Price Models

Price Models

Learning Objec-ves

¨  Models ¤  A mathema-cal or physical representa-on of a hypothesis, theory,

or law ¤  Simplifica-on of reality for decision making and design

¨  Models for dynamical systems n  Determinis-c, Stochas(c, Complex

¨  Security price and return rate models

¨  Review of probability and sta-s-cs

2

A Financial Game

¨  Would you play a financial game with ¤  a 90% probability of gaining $200 and ¤  a 10% probability of losing $100 ?

¨  Alterna-vely, would you take $20 with certainty ?

¨  In probability and sta(s(cs, you studied decision making with uncertainty ¤  But it was actually decision making with risk !

¤  Expected value: $200 ·∙ .90 -‐ $100 ·∙ .1 = $170

¤  What does that calcula-on really mean? Is it ra-onal to play this financial

game? ¤  I would guess that most of you would take the $20 , why ?

3

Ques-ons

¨  Your common sense likely tells you that too much uncertainty and risk remain in the game ¤  How many -mes can I play the game? ¤  What’s the dura-on of the game? ¤  Are there any other alterna-ve games? ¤  Any opportunity cost ? ¤  How much money do I have? ¤  Is there any risk of not ge\ng paid? ¤  Does ‘probability’ (expected value) even apply to a single game? ¤  Other ?

¨  Actually there’s a difference between risk and uncertainty, but we’ll explore that in a later chapter

4

Review Of Probability

¨  Random Variable ¤  Can take on different values unlike determinis(c variables ¤  Values come from experiments, measurements, random processes

n  Say x is a random variable with a -me sequence of values xi produced by a random process, X

¤  Random does not necessarily mean that there is no underlying structure n  The structure is defined by a probability distribu-on characterized by

parameters and sta(s(cs

5

Random process

X

Random variable

x

Random sequence

xi


¨  Expected Value ¤  The expected value of a random variable, x, is the weighted value of m

possible values or outcomes

¤  Example

¤  This calculate implies that each outcome, xi, is independent of the other m-‐1 outcomes n  Thus no condi-onal dependencies between the m outcomes n  How do we determine Pr[xi] ?

6

[ ] [ ] i

m

1ii xxPrxE ⋅=∑

=

[ ] [ ] 170$100$1.200$90.xxPrxE i

2

1ii =⋅−⋅=⋅=∑

=

7


¨  Law of Large Numbers (LLN) ¤  If x is an independent and iden-cally distributed random variable (IID), the sum sn/n

converges to the expected value of x, A, as n approaches infinity

¨  Variance: Average squared error from the mean

¨  Standard Devia-on

8

xn1

ns

x...xxxs

n

1ii

n

n21

n

1iin

∑

∑

=

=

⋅=

+++=≡

[ ] [ ] [ ]( ) 222 BxExExVar ≡−=

[ ] AnsE

AnsE

n As

n

n

⋅→

→⎥⎦⎤

⎢⎣⎡

∞→

[ ] [ ] BxVarxSD ≡=


¨  Independent random variables: No condi-onal dependence

¨  If y is a lagged sequence of x, then

¨  Linear (Pearson) correla-on, ρ, is a measure of linear dependence and is commonly used for ellip(cally distributed random variables, x and y

¨  Ellip-cally distributed random variables include Gaussian, t-‐distribu-on, Cauchy, Laplace, Logis-cs, etc.

¨  Otherwise non-‐correla-on does not imply independence ¨  Checks for independence of an ellip-c random variable x start with checking

auto-‐correla-on of x, and its square, x2, or its de-‐trended square (x-‐E(x))2

9

]Pr[x]x,...,x,x|Pr[x i02i-‐1i-‐i =

[ ] [ ]( ) [ ]( )[ ]yx DSDS

yEyxExEyx,ρ⋅

−⋅−=

Pr[x]y]|Pr[x =

Game Model

¨  Let’s develop a ‘stochas-c model’ for the financial game ¨  Say that your wealth at -me i or ti is Si

¤  Integer periods, i, and discrete -me, ti ¨  You play the game between -me i-‐1 and -me i over -me Δt = ti – ti-‐1 ¨  Your gain or loss (return or change in wealth) is ΔSi for that ith game

¨  Your ini-al wealth is S0 at -me i = 0 or t0=0.0

¨  The implica-on is that the original game is played only once

10

ΔS SS i1i-‐i +=

101 S SS Δ+=

i-‐1 ti-‐1 Si-‐1

ΔSi i ti Si

Game Model

¨  You can also compute the variance, Var, and standard devia-on, SD, of the game

11

[ ] [ ] [ ]( )

[ ] $908,100 ΔSSD

$ 8,10028,900-‐ 1,00036,000

$1700.10$100)(0.90$200

ΔSEΔSEΔSVar

2

222

22

==

=+=

−⋅−+⋅=

−=

-‐100 200 Outcome

Freq

uency

Game Model

¨  Now say that you could play the financial game a number of -mes, n

¨  In each game the probabili-es and payoffs for the game remain the same, thus the wealth increments, ΔSi, are independent and iden(cally distributed (IID)

¨  The variance is also finite, its 8,100 $2, thus ΔS is IID/FV

¨  One of the most important sta-s-cal principles is that sums of IID/ FV random variables, e.g., ΔS approach normal distribu-on as n becomes large (Central Limit Theorem)

12

∑=

+=+=

+++=n

1ii00n

n210n

ΔSSΔS SS

ΔS...ΔSΔSSS

[ ][ ]

[ ][ ][ ] BnΔSDS

BnΔSVar

AnΔSE Bn ,AnNΔS

BA,NnΔS n

ΔSSSΔS...ΔSΔSSS

2

2

2

0n

n210n

⋅→

⋅→

⋅→

⋅⋅→

→∞→

+=

+++=

Central Limit Theorem 13

0

200

400

600

800

1000

1200

1400

$13,000 $14,000 $15,000 $16,000 $17,000 $18,000 $19,000 $20,000

Freq

uency

Gain Per 100 Game Sequence

I wrote a VB program that ran a 100 game sequence 10,000 -mes. I computed the average gain per game sequence and ploled a histogram of the sums. The observed mean sum was $17,002. This is a simula(on. Would you play the game a 100 -mes? Would you play it one -me?


0

200

400

600

800

1000

1200

1400

$130 $135 $140 $145 $150 $155 $160 $165 $170 $175 $180 $185 $190 $195 $200

Freq

uency

Average Gain Per Game

I modified the program to compute the average gain per game in each of 10,000 sequences and ploled a histogram of these average gains. The observed mean gain was $170.02.

Another Game: Coin Flipping

¨  Now lets play another financial game : coin flipping. ¨  Say you gain $10 on heads and pay $10 on tails.

Is each flip an independent event? With the same probabili-es? And has finite variance? Yes, its IID/FV

¨  Coin flipping is a ‘fair game’ since the expected return for each player (counter party) is zero – neither player has an expected advantage ¤  Coin flipping is characterized as a binomial model

15

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

[ ] ( ) [ ] $10$ 100ΔSSD 100$.5$10.5$10ΔSE

$ 100ΔSEΔSEΔSVar $0.5$10.5$10ΔSE

22222

222

===⋅−+⋅=

=−==⋅−+⋅=

Binomial Model for Coin Flipping 16

3 trials 40 trials

0 1 2 3 0 1 2 3 0 1 2 3$30 1 0.125

$20 1 0.250 $10 $10 1 3 0.500 0.375

$0 $0 1 2 1.000 0.500 -‐$10 -‐$10 1 3 0.500 0.375

-‐$20 1 0.250 -‐$30 1 0.125

-‐$ -‐$ -‐$ -‐$ 1 2 4 8 1.000 1.000 1.000 1.000

Flips Flips Flips

Binomial tree

Another Game: Die Rolling

¨  Using a single die, if you roll a 6 then you receive $110, while any other outcome results in you paying $10

17

[ ] ( )

[ ] ( )

[ ] [ ] [ ]( )

[ ] $44.72$ 2,000ΔSD

$ 000,2100100,2ΔSEΔSEΔSar

$ 100,2110$61$10-‐

65ΔSE

10$110$61$10-‐

65ΔSE

2

222

2222

==

=−=−=

=⋅+⋅=

=⋅+⋅=

S

V

0

200

400

600

800

1000

1200

-‐$400 $0 $400 $800 $1,200 $1,600 $2,000 $2,400 $2,800

Freq

uency

Gain Per 100 Roll Sequence

Another Game: Die Rolling 18

I modified the program and ran the 100 die rolling game sequence 10,000 -mes. I computed the sums for the 100 rolls. The mean was $999.

‘Rate Game’

¨  Now let’s consider a game defined by rates of gain or loss (rates of return) ¤  45% chance of losing 1%

¤  55% chance of gaining 1.25%

¨  The Central Limit Theorem also addresses products of IID / FV random variables. For large n, the future value factor, fn, approaches a log-‐normal distribu-on

¨  If f is lognormal, what does that imply about the probability distribu-on of r? ¤  Nothing other than its IID/FV

19

SS r1

SSS r

)r(1SS

1i-‐

ii

1i-‐

1i-‐ii

i1i-‐i

=+

−=

+⋅=

( )...rrr...rrrrrr...rrr1S )r(1....)r(1)r(1 S

)r(1f fS)r(1S S

3213231213210

n210

n

1iinn0

n

1ii0n

+⋅⋅++⋅+⋅+⋅+++++⋅=

+⋅⋅+⋅+⋅=

+=⋅=+⋅= ∏∏==

‘Rate Game’

¨  We can compute the mean, a, and variance, d2, of r

¨  We can also compute the mean, mode, median, and variance of f – but we need to know more about lognormal distribu-ons, so let’s delay for now

20

( ) rof variance arn1d

rof value mean rn1a

n

1i

2i

2

n

1ii

∑

∑

=

=

−⋅≡

⋅≡

NL~ )r(1fn

1iin ∏

=

+=

‘Rate Game’: Log Normal Distribu-on 21

I again modified the VB program for a sequence of 50 rate games. I ran the sequence 10,000 times. I computed the accumulated future value factor for each sequence and plotted this histogram. So there are 10,000 observations in the histogram. The accumulated mean future value factor for 50 games was 1.126.

Another ‘Rate Game’

¨  Now lets play the same rate game again but track the natural log of your wealth, ln(S), instead of your wealth, S

¨  Define the rate of return for natural log wealth, vi, instead of the rate of return, ri, on wealth (r is the simple rate of return)

22

( ) ( )

( ) ( )

)rln(1 SSS1ln

SSln v

SlnSlnv

vSln Sln

i

1i

1ii

1i

ii

1iii

i1i-‐i

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

−=

+=

−

−

−

−

i

i

v1ii

1i

iv

1i

ii

eSS

SSe

SSlnv

⋅=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

−

−

( ) )eln(e S SSln ==

1SS r1i-‐

ii −=

Another ‘Rate Game’ Con-nued

¨  So the equivalent rate game which tracks the natural log of wealth, ln(S), is ¤  45% chance of losing .995% of your natural log wealth = ln(1-‐1%) ¤  55% chance of gaining 1.242% of your natural log wealth = ln(1+1.25%)

¨  The Central Limit Theorem typically addresses sums of IID / FV random

variables. For large n, the sum of natural log rates, sn, approaches a normal distribu-on

23

( ) ( )

( ) ( )

N~vs

vSln Sln

v...vvSln Sln

n

1iin

n

1ii0n

n210n

∑

∑

=

=

=

+=

++++=

Another ‘Rate Game’ Con-nued

¨  The mean, u, and variance, s2, of v can be calculated as before

¨  So v is normally distributed ¨  The normal distribu-on has many nice quali-es including

¤  Dependence is defined by linear correla-on ¤  The parameters that define the PDF are also the sta-s-cs – mean and variance ¤  The sta-s-cs are scalable

24

[ ]2su,N~v

( ) vof variance uvn1s

vof value mean vn1u

n

1i

2i

2

n

1ii

∑

∑

=

=

−⋅≡

⋅≡

[ ]2su,2222N~ ⋅⋅[ ]2su,N~v

If u is the daily mean and s2 is the daily variance of natural log return rate v

Then the monthly rate of return is also normal with mean 22·∙u and variance 22·∙s2

This is not true for a lognormal distributed random variable


( ) ( )

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

)rln(1Sln Sln

n210

n

1ii0n

+++++++=

++= ∑=

The sum of a large number of IID/FV random variables is approximately normally distributed

Sums of v and ln(1+r) -‐ natural log rates of return -‐ approach normal distribu-on

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

)r(1S S

n210

n

1ii0n

+⋅⋅+⋅+⋅=

+⋅= ∏=

The product of a large number of IID/FV random variables is approximately lognormally distributed

Products of (1+r) and ev -‐ future value factors – approach lognormal distribu-on

( ) [ ]

( ) [ ]2i

2n

1ii

s u,N~r1ln

sn u,nN~r1ln

+

⋅⋅+∑=

[ ]2n

1ii sn u,nNL~)r(1 ⋅⋅+∏

=

The same parameters, u and s2,define the lognormal pdf but are not the mean and variance of the lognormal distribu-on

0

200

400

600

800

1000

1200

1400

1600

1/3/1950 11/7/1956 9/12/1963 7/17/1970 5/21/1977 3/25/1984 1/28/1991 12/2/1997 10/6/2004

SPX Price From 1950 to 2011 26

SPX price from 1950 to 2011 15,472 days

Latest Chart

SPX Daily Ln Return Rates 27

15,471 daily natural log return rates from 1950 to 2011

SPX Daily Ln Rate Histogram 28

15,471 daily natural log return rates from 1950 to 2011 Appears to be ‘somewhat normal’, but is leptokur-c and skewed

Mean: Expected value Median: 50% probable value Mode: Highest frequency

Again the CLT says that large sums of natural log daily rates approach a normal distribu-on, but we’ve made no comment on the daily rates themselves other than assume that they’re IID/FV

Stock Inves-ng

¨  The stock prices and returns can be modeled by either ¤  Natural log stock prices, ln(S), and natural log rates of return, v, or ¤  Stock prices, S, and simple rates of return, r

¤  The addi-ve or mul-plica-ve central limit theorem is u-lized. n  approaches a normal distribu-on (for m simula-ons)

n  approaches a lognormal distribu-on (for m simula-ons)

¤  The distribu-on of v and r is not yet specified, but they are assumed IID/FV

29

( ) ( )

( ) ( )

∏∑

∏∑

==

==

+=⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛

+⋅=+=

+⋅=+=

n

1ii

0

nn

1ii

0

n

n

1ii0n

n

1ii0n

i1i-‐ii1i-‐i

)r(1SS v

SSln

)r(1S S vSln Sln

)r(1SS vSln Sln

∑=

n

1iiv

∏=

+n

1ii)r(1

0 1 2 i-‐1 i n-‐1 n

Standard Price Models 30

1i-‐

1i-‐ii

i1i-‐i

SSS r

)r(1SS

−=

+⋅=

[ ]2n

1ii sn u,nNL~)r(1 ⋅⋅+∏

=

If r is IID/FV and n -‐> ∞ If v is IID/FV and n -‐> ∞

[ ]2n

1ii sn u,nN~v ⋅⋅∑

=

( ) ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛=

+=

−1i

ii

i1i-‐i

SSlnv

vSln Sln

0 1 2 i-‐1 i n-‐1 n

n·∙u and n·∙s2 are PDF parameters but not sta-s-cs n·∙u and n·∙s2 are both PDF

parameters and sta-s-cs – the mean and the variance

Standard Price Model

¨  Standard finance theory assumes v and r are IID/FV and use the addi-ve and mul-plica-ve CLTs and resul-ng normal and lognormal pdfs for sums and products

¨  In addi-on, vi and ri , are related as follows ¤  Thus r and v cannot have the same probability distribu-on

¨  So standard finance models make the simplest addi(onal assump-on: Natural log rates, v, are normally distributed which then requires that simple return rates, r, are lognormally distributed

¨  However some finance methods, i.e., single period methods, provide useful results with an assump-on that simple rates, r, are normally distributed, but this assump-on is generally inconsistent with the standard model

31

NL~)r(1f N~ v s n

1iin

n

1iin ∏∑

==

+==

ivi e )r(1 =+

[ ][ ] [ ]2s,uN

2

vi

s,uNLe)r1(

s,uN~v

e )r(1

2

i

==+

=+

Probability Distribu-ons Over Time 32

-‐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.

Three Alterna-ve Models

¨  Relax the finite variance assump-on ¤  4 parameter family of distribu-ons

generated by a ‘Levy stable’ process

¤  Variance doesn’t converge as n increases

¨  Relax the IID assump-on ¤  introduce a simple condi-onally

dependent, stochas-c vola-lity model

¤  GARCH -me series

¨  Use power law frequency distribu-on ¤  Very common distribu-on in nature ¤  Scale invariant

33

Simulated vola-lity

Essen-al Concepts

¨  Systems ¤  Determinis-c: includes chao-c ¤  Stochas-c: sta-onary (IID/FV) and non-‐sta-onary ¤  Complex: including self organized cri-cality and complex adap-ve systems

¨  Standard finance models assume ¤  Natural log rates, v, natural log prices, ln(S), natural log of future value factors,

ln(1+r) are normally distributed

¤  Simple rates, r, future value factors, ev and (1+r), and price, S, are lognormally distributed

¨  Actually the standard model doesn’t fit historical data with any sta-s-cal confidence, but the model is useful, but has limita-ons ¤  Think of Newton’s model of gravity

¨  Alterna-ve models that do fit historical data beler have not been as generally useful as the standard model in a variety of applica-ons

34

Addendum: Links & Sta-s-cs Nota-on

¨  Links ¤  Scien-fic American ¤  A Standard Model Skep-c ¤  Predic-on Markets ¤  TradeKing API

¨  Rate nota-on summary

35

Rate Periodic mean

Annual mean

Periodic standard deviation

Annual 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

Addendum: More Review Of Probability

¨  Random number generator ¤  Actually genera-ng a IID/ FV random variable ¤  Again, random doesn’t only mean IID/FV random ¤  Excel

n  rand() uniform between 0 and 1 n  Normsinv(rand()) normally distributed ~N[0,1] n  Norminv(rand(),µ, σ) normally distributed ~N[µ, σ]

¨  Importance of IID / FV character of a random variable ¤  IID -‐> Law of large numbers -‐> Expected value ¤  IID / FV -‐> Probability density func-ons for random variable

-‐> Central limit theorem -‐> Normal and lognormal distribu-ons for sums and products of random variable regardless of pdf for random variable itself -‐> Produced by a sta-onary random process

36

Addendum: Logarithms and the CLT 37

)xln(xln

)yln()xln()yxln(

n

1ii

n

1ii ∑∏

==

=⎟⎟⎠

⎞⎜⎜⎝

⎛

+=⋅

Natural logs are usually introduced as follows

The rela-onship is generalized as follows

Specializing for standard finance

( )

( ) ( ) [ ]

( ) [ ] [ ]

sn,unNL~e~r1

sn,unN~vr1ln r1ln

r1x

2sn,unNn

1ii

2n

1ii

n

1ii

n

1ii

ii

2

⋅⋅+∴

⋅⋅=+=⎟⎟⎠

⎞⎜⎜⎝

⎛+

+=

⋅⋅

=

===

∏

∑∑∏

Economy & Finance

Price Models