MFE6516 Stochastic Calculus for Finance - NTU€¦ · GeneralProbabilityTheory InformationandConditioning ChangeofMeasure MFE6516StochasticCalculusforFinance WilliamC.H.Leon NanyangBusinessSchool

General Probability TheoryInformation and Conditioning

Change of Measure

MFE6516 Stochastic Calculus for Finance

William C. H. Leon

Nanyang Business School

December 11, 2017

1 / 82 William C. H. Leon MFE6516 Stochastic Calculus for Finance


Change of Measure

1 General Probability TheoryProbability SpaceRandom Variables & DistributionsExpectations

2 Information and ConditioningConditional ProbabilityConditional ExpectationMartingales

3 Change of MeasureRadon-Nikodym DerivativeExpectation



Change of Measure

Probability SpaceRandom Variables & DistributionsExpectations

The Triplet

The axiomatic approach to probability theory is based on a probability

space denoted by a triplet

(Ω,F ,P).

Set Collect of Set Measure



Change of Measure


Sample Space

A sample space Ω is the set of all possible outcomes of an experiment.

Example. Consider tossing a coin 2 times in a row.



Change of Measure


Example

Tossing a coin 2 times in a row produces

Toss Coin ��

��

Head ��

��

Tail��

��

Tail,Tail

Tail,Head

Head,Tail

Head,Head

Sample Space Ω = {TT ,TH,HT ,HH}.



Change of Measure


Event

An event A is a set of outcomes, i.e. A ⊂ Ω.




Change of Measure


Example

Tossing a coin 2 times in a row.


A1 = {TT} is the event that two tail turn up.

A2 = {TH} is the event that a tail turns up before a head turns up.

A3 = {TH,HT} is the event that exactly one head turns up.

A4 = {TT ,HH} is the event that two of a kind turn up.



Change of Measure


Sigma Algebra

A collection F of subsets of Ω is called a σ-algebra if it satisfies thefollowing conditions:

1 ∅ ∈ F ;

2 if A ∈ F then Ac ∈ F .

3 if A1,A2, ... ∈ F , then⋃∞

i=1 Ai ∈ F ;




Change of Measure


Example



F0 ={∅,Ω

}.

G ={∅, {TT ,HH}, {TH,HT},Ω

}.

F∗ ={∅, {TT}, {TH}, {HT}, {HH}, {TT ,TH}, {TT ,HT},{TT ,HH}, {TH,HT}, {TH,HH}, {HT ,HH}, {TT ,TH,HT},{TT ,TH,HH}, {TT ,HT ,HH}, {TH,HT ,HH},Ω

}.



Change of Measure


Sub-Sigma Algebra

A σ-algebra G is a sub-σ-algebra of another σ-algebra F if it is containedin F , i.e. G ⊂ F .




Change of Measure


Example



G ={∅, {TT ,HH}, {TH,HT},Ω

}⊂ F∗

where F∗ is the σ-algebra generated by all possible subsets of Ω, i.e. thepower set of Ω.



Change of Measure


Probability Measure

A probability measure P on (Ω,F) is a function P : F → [0, 1] such that

1 P (Ω) = 1;

2 if A1,A2, . . . is a collection of disjoint members of F , then

P

(∞⋃i=1

Ai

)=

∞∑i=1

P (Ai ) .



Change of Measure


Probability Space

The triplet (Ω,F ,P) is a probability space.



σ-Algebra F∗ = σ({

{TT}, {TH}, {HT}, {HH}})

.



Change of Measure


Exercise

Consider (Ω,F∗). Let p be such that 0 ≤ p ≤ 1. Define

P (A) =

⎧⎪⎨⎪⎩(1− p)2 if A = {TT},p(1− p) if A = {TH} or {HT},p2 if A = {HH}.

Observe that if P is defined for all blocks of the partition that generatesF∗, we can then extend the definition of P to all members of F∗ usingproperty (2) of the definition of a probability measure.

Verify that P is a probability measure on (Ω,F∗).



Change of Measure


Definition of Random Variable

Let (Ω,F ,P) be a probability space.

A random variable (r.v.) is a function X : Ω → R such that

X−1(B) = {X ∈ B} = {ω ∈ Ω : X (ω) ∈ B} ∈ F

for every Borel subset B of R.

Ω

X−1((−∞, x])

X

R

x



Change of Measure


Measurable Functions

In measure theory, we say that the function X is F-measurable ifX−1(B) ∈ F for every B ∈ B(R).

Note. In the above definition, only the pre-images of X must be in Fand it is not necessary that X is defined for all members of F .



Change of Measure


Distribution Functions

The distribution function of a r.v. is the function FX : R → [0, 1] given by

FX (x) = P (X ≤ x) = P ({ω ∈ Ω : X (ω) ≤ x}) .

A distribution function F has the following properties:

1 limx→−∞ F (x) = 0 and limx→∞ F (x) = 1,

2 if x < y then F (x) ≤ F (y),

3 F is right-continuous.



Change of Measure


Example

Consider tossing a coin 2 times in a row.


σ-Algebra F∗ = σ({

{TT}, {TH}, {HT}, {HH}})

.

Probability P (A) =

⎧⎪⎨⎪⎩(1− p)2 if A = {TT},p(1− p) if A = {TH} or {HT},p2 if A = {HH},

where 0 ≤ p ≤ 1.



Change of Measure


Example

Consider (Ω,F∗,P) and the r.v.

X (A) =

⎧⎪⎨⎪⎩0 if A = {TT},1 if A = {TH} or {HT},2 if A = {HH}.

The distribution function of X is

FX (x) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩0 if x < 0,

(1− p)2 if 0 ≤ x < 1,

1− p2 if 1 ≤ x < 2,

1 if x ≥ 2.



Change of Measure


Probability Density Functions

The distribution function of a continuous r.v. X can be expressed as

FX (x) =

∫ x

−∞

fX (u) du

for x ∈ R and for some integrable function fX : R → [0,∞).

The function fX is called the probability density function of X .

From the Fundamental Theorem of Calculus,

fX (x) =dFX (x)

dx.



Change of Measure


Normal Distribution

Selecting a real number.

Ω = (−∞,∞);

X ∈ R;

fX (x) =1√2πσ

e− 1

2 (x−μ

σ)2

for x ∈ R.

We denote X ∼ N (μ, σ2).

Note. Affine transformation of X ∼ N (μ, σ2) is also normally distributedand aX + b ∼ N (aμ+ b, a2σ2).



Change of Measure


Independent Events

Events A1 and A2 are independent if

P (A1 ∩ A2) = P (A1)P (A2) .

More generally, a family {Ai : i ∈ I} is independent if

P

(⋂i∈J

Ai

)=

∏i∈J

P (Ai )

for any finite J ⊂ I.



Change of Measure


Independent Random Variables

Two r.v. X and Y are independent if

P ({X ≤ x} ∩ {Y ≤ y}) = P (X ≤ x)P (Y ≤ y) .

Note. Independence implies pairwise independence; whereas, theconverse relationship is not necessarily true.



Change of Measure


Almost Surely Equal

Two r.v. X and Y are almost surely equal, denoted by X = Y a.s., if

P ({ω ∈ Ω : X (ω) �= Y (ω)}) = 0.

Equivalently,P ({ω ∈ Ω : X (ω) = Y (ω)}) = 1.



Change of Measure


Example

Consider a probability space (Ω,F ,P) where Ω = [0, 1], F is the Borelsigma-algebra and P the Lebesgue measure.

Define a r.v. X such that X (ω) = ω, and a r.v. Y such that

Y (ω) =

{ω if ω �= 1

2 ,

2 if ω = 12 .

ThenP ({ω ∈ Ω : X (ω) �= Y (ω)}) = P

({12

})= 0.



Change of Measure


Expectations

The expectation or expected value of a function g of a r.v. X is

E (g(X )) =

∫Ω

g(X (ω)) dP (ω) =

∫ ∞

−∞

g(x) dFX (x).

For discrete r.v. X ,

E (g(X )) =∑x

g(x)px =∑x

g(x)fX (x).

For continuous r.v. X ,

E (g(X )) =

∫ ∞

−∞

g(x)fX (x) dx .



Change of Measure


Results

Let X and Y be r.v., a and b be constants, and f and g be real-valuedfunctions. Then

1 E (aX + b) = aE (X ) + b,

2 E (X + Y ) = E (X ) + E (Y ),

3 E (f (X )g(Y )) = E (f (X )) E (g(Y )) if X and Y are independent.

Jensen’s Inequality. Let X be a r.v. and f be a real-values convexfunction. Then

E (f (X )) ≥ f(E (X )

).



Change of Measure


Mean & Variance

The mean μX of a r.v. X is

μX = E (X ) .

The variance σ2X of a r.v. X is

σ2X = Var (X ) = E

((X − μX )

2)= E

(X 2

)− μ2X .



Change of Measure


Exercise

Suppose X ∼ N (μ, σ2).

Show thatE (X ) = μ

andVar (X ) = σ2.



Change of Measure


Answer



Change of Measure


Covariance

The covariance Cov (X ,Y ) of two r.v. X and Y is

Cov (X ,Y ) = E ((X − μX )(Y − μY )) = E (XY )− μXμY .

Result. If two r.v. X and Y are independent, then

Cov (X ,Y ) = 0.



Change of Measure


Correlation

The correlation coefficient ρ(X ,Y ) of two r.v. X and Y is

ρ(X ,Y ) =Cov (X ,Y )√

Var (X ) Var (Y ).

Result. If two r.v. X and Y are independent, then

ρ(X ,Y ) = 0.



Change of Measure


Multivariate Random Variables

Given a probability space (Ω,F ,P).

A multivariate random variable is a function X : Ω → Rd with theproperty that {ω ∈ Ω : X (ω) ∈ A} ∈ F for each Borel set A ⊂ Rd .



Change of Measure


Joint Distribution Function

The joint distribution function of a multivariate r.v.X = (X1,X2, . . . ,Xd )

� is the function F : Rd → [0, 1] given by

FX (x) = FX1,X2,...,Xd(x1, x2, . . . , xd )

= P ({X1 ≤ x1} ∩ {X2 ≤ x2} ∩ · · · ∩ {Xd ≤ xd}) .



Change of Measure


Joint Density Function

The distribution function of a continuous multivariate r.v.X = (X1,X2, . . . ,Xd )

� can be expressed as

FX (x) = FX1,X2,...,Xd(x1, x2, . . . , xd )

=

∫ x1

−∞

∫ x2

−∞

· · ·∫ xd

−∞

fX1,X2,...,Xd(u1, u2, . . . , ud ) dud . . . du2 du1.

The joint density function of X is given by

fX1,X2,...,Xd(x1, x2, . . . , xd ) =

∂d

∂x1∂x2 · · · ∂xd FX1,X2,...,Xd(x1, x2, . . . , xd ).



Change of Measure


Marginal Density Function

For a continuous multivariate r.v. X = (X1,X2, . . . ,Xd )�, the marginal

density function of its component Xi is

fXi(xi ) =∫ ∞

−∞

· · ·∫ ∞

−∞

fX (u1, . . . , ui−1, xi , ui+1, . . . , ud ) dud . . . dui−1dui+1 . . . du1.



Change of Measure


Expectation

The expectation of a function g : Rd → R of a multivariate r.v.X = (X1,X2, . . . ,Xd )

� is

E (g(X )) =

∫ ∞

−∞

· · ·∫ ∞

−∞

g(u1, . . . , ud ) dFX (u1, . . . , ud ).

The expectation of a multivariate function g : Rd → Rk , whereg = (g1, g2, . . . , gk)

�, of a multivariate r.v. X is

E (g(X )) =(E (g1(X )) , E (g2(X )) , . . . , E (gk(X ))

)�.



Change of Measure


Mean

The mean μX of a multivariate r.v. X = (X1,X2, . . . ,Xd )� is

μX = E (X ) =

⎛⎜⎜⎜⎝μX1

μX2

...μXd

⎞⎟⎟⎟⎠where μXi

= E (Xi ) for i = 1, 2, . . . , d .



Change of Measure


Variance

The variance ΣX of a multivariate r.v. X = (X1,X2, . . . ,Xd )� is

ΣX = Var (X )

= E((X − μX )(X − μX )

�)= E

(XX�

)− μXμ�X

= E(((Xi − μXi

)(Xj − μXj)))

=(Cov (Xi ,Xj)

).



Change of Measure


Multivariate Normal Distribution

A multivariate r.v. X = (X1,X2, . . . ,Xn)� is said to be normally

distributed if

fX (x) =1√

(2π)n det(Σ)e− 1

2 (x−μ)�Σ−1(x−μ)

where μ is a n × 1 column vector and Σ is a n × n symmetric positivedefinite square matrix.

We denote X ∼ MVN (μ,Σ).



Change of Measure


Suppose X = (X1,X2, . . . ,Xn)� ∼ MVN (μ,Σ). Then

E (X ) = (E (X1) , E (X2) , . . . , E (Xn))� = μ;

Var (X ) =(Cov (Xi ,Xj)

)= Σ.

In addition,

fXi(xi ) =

∫ ∞

−∞

· · ·∫ ∞

−∞

fX (u1, . . . , ui−1, xi , ui+1, . . . , un) dun . . . du1

=1√2πσ2

Xi

e

− 12

(xi−μXi

σ2Xi

)2

.

That is Xi ∼ N (μXi, σ2

Xi).



Change of Measure


Moment Generating Function

Let X be a random variable, then the moment generating function

(m.g.f.) of X isMX (u) = E

(euX

)provided the expectation is finite for some values of u in an open intervalcontaining zero.

The distribution of a r.v. X is uniquely associated with its m.g.f. MX if itexists. Therefore, we can identify the distribution of X from its m.g.f..



Change of Measure


Exercise

Let X ∼ N (μ, σ2).

Show that

MX (u) = E(euX

)= e

μu+12σ

2u2

.



Change of Measure


Answer



Change of Measure


Results

Let X and Y be independent r.v. whose m.g.f. exist, a and b beconstants, and n be a positive integer. Then

1 MaX+b(u) = E(e(aX+b)u

)= e

buMX (au).

2 E (X n) = dnMX (u)dun

∣∣∣u=0

.

3 MX+Y (u) = MX (u)MY (u).



Change of Measure


Proof of (1):

MaX+b(u) =

∫ezufaX+b(z) dz

=

∫e(ax+b)ufX (x) dx

= E

(e(aX+b)u

)= e

bu E

(e(au)X

)= e

buMX (au).



Change of Measure


Proof of (2):

MX (u) =

∫ezufX (z) dz ,

dMX (u)

du=

∫z e

zufX (z) dz ,

dMX (u)

du

∣∣∣∣u=0

=

∫z fX (z) dz = E (X ) ;

...

dnMX (u)

dun=

∫zn e

zufX (z) dz ,

dnMX (u)

dun

∣∣∣∣u=0

=

∫zn fX (z) dz = E (X n) .



Change of Measure


Proof of (3):

MX+Y (u) =

∫ezufX+Y (z) dz

=

∫e(z−x+x)u

∫fX (x)fY (z − x) dx dz

=

∫ ∫exufX (x) e

(z−x)ufY (z − x) dx dz

=

∫ ∫exufX (x) e

yufY (y) dx dy

= MX (u)MY (u).



Change of Measure

Conditional ProbabilityConditional ExpectationMartingales

Conditional Probability of Discrete Random Variables

Let X and Y be discrete r.v. defined on a probability space (Ω,F ,P).

The conditional probability of X given Y = y is

P (X = x | Y = y) =

⎧⎪⎨⎪⎩P ({X = x} ∩ {Y = y})

P (Y = y)if P (Y = y) > 0,

0 otherwise.



Change of Measure


Example

Suppose a r.v. (X ,Y ,Z ) is equally likely to take any one of the values(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 0, 1) or (1, 1, 1).

P (X = 0 | Y = 0) =P ({X = 0} ∩ {Y = 0})

P (Y = 0)=

1/5

3/5=

1

3.



Change of Measure


Example


P (X = 1 | Y = 0) =P ({X = 1} ∩ {Y = 0})

P (Y = 0)=

2/5

3/5=

2

3.



Change of Measure


Example


P (X = 0 | Y = 1) =P ({X = 0} ∩ {Y = 1})

P (Y = 1)=

1/5

2/5=

1

2.



Change of Measure


Example


P (X = 1 | Y = 1) =P ({X = 1} ∩ {Y = 1})

P (Y = 1)=

1/5

2/5=

1

2.



Change of Measure


Conditional Probability of Continuous Random Variables

Let X and Y be continuous r.v. defined on a probability space (Ω,F ,P)with joint density function fX ,Y (x , y), and marginal density functionsfX (x) and fY (y).

The conditional density of X given Y is

fX |Y (x | y) =

⎧⎪⎨⎪⎩fX ,Y (x , y)

fY (y)if fY (y) > 0,

0 otherwise.



Change of Measure


Conditional Expectation of Discrete Random Variables


The conditional expectation of X given Y = y is

E (X | Y = y) =∑x

x P (X = x | Y = y) .



Change of Measure


Example


P (X = 0 | Y = 0) =1

3,

P (X = 1 | Y = 0) =2

3,

E(X | Y = 0) =

1∑x=0

x P (X = x | Y = 0) =2

3.



Change of Measure


Example


P (X = 0 | Y = 1) =1

2,

P (X = 1 | Y = 1) =1

2,

E(X | Y = 1) =

1∑x=0

x P (X = x | Y = 1) =1

2.



Change of Measure


Conditional Expectation & Random Variable


Define Z : Ω → R by

Z (ω) = E (X | Y = Y (ω)) .

Then Z is a r.v. w.r.t the σ-algebra F .

We shall denote the r.v. Z by E (X | Y ).



Change of Measure


Conditional Expectation & Sigma-Algebra

The coarsest σ-algebra for which E (X | Y ) is a r.v. is the σ-algebragenerated by the r.v. Y given by

σ(Y ) ={A ⊂ Ω | A = Y−1(B) for some B ∈ B(R)

}.



Change of Measure


Definition of Conditional Expectation

Let (Ω,F ,P) be a probability space, X a r.v. with E (|X |) < ∞, and Gbe a sub-σ-algebra of F .

Define the conditional expectation of X given G, denoted by E (X | G),to be any r.v. Y such that

1 Y is G-measurable, i.e. Y ∈ G,

2∫GY dP =

∫GX dP for each set G ∈ G.

If a r.v. Y has the above properties, then Y is a version of theconditional expectation of X given G, i.e. Y = E (X | G) a.s..



Change of Measure


Conditional Expectation of Continuous Random Variables

Let X and Y be continuous r.v. defined on a probability space (Ω,F ,P)with conditional density function fX |Y (x | y).

Let h be a function such that E (|h(X )|) < ∞ and set

g(y) =

∫ ∞

−∞

h(x)fX |Y (x |y) dx .

Then g(Y ) is a version of the conditional expectation of h(X ) givenσ(Y ), i.e.

g(Y ) = E(h(X ) | σ(Y )) a.s..



Change of Measure


Proof

Let G ∈ σ(Y ). Then Y (G ) = A ⊂ R and

∫G

h(X ) dP =

∫Ω

h(X (ω)

)1G (ω) dP(ω)

=

∫R

∫R

h(x) 1A(y)fX ,Y (x , y) dx dy

=

∫R

∫R

h(x) 1A(y)fX |Y (x |y)fY (y) dx dy

=

∫R

(∫R

h(x)fX |Y (x |y) dx)

1A(y)fY (y) dy



Change of Measure


Proof

Let G ∈ σ(Y ). Then Y (G ) = A ⊂ R and

∫G

h(X ) dP =

∫R

(∫R

h(x)fX |Y (x |y) dx)

1A(y)fY (y) dy

=

∫R

g(y) 1A(y)fY (y) dy

=

∫Ω

g(Y (ω)

)1G (ω) dP(ω)

=

∫G

g(Y ) dP.



Change of Measure


Some Mathematical Results

Let (Ω,F ,P) be a probability space, a and b constants, X and Y

integrable r.v., and G and H sub-σ-algebras of F .

1 E (Y ) = E (X ) if Y is a version of E (X | G).

2 E (X | G) = X if X is G-measurable.

3 E (Y X | G) = Y E (X | G) if Y is G-measurable.

4 E (X | σ(G,H)) = E (X | G) if H is independent of σ(σ(X ),G).

5 E (aX + bY | G) = aE (X | G) + bE (Y | G)(Linearity Property).



Change of Measure


6 E (X | G) ≥ E (Y | G) if X ≥ Y

(Monotonicity).

7 E (E (X | G) | H) = E (X | H) if H ⊂ G(Tower Property).

8 E (ψ(X ) | G) ≥ ψ(E (X | G)) if ψ is a convex function (Jensen’sInequality).

9 lim infn→∞ E (Xn | G) ≥ E (lim infn→∞ Xn | G) if Xn ≥ 0(Fatou’s Lemma).

10 E (Xn | G) ↑ E (X | G) if Xn ≥ 0 and Xn ↑ X

(Monotone Convergence Theorem).

11 E (Xn | G) → E (X | G) if |Xn| ≤ Y for all n and Xn → X

(Dominated Convergence Theorem).



Change of Measure


(Linearity) E (aX + bY | G) = aE (X | G) + bE (Y | G).

Proof. Since

1 aE (X | G) + bE (Y | G) ∈ G,

we need only to verify the condition

3

∫G

aE (X | G) + bE (Y | G) dP =

∫G

aX + bY dP for G ∈ G.



Change of Measure


(Linearity) E (aX + bY | G) = aE (X | G) + bE (Y | G).

Proof. Let G ∈ G.∫G

aE (X | G) + bE (Y | G) dP

= a

∫G

E (X | G) dP+ b

∫G

E (Y | G) dP

= a

∫G

X dP+ b

∫G

Y dP

=

∫G

aX + bY dP.



Change of Measure


(Monotonicity) E (X | G) ≥ E (Y | G) a.s. if X ≥ Y .

Proof. For any G ∈ G,∫G

E (X | G) dP =

∫G

X dP ≥∫G

Y dP =

∫G

E (Y | G) dP,∫G

E (Y | G)− E (X | G) dP ≤ 0.

Let ε > 0. Then

Aε ={E (Y | G)− E (X | G) > ε

}∈ G

andP (Aε) = 0.



Change of Measure


(Jensen’s Inequality) E (ψ(X ) | G) ≥ ψ(E (X | G)) a.s. if ψ : R → R is aconvex function.

Proof. Let L ={(a, b) ∈ Q2 | ax + b ≤ ψ(x) ∀ x

}. Then

ψ(x) = sup{ax + b | (a, b) ∈ L

}.

If ψ(x) ≥ ax + b,

E (ψ(X ) | G) ≥ aE (X | G) + b.

Taking the supremum over (a, b) ∈ L gives

E (ψ(X ) | G) ≥ ψ (E (X | G)) .



Change of Measure


If Y is a version of E (X | G), then E (Y ) = E (X ).

Proof. For any A ∈ G,∫Ω

1AY dP =

∫A

Y dP =

∫A

X dP =

∫Ω

1AX dP,

E ( 1AY ) = E ( 1AX ) .

In particular, for A = Ω ∈ G,

E (Y ) = E (X ) .



Change of Measure


E (X | G) = X a.s. if X is G-measurable

Proof. Since

1 X is G-measurable,

2∫GX dP =

∫GX dP for G ∈ G,

we haveX = E (X | G) .



Change of Measure


E (Y X | G) = Y E (X | G) a.s. if Y is G-measurable.

Proof. Since

1 Y E (X | G) ∈ G,

we need only to verify the condition

3

∫G

Y E (X | G) dP =

∫G

Y X dP for G ∈ G.



Change of Measure



Proof. Let G ∈ G. Suppose Y = 1B for some B ∈ G.∫G

1B E (X | G) dP =

∫G∩B

E (X | G) dP =

∫G∩B

X dP

=

∫G

1BX dP.

If Y ≥ 0, let {Yn} be a sequence of simple r.v. that converges to Y , anduse the monotone convergence theorem to show that∫

G

Y E (X | G) dP =

∫G

Y X dP.



Change of Measure



Proof. If Y ≥ 0, let {Yn} be a sequence of simple r.v. thatconverges to Y , and use the monotone convergence theorem to concludethat ∫

G

Y E (X | G) dP =

∫G

Y X dP.

To prove the result in general, split Y into positive and negative parts.



Change of Measure


(Tower Property) E (E (X | G) | H) = E (X | H) a.s. if H ⊂ G.

Proof. For any r.v. Y and any A ∈ H,

E ( 1A E (Y | H)) = E (E ( 1AY | H)) = E ( 1AY ) .

Since A ∈ H ⊂ G,

E

(1A E

(E (X | G)︸︷︷︸ | H

))= E

(1A E (X | G)︸︷︷︸

)= E ( 1AX ) .



Change of Measure


Filtration & Filtered Space

Given a probability space (Ω,F ,P), a filtration is a nested family{Ft | t ∈ N} of sub-σ-algebras of F where

F0 ⊂ F1 ⊂ · · · ⊂ F .

The probability space with the filtration {Ft | t ∈ N} is called a filtered

probability space (Ω,F , {Ft},P

).



Change of Measure


Adapted Process

A process X = {Xt} is adapted to the filtration {Ft} if Xt isFt-measurable for each t, i.e. Xt ∈ Ft .

Recall that Xt is Ft-measurable means

X−1t (B) =

{ω ∈ Ω | Xt(ω) ∈ B

} ∈ Ft ∀B ∈ B(R).



Change of Measure


Martingale

A process X = {Xt} is a martingale relative to a filtered space(Ω,F , {Ft},P

)if

1 X is adapted,

2 E (Xt | Fs) = Xs a.s. for t ≥ s.



Change of Measure


Discrete Time Martingale

For discrete time process X = {Xt}, condition 2 for a martingale can berestated as

2 E (Xt | Ft−1) = Xt−1 a.s. for t ≥ 1.

E (Xt | Ft−1) = Xt−1

E (Xt | Ft−2) = E (E (Xt | Ft−1) | Ft−2)

= E (Xt−1 | Ft−2) = Xt−2

...

E (Xt | Ft−k) = E (E (Xt | Ft−k+1) | Ft−k)

= E (Xt−k+1 | Ft−k) = Xt−k



Change of Measure


Supermartingale & Submartingale

A process X = {Xt} is a supermartingale relative to a filtered space(Ω,F , {Ft},P

)if

1 X is adapted,

2 E (Xt | Fs) ≤ Xs a.s. for t ≥ s.

And X is a submartingale if condition 2 is replaced with

2 E (Xt | Fs) ≥ Xs a.s. for t ≥ s.



Change of Measure


Exercise

Suppose it is equally likely to get a head or a tail when a coin is toss. Let

Xi =

{1 if the ith toss results in a head,

−1 if the ith toss results in a tail.

Consider the stochastic process {Mn} defined by M0 = 0 and

Mn =

n∑i=1

Xi , for n ≥ 1.

1 Show that {Mn} is a martingale, i.e. E(Mn | Mn−1) = Mn−1 forn ≥ 1.



Change of Measure


Answer



Change of Measure


Exercise

Let ζ be an integrable r.v. define on a probability space (Ω,F ,P), and{Ft} be a filtration.

DefineMt = E (ζ | Ft) for all t.

Show that the stochastic process {Mt} is a martingale.



Change of Measure

Radon-Nikodym DerivativeExpectation

Absolute Continuous & Equivalent Measure

A probability measure Q is absolutely continuous w.r.t. measure P,denoted by Q � P, if

P (A) = 0 =⇒ Q (A) = 0 ∀ A ∈ F .

The probability measures P and Q are equivalent measure, denoted byP ∼ Q, if

Q � P and P � Q.



Change of Measure


Changing Probability Measure

Let (Ω,F ,P) be a probability space, and let Z ≥ 0 a.s. with EP(Z ) = 1.For A ∈ F , define

Q (A) =

∫A

Z (ω) dP (ω) .

Then Q is a probability measure. Furthermore,

EQ(X ) = EP(ZX )

for any integrable r.v. X . If Z > 0 a.s., we also have

EP(Y ) = EQ

(Y

Z

)for any integrable r.v. Y .



Change of Measure


Radon-Nikodym Derivative

Suppose Q � P. Then there exists a r.v. Y ∈ F such that

1 Y is positive P-a.s.

2 EP(Y ) = 1.

3 Q (A) =

∫A

Y ({ω}) dP ({ω}) for A ∈ F .

The r.v. Y is the Radon-Nikodym derivative of Q w.r.t. P, denoted bydQ

dP.

Moreover, if P ∼ Q then Y is strictly positive P-a.s. and Q-a.s..



Change of Measure


Change of Measure & Expectation

Let X be an integrable r.v. defined on a probability space (Ω,F ,P), andQ ∼ P.

1 EQ(X ) = EP

(X

dQ

dP

).

2 EQ (X | Ft) =

EP

(X

dQ

dP

∣∣∣∣Ft

)EP

(dQ

dP

∣∣∣∣Ft

) .



Change of Measure


Proof

1 EQ(X ) = EP

(X

dQ

dP

).

EQ(X ) =

∫Ω

X dQ =

∫Ω

XdQ

dPdP

= EP

(X

dQ

dP

).



Change of Measure


Proof

2 EP

(X

dQ

dP

∣∣∣∣Ft

)= EQ (X | Ft) E

P

(dQ

dP

∣∣∣∣Ft

).

Let A ∈ Ft .∫A

XdQ

dPdP =

∫A

X dQ =

∫A

EQ (X | Ft) dQ

=

∫A

EQ (X | Ft)dQ

dPdP

=

∫A

EP

(EQ (X | Ft)

dQ

dP

∣∣∣∣Ft

)dP

=

∫A

EQ (X | Ft) EP

(dQ

dP

∣∣∣∣Ft

)dP.


Documents

MFE6516 Stochastic Calculus for Finance - NTU€¦ · GeneralProbabilityTheory InformationandConditioning ChangeofMeasure MFE6516StochasticCalculusforFinance WilliamC.H.Leon NanyangBusinessSchool