Constructing Brownian Motions and Radon-Nikodym Derivativessdunbar1/Mathematical...Binomial Trees...

ConstructingBrownian

Motions andRadon-Nikodym

Derivatives

Steven R.Dunbar

StandardBrownianMotion

Binomial Trees

Using theRadon-Nikodymderivative

Constructing Brownian Motions andRadon-Nikodym Derivatives

Steven R. Dunbar

February 5, 2016

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Derivatives

Steven R.Dunbar


Binomial Trees


Outline

1 Standard Brownian Motion

2 Binomial Trees

3 Using the Radon-Nikodym derivative

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Derivatives

Steven R.Dunbar


Binomial Trees


Comment

I tend to use Brownian Motion and Wiener Processinterchangeably, but

I like to use Brownian Motion for a physicalmanifestation (stock prices, motion of a particle in afluid)I like to use Wiener Process for the mathematicalmodel.

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Derivatives

Steven R.Dunbar


Binomial Trees


Definition

Wiener ProcessThe Standard Wiener Process is a stochastic processW (t), for t ≥ 0, with the following properties:

1 Increments W (t)−W (s) are normally distributedW (t)−W (s) ∼ N(0, t− s).

2 For t1 < t2 ≤ t3 < t4, increments W (t4)−W (t3)and W (t2)−W (t1) are independent randomvariables.

3 W (0) = 0.4 W (t) is continuous for all t (with probability 1).

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Derivatives

Steven R.Dunbar


Binomial Trees


Random Walk

Let

Yi =

+1 with probability 1/2−1 with probability 1/2

be a sequence of

independent, identically distributed Bernoulli randomvariables. Let Y0 = 0 for convenience and let

Tn =n∑

i=0

Yi

Note that Var [Yi] = 1, which we will need to use in amoment.

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Derivatives

Steven R.Dunbar


Binomial Trees


Random Walk Graph

n 0 1 2 3 4 5 6 7 8 9 10Yn 0 1 1 1 -1 -1 -1 1 -1 -1 -1Tn 0 1 2 3 2 1 0 1 -1 -2 -3

0 2 4 6 8 10-2

-1

0

1

2

3

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Derivatives

Steven R.Dunbar


Binomial Trees


Random Walk as continuous function

Sketch the random fortune Tn versus time using linearinterpolation between the points (n− 1, Tn−1) and(n, Tn).

The interpolation defines a function W (t) defined on[0,∞) with W (n) = Tn.

The notation W (t) reminds us of the piecewise linearnature of the fu

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Steven R.Dunbar


Binomial Trees


Random Walk Continuous FunctionExample

0 2 4 6 8 10-2

-1

0

1

2

3WcaretN

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Derivatives

Steven R.Dunbar


Binomial Trees


Scaling Random Walk

Compress time, rescale the space in a connected way. LetN be a large integer, and consider the rescaled function

WN(t) =

(1√N

)W (Nt).

This has the effect of taking a step of size ±1/√N in

1/N time unit. For example,

WN(1/N) =

(1√N

)W (N · 1/N) =

T1√N

=Y1√N.

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Derivatives

Steven R.Dunbar


Binomial Trees


Distribution of the Scaled Walk

Now consider

WN(1) =W (N · 1)√

N=W (N)√

N=

TN√N.

According to the Central Limit Theorem, this quantity isapproximately normally distributed, with mean zero, andvariance 1. More generally,

WN(t) =W (Nt)√

N=√tW (Nt)√

Nt

If Nt is an integer, WN(t) is normally distributed withmean 0 and variance t. Furthermore, WN(0) = 0 andWN(t) is a continuous function.10 / 28



Derivatives

Steven R.Dunbar


Binomial Trees


Limit Theorem

Theorem (essentially Donsker, 1951)The limit of the rescaled random walk definingapproximate Brownian Motion is Brownian Motion in thefollowing sense:

P[WN(t1) < x1, WN(t2) < x2, . . . WN(tn) < xn

]→

P [W (t1) < x1,W (t2) < x2, . . .W (tn) < xn]

as N →∞ where t1 < t2 < · · · < tn.

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Derivatives

Steven R.Dunbar


Binomial Trees


Example

Octave Source (compressed for space)p = 0.5;global N = 400; global T = 1; global SS = zeros(N+1, 1);S(2:N+1) = cumsum( 2 * (rand(N,1)<=p) - 1);function retval = WcaretN(x)

global N; global T; global S;Delta = T/N;prior = floor(x/Delta) + 1; # add 1 since arrays are 1basedsubsequent = ceil(x/Delta) + 1;retval = sqrt(Delta)*(S(prior) + ((x/Delta+1) - prior).*(S(subsequent)-S(prior)));

endfunctionfplot(@WcaretN, [0,T])

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Derivatives

Steven R.Dunbar


Binomial Trees


Example

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Derivatives

Steven R.Dunbar


Binomial Trees


The goal

Build more intuition about Brownian Motions by lookingat probability on several discrete binomial trees.

The goal is to be very strongly intuitive and motivationalin contrast to rigorous, but still to do honest examples.

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Derivatives

Steven R.Dunbar


Binomial Trees


A standard binomial tree

Note that the expected value at time n is 0.

H

1/2

T

1/2

HH

1/2

HT

1/2

TH1/2

TT

1/2

HHH

1/2

HHT

1/2

HTH1/2

HTT

1/2

THH1/2

THT

1/2

TTH1/2

TTT

1/2

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Derivatives

Steven R.Dunbar


Binomial Trees


A skewed binomial tree

The expected value at n is 13n, this process is “drifting”

upward.

H

2/3

T

1/3

HH

2/3

HT

1/3

TH2/3

TT

1/3

HHH

2/3

HHT

1/3

HTH2/3

HTT

1/3

THH2/3

THT

1/3

TTH2/3

TTT

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Derivatives

Steven R.Dunbar


Binomial Trees


A general binomial tree

Instead of branch probabilities, give the path probabilitymeasure instead. Each can be recovered from the other.

πH

p11

πT

p10

πHH

p21

πHT

p20

πTHp21

πTT

p20

πHHH

p31

πHHT

p30

πHTHp31

πHTT

p30

πTHHp31

πTHT

p30

πTTHp31

πTTT

p3017 / 28



Derivatives

Steven R.Dunbar


Binomial Trees


Another general binomial tree

Another probability measure. What is the relation to theprevious measure?

0

φH

φT

φHH

φHT

φTH

φTT

φHHH

φHHT

φHTH

φHTT

φTHH

φTHT

φTTH

φTTT

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Derivatives

Steven R.Dunbar


Binomial Trees


The Radon-Nikodym derivative on the tree

The Radon-Nikodym derivative measures the likelihoodratio.

φH⁄πH

φT ⁄πT

φHH⁄πHH

φHT⁄πHT

φTH⁄πTH

φTT ⁄πTT

φHHH⁄πHHH

φHHT ⁄πHHT

φHTH ⁄πHTH

φHTT ⁄πHTT

φTHH ⁄πTHH

φTHT ⁄πTHT

φTTH⁄πTTH

φTTT ⁄πTTT

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Steven R.Dunbar


Binomial Trees


Recover φ from π

Given P on paths, and the R-N derivative dQdP ,

Q =dQdP

P

Note also: the R-N derivative dQdP is defined on paths, so

it too is a random variable on the space Ω.

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Steven R.Dunbar


Binomial Trees


Equivalent Measures

Equivalent measuresTwo probability measures on the space Ω with σ-algebraF are equivalent if for any set B ∈ F , P [B] > 0 if andonly if Q [B] > 0. This is, the probability measures areequivalent if P Q and Q P.

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Derivatives

Steven R.Dunbar


Binomial Trees


Expectation

The Radon-Nikodym derivative is defined on paths, andis F -measurable, so it is also a random variable.

Even more, the Radon-Nikodym derivative is Ft-adapted.

Let ζt = dQdP folowing paths to time t.

ζt = EP

[dQdP|Ft

]for every t.

The expectation, knowing the information up to time tmeasured by P, represents the amount of change ofmeasure so far up to time t along the current path as ζt.

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Derivatives

Steven R.Dunbar


Binomial Trees


Using the R-N Derivative

If we want to know EQ [F (Xt)], it would be EP [ζtF (Xt)].

If we want to know EQ [F (Xt)|Fs], then we would needthe amount of change from time s to t which is justζt/ζs, which is change up to time t with the change upto time s removed. In other words

EQ [F (Xt)|Fs] = ζ−1s EP [ζtF (Xt)|Fs] .

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Steven R.Dunbar


Binomial Trees


Passing to the Limit

Suppose P and Q are equivalent measures on a tree ofpaths. Given scaled path points (t1, x1), . . . (tn, xn) withtn = 1 then dQ

dP up to time T = 1 is the limit of thelikelihood ratios:

dQdP

= limn→∞

fnQ(x1, . . . , xn)

fnP (x1, . . . , xn)

Then for Brownian Motions let ζt = dQdP . Then

ζt = EP

[dQdP|Ft

]and

EQ [F (Xt)|Fs] = ζ−1s EP [ζtF (Xt)|Fs] .24 / 28



Derivatives

Steven R.Dunbar


Binomial Trees


Simple Example

The point of the example is to apply the defintions andnotations to the pair of binomial trees with P defined byP [H] = 1

2and P [T ] = 1

2and Q [H] = 2

3and Q [T ] = 1

3.

This has minimal mathematical content, but is a goodillustration of defintions and notation.

ζt =

(4

3

) t+Xt2

·(

2

3

) t−Xt2

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Derivatives

Steven R.Dunbar


Binomial Trees


Simple Example Continued

EQ[1[X3≥1]|F2

]Node Expectation X2 ProbHH 2

3· 1 + 1

3· 1 2 1

HT 23· 1 + 1

3· 0 1 2

3

TH 23· 1 + 1

3· 0 1

TT 23· 0 + 1

3· 0 0 0

Conditional probabilities the “old-fashioned way”

EQ[1[X3≥1]|F2

]X=0

= Q [X3 = 1 |HT,HT ]

= Q [HHH |HT, TH] =Q [HHH]

Q [HT, TH]

=8/27

4/9=

2

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Derivatives

Steven R.Dunbar


Binomial Trees


Simple Example Continued

EP[1[X3≥1]ζ3|F2

]Node Expectation X2 ProbHH 1

2· 1 ·

(43

)3+ 1

2· 1 ·

(43

)2 (23

)2(43

)2HT 1

2· 1 ·

(43

)2 (23

)+ 1

2· 0 ·

(43

) (23

)2 1 12

(43

)2 23

TH 12· 1 ·

(43

)2 (23

)+ 1

2· 0 ·

(43

) (23

)2 1 12

(43

)2 23

TT 12· 0 ·

(43

) (23

)2+ 1

2· 0 ·

(23

)3 0 0

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Derivatives

Steven R.Dunbar


Binomial Trees


Simple Radon-Nikodym DerivativeExample Summarized

EQ [F (Xt)|Fs] = ζ−1s EP [ζtF (Xt)|Fs] .

X2 EQ[1[X3≥1]|F2

]EP[1[X3≥1]ζ3|F2

]ζ2 Prob

2 1(43

)2 (43

)2 11 2

3

(12

) (43

)2 23

(43

) (23

)23

0 0 0(23

)2 0

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Constructing Brownian Motions and Radon-Nikodym Derivativessdunbar1/Mathematical...Binomial Trees...

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