advanced financial models

Advanced Financial Models under construc2on

Learning Objec-ves

¨  Lognormal Distribu-ons ¨  Rela-ons between

¤  Normal & lognormal n  Pdfs n  Sta-s-cs

¤  Simple and natural log rates

2

Hypotheses and Models

¨  Explana-ons of phenomenon ¤  Hypothesis

n  A proposed explana-on for a phenomena

¤  Law n  Statement of a cause and effect

without explana-on n  Newton’s law of gravity

¤  Theory n  A well-‐established explana-on for

a phenomenon n  Einstein’s theory of gravity

¨  A model is a mathema-cal or physical representa-on of a phenomenon ¤  The “Bohr atomic model” ¤  Newton’s inverse square law of

gravity

¤  Einstein’s Theory of General Rela-vity

3

221

rmmGF ⋅

⋅=

SPX Daily Ln Rate Histogram: Zoom 4

SPX Daily Ln Rate Histogram: More Zoom 5

Again this histogram includes daily return rates from 1950 <-4.5% should happen less than once in a thousand years, but there have been 31 such days since 1950 or about once every two years -22.9% day should not have happened (Oct 19, 1987)

SPX Daily Ln Rate: August – December 2008 6

SPX Daily Ln Rate: Mean 7

-‐350%

-‐250%

-‐150%

-‐50%

50%

150%

250%

1/5/51 11/9/57 9/13/64 7/19/71 5/23/78 3/27/85 1/30/92 12/4/98 10/8/05

Annualized mean 22 day annualized tailing mean 252 day annualized tailing mean Long term annualized tailing mean

SPX Daily Ln Rate: Mean 8

-‐80%

-‐60%

-‐40%

-‐20%

0%

20%

40%

60%

80%

1/2/90 3/12/92 5/21/94 7/29/96 10/7/98 12/15/00 2/23/03 5/3/05 7/12/07 9/19/09

Zoom in on Annualized mean 252 day annualized tailing mean Long term annualized tailing mean

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1/3/1950 3/22/1958 6/8/1966 8/25/1974 11/11/1982 1/28/1991 4/16/1999 7/3/2007

SPX Daily Ln Rate: Standard Devia-on 9

Annualized standard devia-ons (‘vola-lity’) 22 day annualized trailing vola-lity 252 day annualized trailing vola-lity Long term annualized trailing vola-lity

0%

5%

10%

15%

20%

25%

30%

1/2/1990 9/28/1992 6/25/1995 3/21/1998 12/15/2000 9/11/2003 6/7/2006

SPX Daily Ln Rate: Standard Devia-on 10

Zoom in on Annualized standard devia-ons (‘vola-lity’) 252 day annualized trailing vola-lity Long term annualized trailing vola-lity

SPX Daily Ln Rate: Autocorrela-on Cluster 11

SPX Daily Ln Rate: Autocorrelogram 12

Natural log daily return rates for SPX, v 1950 – 2011 15471 days Rates do look rather uncorrelated

SPX Daily Ln Rate: Autocorrelogram 13

Natural log daily de-‐trended squares of return rates (variance) for SPX, (v-‐u)2 1950 – 2011 15471 days There is some posi-ve autocorrela-on (persistence) Might even be greater persistence over shorter periods

SPX Daily Ln Rate: Histogram of Annualized Daily Variance

14

Histogram of Annualized Daily Variance

SPX: Annual Accumula-on of Daily Returns 15

10,000 annual sums of 252 day (1 year) con-guous return rates randomly selected from 1950 to 2011 This histogram doesn’t look normal at all as the addi-ve CLT would indicate So the rates are not IID/ FV

SPX: Ln Rate Q-‐Q Plot 16

A Q-‐Q plot compares the measured rates to ideal normal rates from measured mean and variance

Natural Log Rate – More Tests

¨  Jarque Bera normality test ¤  JB is a Chi Squared sta-s-c with 2 dof ¤  Normality via chi squared considera-on of s

skew, S, and kurtosis, K, the 3rd and 4th moments of distribu-on which measure asymmetry

17

440,657 4

3)(29.06001.056216

15471

43)(KS

6nJB

22

22

=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

JB (χ2

statistic)

If normality is rejected, what is the probability of a rejection error

0.0000 100.00%4.6051 10.00%5.9914 5.00%9.2103 1.00%10.0000 0.67%15.0000 0.06%20.0000 0.00%25.0000 0.00%30.0000 0.00%35.0000 0.00%40.0000 0.00%45.0000 0.00%50.0000 0.00%

So there is ~0% probability of incorrectly rejec-ng the normal hypothesis

Natural Log Rate – Tests For Normality 18

Stock Return Rate Summary

¨  Historical stock return rates, r and v, are characterized by ¤  Leptokurtosis

n  Fat or heavy tails: more extreme events than ‘normal’ n  More return rates near the mean than ‘normal’

¤  Nega-ve skew n  More extreme downside events than upside

¨  Dependence in return rate vola-lity ¤  Rate vola-lity clustering, short term persistence then reversion to mean

¨  Less frequent sampling e.g., weekly and monthly would show some smoothing, but s-ll not normal ¤  However, quarterly or annual sampling would ignore important rate of

return informa-on

19

Lognormal Pdf 20

The lognormal pdf is asymmetric, is not nega-ve, over -me the mean, mode, and median drii further apart, and the distribu-on skews more posi-vely.

Lognormal Pdf 21

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Mode Median Mean (expected)

22

( )

( )

∞>>⋅⋅

=

−

⋅

−−

x 0

eπ2σx

1σ μ, |x f

1s,uNL~r

σ2μ)(lnx

x

2

2

2

( )

( )

∞>>∞⋅

= ⋅

−−

x -‐

eπ2σ

1σ μ, |x f

s,uN~v

σ2μ)(x

x

2

2

2

u is mean, median, and mode The parameters is the normal pdf above are also the sta-s-cs – the mean and variance

The mean, mode, and median are all different The parameters is the lognormal pdf are the same as for the normal pdf, but they are not the sta-s-cs, not the mean or variance

¨  Why simple returns can’t really be normal ¤  Simple returns are compounded over -me increments, but normal random variables are mul-plied

¤  (1+r)n ¤  u·∙n

23

( )r1lnv +=

...3r

2rr)r1ln(v

)1x( ...3x

2xx)x1ln(

32

32

−+−=+=

−≠−+−=+

Variance of Simple and Log Returns 24

[ ] [ ][ ] [ ]

( )

( )

( )

[ ]( ) ( )

1er1E

1ee

1ee

1ee

ee

ee

xExE

dr1VarrVar

2

2

2

2

2

22

22

222

s2

s

2

2su

s2su2

ssu2

s2us2u2

2

2su

2s2u2

22

2

−⋅+=

−⋅⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

−⋅=

−=

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−=

=+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅

+⋅

+⋅+⋅

+⋅

+⋅

[ ]( ) ( )( ) ( )

( )( )

( )

1)s 1,(a sd

1ed

d1e

d1lns

1)(a 1 a1

a1d1lns

1ea1

1er1E d

22

s2

2s

22

2

2

22

s2

s22

2

2

2

2

<<<<≈≈

−≈

+≈

+≈

<<≈+

⎟⎟⎠

⎞⎜⎜⎝

⎛

++=

−⋅+=

−⋅+=

( )[ ]

[ ]

[ ][ ] 2

2

22

s2u22

2su

2skukk

2

v

e xE

e xE

e xE

su,NL~

er1X

⋅+⋅

+

⋅+⋅

=

=

=

=+=

Variance of Simple and Log Returns 25

Future Value Factor: 1+r = ev

[ ] ( ) 1eer1var22 ssu2 −⋅=+ +⋅

[ ] *

2

u2s

ueer1E ≡=+

+

[ ] uer1M =+

26

( ) ( )

[ ]

[ ] [ ][ ] ( )[ ]

[ ] ondistributi normal log for Median er1MM[x]

ondistributi lognormal for moment 2 er1E xE

ondistributi normal log for (mean) moment 1 er1E xE

ondistributi normal log for moment k exE

su,NL~ er1

u

nds2u222

st2s

u

th2sk

ukk

2v

2

2

22

=+=

=+=

=+=

=

=+

⋅+⋅

+

⋅+⋅

Central Limit Theorem 27

( )

( )2n

n

n

1ii

n

1ii0n

s,uN~ny

uny

n

v

vSln)Sln()Sln(

→=

=Δ=−

∑

∑

=

=( )

1)x,x(NL~r

g1f)r(1

fr1SS

n1

n

n1

n

1ii

n

n

1ii

0

n

−

+→=⎥⎦

⎤⎢⎣

⎡+

≡+=

∏

∏

=

=

Assume that n is large and r and v are IID/FV

28

( ) ( )

( )

∑∏

∑∏

∑∏

==

==

==

=⎥⎦

⎤⎢⎣

⎡

+=⎥⎦

⎤⎢⎣

⎡

+=⎥⎦

⎤⎢⎣

⎡+

n

1ii

n

1i

v

n

1ii

n

1i

v

n

1ii

n

1ii

veln

r1lneln

r1lnr1ln

i

i

n21

i

vvv0

n

1i

v0n

n210

n

1ii0n

e ...e e S

eS S

)r(1....)r(1)r(1 S

)r(1S S

⋅⋅⋅⋅=

⋅=

+⋅⋅+⋅+⋅=

+⋅=

∏

∏

=

=

( ) ( )

( )

( ) ( )

( ) n210

n

1ii0n

n210

n

1ii0n

v...vvSln

vSln Sln

)rln(1....)rln(1)rln(1Sln

)rln(1Sln Sln

++++=

+=

+++++++=

++=

∑

∑

=

=

Mean Natural Log Return Rates 29

Example: v is distributed uniformly from -‐10% to +20% Average of sums of vi are normal (sum of n rates)

( )2n

1ii

s,uN~n

v∑=

Simple Future Value Factors 30

Example: r is distributed uniformly from -‐10% to +20%

)x,x(NL~)r(1fn1

n

1ii

n1

n ⎥⎦

⎤⎢⎣

⎡+= ∏

=

Simple Future Value Factors 31

( ) )x,x(NL~r1fn

1iin ∏

=

+=

Example: r is distributed uniformly from -‐10% to +20% f is distributed lognormal

We did plot a histogram of natural return rates, v, for the SPX. It did have the general appearance of normality. But a Levy stable seems like a bemer fit, but has disadvantages. However, the typical assump-on in finance is that v is normally distributed which has a number of advantages. One advantage is that the loca-on sta-s-cs are iden-cal – mode, median, and mean – it’s a symmetric

32

v)r1(ln e)r1(

v)r1(ln e)r1(v

iiv

ii

=+=+

=+=+

For stocks or other financial assets, so far there has been no assump-on on the distribu-ons of v and r other than being IID/FV But the rela-onship between r and v has been defined as

( ) ( )2s,uN~r1lnv +=

33

( ) ( )( ) ( ) ( )

( ) ( ) 1s,uNL1e~r

s,uNLe~r1

s,uN~r1lnv

2s,uN

2s,uN

2

2

2

−≡−

≡+

+=

Another advantage is the normal distribu-on scale linearly in -me. The mean driis to the right while the variance increases.

( )2sn,unN~vn ⋅⋅⋅

Another advantage is the normal distribu-on scale linearly in -me. The mean driis to the right while the variance increases. Yet another advantage is the rela-on between the normal and lognormal distribu-ons is similar to the rela-on between the na-ral log rate and simple rate

Therefore the simple rate, r, is lognormal under assump-on that the natural log rate is lognormal

Natural Log Rate Autocorrelogram 36

Natural log daily absolute return rates for SPX, |v| Daily range 1950 – 2011 15471 days

Common PDFs in Finance

¨  Gaussian / Normal ¤  IID / FV, two parameters ¤  CLT for sums of IID/FV random

variables ¤  Special case Levy stable and ellip-c

distribu-ons

¨  Ellip-c ¤  IID / FV, two parameters ¤  unimodal, no skew, no kurtosis other

than Gaussian case ¤  Linear correla-on defines linear

dependence ¤  Used in MPT and CAPM ¤  Includes Gauss, Cauchy, t-‐distr,

Laplace, symmetric Levy Stable

¨  Lognormal ¤  IID / FV ¤  CLT for products of IID/FV random

variables ¤  Posi-ve ¤  Mode, median, mean non-‐coincident

¨  Levy stable ¤  IID, not generally FV, 4 parameters

¤  Unimodal, skew, kurtosis other than Gaussian case

¤  Central limit theorem for IID and stable but not FV random variables converges to a Levy stable distribu-on

¤  Includes Gaussian, Cauchy, Levy

37

More on Covar & Corre 38

[ ] [ ] [ ] [ ]

yExEyxEy,xCov ⋅−⋅=

Monthly Idealized PDFs From SPX History 39

ln(1+r) = v Normally distributed N(u,s2) u and s2 are normal pdf parameters and sta-s-cs -‐ mean and variance

(1+r)= ev Lognormally distributed NL(u,s2)

Same pdf parameters, but different mean and

variance

Monthly Idealized PDFs From SPX History 40

Future value factor, (1+r) = ev lei shiied by -‐1, r

Return Rate PDFs: Sta-s-cs Increase With Time 41

-‐75% -‐50% -‐25% 0% 25% 50% 75% 100% 125% 150% 175% 200% 225% 250% 275% 300%

Natural log rates (v) are assumed normal. The mean and variance of a normal distribu-on scale linear in -me

The future value factors (1+r) are assumed log normally distributed. The mean and variance do not scale linearly in -me.

42

Future Value Factor: 1+r = ev

[ ] ( ) 1eer1var22 ssu2 −⋅=+ +⋅

[ ] *

2

u2s

ueer1E ≡=+

+

[ ] uer1M =+

43

[ ] [ ] [ ] [ ]

( )

( )

( )

[ ]( ) ( )

1er1E

1ee

1ee

ondistributi normal logof Variance 1ee

ee

ee

xExEr1VarrVar

2

2

2

2

2

22

22

222

s2

s

2

2s

u

s2s

u2

ssu2

s2us2u2

2

2s

u2s2

u2

22

−⋅+=

−⋅⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

−⋅=

−=

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−=

−=+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅

+⋅

+⋅+⋅

+⋅

+⋅

-‐1.25 -‐1.00 -‐0.75 -‐0.50 -‐0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

Idealized PDFs for 36 Months 45

Forecast 36 month natural log rate of return normally distributed N(36·∙u, 36·∙s2)

Forecast 36 month future value factor lognormally distributed

Forecast 36 month simple rate of return lognormally distributed

Lognormal Distribu-on 46

( ) ( )

[ ]

[ ] [ ][ ] ( )[ ]





su,NL~ er1

u

nds2u222

st2s

u

th2sk

ukk

2v

2

2

22

=+=

=+=

=+=

=

=+

⋅+⋅

+

⋅+⋅

( ) ( )

( )

( ) ( )

( ) n210

n

1ii0n

n210

n

1ii0n

v...vvSln

vSln Sln

)rln(1....)rln(1)rln(1Sln

)rln(1Sln Sln

++++=

+=

+++++++=

++=

∑

∑

=

=

SSS

r

)r1(SS

rate return Simple

1i

1iii

i1ii

−

−

−

−=

+⋅=

( ) ( )

)r1ln(

SlnSlnSS

lnv

eSS

rate return log Natural

i

1ii1i

ii

v1ii

i

+=

−=⎟⎟⎠

⎞⎜⎜⎝

⎛=

⋅=

−−

−

Lognormal Distribu-on 47

( ) ( )

[ ]

[ ] [ ][ ] ( )[ ]





su,NL~ er1

u

nds2u222

st2s

u

th2sk

ukk

2v

2

2

22

=+=

=+=

=+=

=

=+

⋅+⋅

+

⋅+⋅

GARCH Time Series

¨  Similar to historic vola-lity ¤  Simple condi-onal dependence in the second moment (vola-lity)

n  Vola-lity clustering or persistence

¨  The GARCH vola-lity has three contribu-ons ¤  Long term average vola-lity, s2, so there’s a reversion of the mean ¤  Short term dependence on recent square of return rate, v2 ¤  Short term dependence on recent Garch vola-lity, h

¨  To Do n  Is there a probability distribu-on? Maybe not n  Plot the resul-ng rates and look for fat tails n  So it looks good historically, but how can it be used in decision making ?

48

GARCH Time Series

¨  The GARCH(1,1) vola-lity model with the natural log rate process model vola-lity has three contribu-ons

49

( )

0βλ,α,1βαγ

β ,α ,γ :weights

hβvαsβα1

hβvαsγh

zh uv

1i21i

2

1i21i

2i

iii

>

=++

⋅+⋅+⋅−−=

⋅+⋅+⋅=

⋅+=

−−

−−

The Gaussian rate process is vi = u + s ·∙zi s is the (tradi-onal) long term average standard devia-on z is the standard normal random variable h is the Garch variance v is the nat log return rates Example: α = .85 , β = .1 , γ = .05

GARCH Time Series 50

Single simulated GARCH(1,1) vola-lity for 15,461 days

Adendum: Nat Log & Exp 53

( ) ( ) ( )

y+xyxyxyx

32

32

x

)xln(

e = e e )(e = e

...31x

21x1x)xln(

)1x( ...3x

2xx)x1ln(

x1)xln(

dxd

)yln()xln(yxln

)yln()xln()yxln(x)eln(

)0x( xe

⋅

−−

+−

−−=

−≠−+−=+

=

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

+=⋅

=

>=

⋅

Addendum 54

RatePeriodic mean

Annual mean

Periodic standard deviation

Annual standard deviation

a α

g γ

vi u µ s σ

d = Var(r) = Var(1+r)

ri d δ

Addendum 55

dwσdt2σμ dln(S)2

* ⋅+⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−=( )

( )Tσ

Tσ.5rKSln

d

Tσ

Tσ.5rKSln

d

2*0

2

2*0

1

⋅

⋅⋅−+⎟⎠⎞

⎜⎝⎛

=

⋅

⋅⋅++⎟⎠⎞

⎜⎝⎛

=

[ ] ( ) 1eer1var22 ssu22 −⋅=+=δ +⋅

[ ] *

2

u2s

ueer1E ≡=+

+

( ) ( )[ ]

[ ]

tsz1ii

s,0N

1i

i

v1ii

2i

i1i-‐i

eSS

e~SS

eSS

s,0N~v

vSln Sln

2

i

Δ⋅⋅−

−

−

⋅=

⋅=

+=

tszi

itsz

i1ii

i)rln(1v

ii

ii

i

ii

er

)r(1e

)r(1SS)r(1ee

tszv

)rln(1 v

Δ⋅⋅

Δ⋅⋅

−

+

=

+=

+⋅=

+==

Δ⋅⋅=

+=

Addendum 56

( )[ ] [ ]

( )

( ) ( )

−+−=+

+Δ⋅⋅

+Δ⋅⋅

+Δ⋅⋅+==

−+−=+

++++=

=⋅δ⋅+

=+≡δ

⋅δ⋅+⋅=

⋅δ⋅==

Δ⋅⋅

Δ⋅⋅

3r

2rr)r1ln(

...6

tsz2

tsztsz1ee

3x

2xx)x1ln(

6x

2xx1e

e ΔtZ1

rSDr)(1SD

ΔtZ1SS

ΔtZ?r

3i

2i

ii

3

i

2

ii

tszv

32

32x

tszi

itt

ii

ii

1-‐ii

Not yet ready to related normal and lognormal distribu-ons. Need lognormal sta-s-cs and Ito’s Lemma Normal

Natural log rates Natural log prices

Lognormal Simple rates Future value factors Prices

Levy Stable Distribu-on

¤  Bemer fits historical rates of return n  Can model Leptokurtosis and skew

n  Constant parameters

n  Generalized Central Limit Theorem n  Normal distribu-on is a special case n  Problems included

n  Infinite variance n  Variance cant be used as a measure of risk or vola-lity n  CAPM, MPT, B-‐S n  PDF models not applicable

n  Generally no analy-c representa-on ¤  To Do

n  Fit data to a distribu-on and graph n  Why does FMH without IID invoke this model n  How does it relate to power law model (Has an α > 2 ?)

57

Levy Stable Distribu-ons 58

[ ]( )( ) parameter location ,μ

parameter scale 0,cparameter skewness 1,1β

parameter stability (0,2]αParameters

∞∞−∈

∞∈

−∈

∈

undefined otherwise2,α when 0 :kurtosis excessundefined otherwise2,α when 0:skew

infinite otherwise2,α when c2 :variance

undefined otherwise 1,α whenμ :mean2

=

=

=⋅

>

Levy Stable Distribu-ons

¨  DJIA: α = 1.5958 β = -‐.0995 µ = .0002 σ = .0056 ¤  (5/26/1896 – 1/16/2004 daily)

¨  SPX = α = 1.6735 β = .1064 µ = -‐.0002 σ = .0049 ¤  (3/1/1950 – 5/27.2005 daily)

¨  Only three sets of parameters result in closed form ¤  Gaussian

n  Actually two of the four parameters are zero (?) or reduced to different 2 ? n  Finite variance

¤  Levy ¤  Cauchy

59

[ ]( )( ) parameter location ,μ

parameter scale 0,cparameter skewness 1,1β

parameter stability (0,2]αParameters

∞∞−∈

∞∈

−∈

∈

Power Law 60

Power Law Method

¨  Coopera-on, herding, cri-cality ¤  How Nature Works – Bak ¤  Ubiquity – Buchanan

¨  The ubiquity of scale-‐free behavior and self-‐organiza-on in Nature led Bak, Tang and Wiesenfeld (BTW) to coin the term Self-‐Organized Cri-cality (SOC) to explain the emergence of complexity in dynamical systems with many interac-ng degrees of freedom without the presence of any external agent ; SOC was devised to be a sort of supergeneral theory of complexity.

61

Power Law

¨  Confusion based on Fractal Market Hypothesis: Is it stable or power law?? ¨  Hurst soiware shows a random series to be persistent ??

¨  Hurst exponent ¤  0.5 is Brownian t1/2 √t

¤  0 < H < 0.5 : an--‐persistent, mean rever-ng ¤  .5 < H ≤ 1.0 : persistent

¨  Stability parameter ¤  α = 1 / H, example Gaussian: α = 2, H = .5

¨  Correla-on (?) C = 22H-‐1 – 1

¨  Example ¤  SPX: 3/1/1950 – 5/27/2005 daily

n  α = 1.6735 β = .1064 µ = -‐.0002 σ = .0049 n  H = .5976 C= .1448

¤  SPX: 1/3/1950 – 6/24/2011 daily n  H= .562 α= 1.779 C= .090

62

Reference: Nat Log & Exp 63

)1x( ...3x

2xx)x1ln(

x1)xln(

dxd

)yln()xln(yx

ln

)yln()xln()yxln(x)eln(

)0x( xe

32

x

)xln(

−≠−+−=+

=

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

+=⋅

=

>= ( ) ( ) ( )

++++=

⋅

−−

+−

−−=

⋅

6x

2xx1e

e = e e )(e = e

...31x

21x1x)xln(

32x

y+xyxyxyx

32

RatePeriodic mean

Annual mean


Period standard deviation

Rate pdf

a α

g γ

vi u µ s σ Normal

d = SD(r) = SD(1+r)

ri d δ Log normal

Related Concepts

¨  Expected Rate of Return On Equity ¤  CAPM requires that the return rate is normally distributed with a trend ¤  Ordinary least squares

¨  Theore-cal basis for r being an independent random variable ¤  Efficient Market Hypothesis

¨  Theore-cal basis for r being an independent random variable with a trend ¤  Ra-onal Market Hypothesis

64

( ) ( )( )

( )FMFEE

iE1ii

E1ii

rrβr k r zsr1SS

r1SSE

−⋅+==

⋅++⋅=

+⋅=

−

−

Geometric Brownian Mo-on 65

( ) ( )[ ]

[ ]

tsz1ii

s,0N

1i

i

v1ii

2i

i1i-‐i

eSS

e~SS

eSS

s,0N~v

vSln Sln

2

i

Δ⋅⋅−

−

−

⋅=

⋅=

+=

tszi

itsz

i1ii

i)rln(1v

ii

ii

i

ii

er

)r(1e

)r(1SS)r(1ee

tszv

)rln(1 v

Δ⋅⋅

Δ⋅⋅

−

+

=

+=

+⋅=

+==

Δ⋅⋅=

+=

dwσdt2σμ dln(S)2

* ⋅+⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−=( )

( )Tσ

Tσ.5rKSln

d

Tσ

Tσ.5rKSln

d

2*0

2

2*0

1

⋅

⋅⋅−+⎟⎠⎞

⎜⎝⎛

=

⋅

⋅⋅++⎟⎠⎞

⎜⎝⎛

=

[ ] ( ) 1eer1var22 ssu22 −⋅=+=δ +⋅

[ ] *

2

u2s

ueer1E ≡=+

+

Geometric Brownian Mo-on 66

( )[ ] [ ]

( )

( ) ( )...

6tsz

2tsztsz1ee

3x

2xx)x1ln(

6x

2xx1e

e ΔtZ1

rSDr)(1SD

ΔtZ1SS

ΔtZ?r

3

i

2

ii

tszv

32

32x

tszi

itt

ii

ii

1-‐ii

+Δ⋅⋅

+Δ⋅⋅

+Δ⋅⋅+==

−+−=+

++++=

=⋅δ⋅+

=+≡δ

⋅δ⋅+⋅=

⋅δ⋅==

Δ⋅⋅

Δ⋅⋅

Not yet ready to related normal and lognormal distribu-ons. Need lognormal sta-s-cs and Ito’s Lemma Normal

Natural log rates Natural log prices

Lognormal Simple rates Future value factors Prices

u, s µ, σ r, d α, δ g, γ,

Alterna-ves

¨  Fat Tail Models ¤  Power law not exponen-al tails ¤  Leptokurtosis, finite variance ? ¤  Examples

n  Student t – no skew n  Levy stable – skew

¨  Non IID Models – non-‐sta-onary process ¤  Correla-on in rate vola-lity, but not in rate, so s-ll ‘unpredictable’

ARCH models ¤  Used with normal or other distribu-on

67

Addendum 68

( ) ( )

( ) ( ) ( )SlnSlnSln-‐Sln

eSS

Δtσz Δt2σμ 1SS

Δtσz ΔtμSlnSln

1-‐ii

it

2

ii

i1-‐ii

i1-‐ii

tt

Δtσz Δt2σμ

1tt

t

2

tt

ttt

Δ≠Δ=

=

⎥⎦

⎤⎢⎣

⎡⋅⋅+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛++⋅=

⋅⋅+⋅+=

⋅⋅+⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅−

( ) ( ) nszunSlnSln 1i-‐1-‐ni ⋅⋅+⋅+=+( ) ( )[ ]

[ ]

tsztu1ii

s,uN

1i

i

v1ii

2i

i1i-‐i

eSS

e~SS

eSS

s,uN~v

vSln Sln

2

i

Δ⋅⋅+⋅−

−

−

⋅=

⋅=

+=

69

[ ][ ]

[ ]

[ ]2εii1i-‐i

2εii1i-‐i

1i-‐i

2εii1i-‐i

s0,N~ε εSS

s0,IID~ε εSS

SSE s0,~ε εSS

+=

+=

=

+=

( ) ( )[ ]

( ) ( )

( )[ ] ( )

Δtσz 1tt

tt

tt

1i-‐1-‐ni

i

ii

1ii

1ii

eSS

SlnSlnE

tszSlnSln

1,0N~z nszSlnSln

⋅⋅⋅−

+

=

=

Δ⋅⋅+=

⋅⋅+=

−

−

Generally Rate

Periodic mean

Annual mean


Period standard deviation

Rate pdf

a α

g γ

v u µ s σ Normal

d = SD(r) = SD(1+r)

r d δ Log normal

70

[ ][ ]

tΔBzSS

tΔB ,0N~ε

tttΔ tΔ ,0N~ε εSS

i1-‐ii

i

ii1-‐ii

ttt

2t

1i-‐itttt

⋅⋅+=

⋅

−=+=

[ ]

[ ]

mszum1i1mi

sm,umNm

1i

v

2m

1ii

i

2i

eSS

e~e

sm,umN~v

increment d,multiperio

⋅⋅+⋅−−+

⋅⋅

=

=

⋅=

⋅⋅

∏

∑[ ] [ ]

it

1i

i

1ii

eS

eSS

tzt,N~sm,umN~

t

tzttt

tt

22t

µ

Δ⋅σ⋅+Δ⋅µ

⋅=

⋅=

Δ⋅σ⋅+Δ⋅µ=µ

σµ⋅⋅µ

−

−

( )ΔtσzΔtμ

SS

ΔtσzΔtμ1SS

t*

i*

tt 1-‐ii

⋅⋅+⋅=Δ

⋅⋅+⋅+⋅=

ΔtzΔw

tttΔ

SSSΔ

1i-‐i

tt 1-‐ii

⋅=

−=

−=

71

( )

1)r(1g

1fg

)r(1f

fSS

)r(1 SS

n1

n

1iin

n1

nn

n

1iin

n0

n

n

1ii

0

n

−⎟⎟⎠

⎞⎜⎜⎝

⎛+=

−=

+=

=

+=

∏

∏

∏

=

=

=

ns

u

)r(1lns

vSSln

nn

n

1iin

n

1ii

0

n

=

+=

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∑

∑

=

=

Power Law

¨  Power law with rescaled range ¨  Many natural phenomena modeled

with power law ¨  Nonlinear feedback ¨  Hurst exponent is the slope ¨  Fractal and self similar ¨  Complexity ¨  How can it be used in decision

making? ¨  The rescaled range follows a

power law

75

76

[ ][ ]

tΔBzSS

tΔB ,0N~ε

tttΔ tΔ ,0N~ε εSS

i1-‐ii

i

ii1-‐ii

ttt

2t

1i-‐itttt

⋅⋅+=

⋅

−=+=

Rate based process is Geometric Brownian Motion (GBM)

[ ]

[ ]

mszum1i1mi

sm,umNm

1i

v

2m

1ii

i

2i

eSS

e~e

sm,umN~v

increment d,multiperio

⋅⋅+⋅−−+

⋅⋅

=

=

⋅=

⋅⋅

∏

∑[ ] [ ]

it

1i

i

1ii

eS

eSS

tzt,N~sm,umN~

t

tzttt

tt

22t

µ

Δ⋅σ⋅+Δ⋅µ

⋅=

⋅=

Δ⋅σ⋅+Δ⋅µ=µ

σµ⋅⋅µ

−

−

( )ΔtσzΔtμ

SS

ΔtσzΔtμ1SS

t*

i*

tt 1-‐ii

⋅⋅+⋅=Δ

⋅⋅+⋅+⋅=

ΔtzΔw

tttΔ

SSSΔ

1i-‐i

tt 1-‐ii

⋅=

−=

−=

Appendix: Exponen-als and Natural Logs 77

( )

( )dxdy

y1

dxln(y)d

edxdy

dxed yy

⋅=

⋅=

+++++=

⎟⎠⎞

⎜⎝⎛ +=

∞→

!4x

!3x

!2xx1e

n11lime

432x

n

n

xlndxX1

ea1dxe xaxa

=

⋅=

∫

∫ ⋅⋅

Appendix: Exponen-als and Natural Logs 78

Price as a Stochas-c Diff Eqn 79

( )( )1eSSd

1eSS

eSSS

eSSS

eSS

eSS

dwtd

tztt

tzttt

tzt

t

t

tzt

t

t

tzttt

i

1i

i

1i1i

i

1i

1i

i

1i

i

i

1ii

−⋅=

−⋅=Δ

⋅=+Δ

=+Δ

=

⋅=

⋅σ+⋅µ

Δ⋅σ⋅+Δ⋅µ

Δ⋅σ⋅+Δ⋅µ

Δ⋅σ⋅+Δ⋅µ

Δ⋅σ⋅+Δ⋅µ

Δ⋅σ⋅+Δ⋅µ

−

−−

−

−

−

−

( )SfF =

80

( )[ ] ( )[ ] ... 1eSSF

211eS

SFdt

tFdF

... dSSF

21dS

SFdt

tFdF

2dwtd2

2dwtd

22

2

+−⋅∂

∂⋅+−⋅

∂

∂+

∂

∂=

+∂

∂⋅+

∂

∂+

∂

∂=

⋅σ+⋅µ⋅σ+⋅µ

( )

( )

( )dxdy

y1

dxdy

y1

dxln(y)d

edxyd

dxed

edxdy

dxed

2

y2

2

2

y2

yy

⋅=⋅=

⋅=

⋅=

dxdS

S1

dxdS

S1

dx d

dxdS

S1

dxln(S) d

2 ⋅−=⎟⎠⎞

⎜⎝⎛ ⋅

⋅=

( )

n0

nu0

tμ0t

*

*

n*

nnu

n0

nu0

)a1(SeSeS]E[S

)a1ln(u

)a1ln(nn1u

)a1(lnnu

)a1(e

)a1(SeS

**

*

*

+⋅=⋅=⋅=

+=

+⋅⋅=

+=⋅

+=

+⋅=⋅

⋅⋅

⋅

⋅

¨  Actually, they [power laws] aren’t special at all. They can arise as natural consequences of aggrega-on of high variance data. You know from sta-s-cs that the Central Limit Theorem says distribu-ons of data with limited variability tend to follow the Normal (bell-‐shaped, or Gaussian) curve. There is a less well-‐known version of the theorem that shows aggrega-on of high (or infinite) variance data leads to power laws. Thus, the bell curve is normal for low-‐variance data and the power law curve is normal for high-‐variance data. In many cases, I don’t think anything deeper than that is going on.

81

advanced financial models

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