financial mathematics class1
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8/7/2019 Financial Mathematics Class1
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Mike KokalariEmail: kokalari@post.harvard.edu
Email Work:michaelkokalari@vinafund.com
Cell: 090-797-4408
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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
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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
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Ton ti chnh
2 ti chnh:
- Portfolio Optimization
- Derivatives Pricing(Options Pricing)
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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
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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)
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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)
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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
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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
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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,
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Future Value, Present Value
r = Interest Rate
PV= Present Value (money you have today)
FV= Future Value (money you will have in T years)
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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)
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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 !
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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
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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!
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Interest Rates
Interest Rates Change Every Day
Inflation, Rates
Recession, Rates
o
q
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( , ) (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!
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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 !
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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
!
!
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Terminology
This is called Modified Duration
There are several different Duration
calculations
All Basically B(r)
'( )%
B rB
B! (
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( 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
!
!
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Terminology
"( )B r
B
"Bincluding
B
is called convexity
In finance f(x) is called Gamma or Convexity
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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 "
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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
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Compound Interest
365(1 ) 1.1057365
rFV PV ! !
Daily Compounding, for 5 Years
365 5(1 ) 1.6486365
rFV PV
! !
Daily Compounding
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.(1 )n Tr
FV PV n
!
.lim(1 )n T rTn
re
npg !
365.10(1 ) 1.105156365
!
.101 1.105171e !
Continous Compunding.pdf
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rTFV PV e!
rTPV FV e
!
( , ) rTDF T r e!
rTDF Ter
x ! x
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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
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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|
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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)
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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 ( !
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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
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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
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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!
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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)
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? A( ) (0)E S T S!
? A 0E dS !
Because
Then
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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
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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%
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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
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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)
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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
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Moment Generating Functions
? A'(0)m E x
!%
2"(0)m E x ! %
3"'(0)m E x ! %
( ) (0) [ ]k km E x! %
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~ ( , )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%
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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:
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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:
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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 !
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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
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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
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21( ) ( )
2( ) (0)T Z T
S T S eQ W W
!
? A ? A( ) ( )E f x f E xu% %
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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%
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