM.

2 .10W ! 2 .15W !


47/50

Summery

"( )B r

(1 )nTr

FV PV n!

(1 )nTr

PV FV n

!

rT

FV PV e!

rTPV FV

e

!

( , )PV FV DF r T!

'( )B r duration convexity


48/50

dA rAdt!

dS Sdt SdZQ W!

( ) ~ (0,1)Z t N

~ (0, )dZ N dt

Money Growth ODE

Stock Price Growth

? A~ ( ) ( )Z Z t t Z t( (

Wiener Process


49/50

21( ) ( )

2( ) (0)T Z T

S T S eQ W W

!

? A ? A( ) ( )E f x f E xu% %


50/50

Files

Stock Simulation.xls

Continuous Compounding.pdf

Upcoming Classes:

Oct 7: Itos Lemma

Oct 21: Deriving the Black-Scholes PDE

Solving the Black-Scholes PDE

Th Bi i l M d l

? A2dZ dt!

( ( ))df S t%

financial mathematics class1

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