Conditional Expectation on an Event
Definition
Introduction to Mathematical Finance:
Let X be a r.v. on (Ω, F, P), where Ω is finite, F = 2Ω , P(ω) > 0
for all ω ∈ Ω and A ∈ F, with 0 < P(A) ≤ 1, we let
Part I: Discrete-Time Models
E [X |A] =
African Institute for Mathematical Sciences (AIMS)
X
X (ω) P({ω}|A)
ω∈Ω
where
and
P({ω} ∩ A)
=
P({ω}|A) =
P(A)
Stellenbosch University
(
P({ω})/P(A)
0
if ω ∈ A
if ω ∈
/A
Remark
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If A = Ω, it follows
E [X |Ω] = E [X ]
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Conditional Expectation on an Event
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Conditional Expectation on an Event
Remark
E [X |A] =
X
Remark
X (ω) P({ω}|A)
In particular, the conditional probability of event B given A is
ω∈Ω
=
X
X (ω)
ω∈A
P({ω})
P(A)
P(B|A) = E [1B |A] =
1
X (ω) dP(ω)
P(A) A
Z
Z
1
1
X dP =
X 1A dP
P(A) A
P(A)
E [X 1A ]
P(A)
Z
=
=
=
=
=
1
1B (ω) dP(ω)
P(A) A
Z
1
dP
P(A) A∩B
P(A ∩ B)
P(A)
Z
Remember: A sum is a discrete integral
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Conditional Expectation on a Filtration
Conditional Expectation on a Filtration
A slightly more subtle and general concept is that of
expectations which are conditional on the information
contained in a partition P (and it related algebra/filtration) A
Assume first that the partition is P =
A = {∅, Ω, A, Ac })
{A, Ac }
Remark
Therefore, for A = {∅, Ω, A, Ac } with 0 < P(A) < 1, we have
(hence
E [X |A](ω) := E [X |A] 1A (ω) + E [X |Ac ] 1Ac (ω)
Definition
The expectation conditional on P or the conditional
expectation given A = {∅, Ω, A, Ac }) is the random variable
∀ω ∈ Ω
in other way, we have
E [X |A](ω) =
E [X |A] : Ω → R

 1
R
P(A) A X dP =: E [X |A],
R
c
 1c
P(A ) Ac X dP =: E [X |A ],
ω∈A
ω ∈ Ac
given by
E [X |A](ω) := E [X |A] 1A (ω) + E [X |Ac ] 1Ac (ω)
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∀ω ∈ Ω
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Conditional Expectation on a Filtration
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Example
If P = {Ω}, hence A(P) = {∅, Ω} = F0 , we have
Definition
The expectation conditional on P = {A1 , A2 , · · · , Am } or the
conditional expectation given A(P)) is the random variable
E [X |{∅, Ω}](ω) = E [X |F0 ](ω) = E [X ]
In particular, in our one-period Binomial model, we have seen,
under the risk-neutral probability measure Q
E [X |A(P)] : Ω → R
given by
C0 =
E [X |A(P)](ω) := E [X |A1 ] 1A1 (ω)+· · ·+E [X |Am ] 1Am (ω)
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Conditional Expectation on a Filtration
More generally, if a partition P is given by {A1 , A2 , · · · , Am } (and
its associate algebra/filtration A(P)), we define
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∀ω ∈ Ω
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1
1
E Q C1 =
E Q C1 |F0
1+r
1+r
for any alternative C
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Characterization of Conditional Expectation
Characterization of Conditional Expectation
For any i we have
Z 1Ai = E [X |Ai ] 1Ai
Proposition
Thus we obtain
Let a r.v. X be given. For any partition P = {A1 , A2 , · · · , Am }
(equivalently, associate filtration A(P)), its conditional expectation
E [X |A(P)] is A(P)-measurable (equivalently, P-measurable.)
Moreover, it is the only A(P)-measurable r.v. Y satisfying
E [Y 1Ai ] = E [X 1Ai ] for all
E [Z 1Ai ] = E [X |Ai ] E [1Ai ] =
Let Y be another A(P)-measurable r.v. satisfying
E [Y 1Ai ] = E [X 1Ai ]
i = 1, 2, · · · , m
for all
for all
i = 1, 2, · · · , m
E [Y 1Ai ] = E [X 1Ai ]
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i = 1, 2, · · · , m
The A(P)-measurable of Z and Y imply that they are constant on
atoms of P. From this we infer
E [Z 1Ai ]
E [Y 1Ai ]
Z (Ai ) =
=
= Y (Ai )
P(Ai )
P(Ai )
holds for all i. It follows Z = Y , i.e. it is the only
A(P)-measurable r.v. Y satisfying
Proof. Let Z := E [X |A(P)]. As a linear combination of the
A(P)-measurable r.vs 1A1 , · · · , 1Am it is clear that Z is also
A(P)-measurable. We now show that Z satisfies
E [Y 1Ai ] = E [X 1Ai ]
1
E [X 1Ai ] P(Ai ) = E [X 1Ai ]
P(Ai )
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Characterization of Conditional Expectation
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for all
i = 1, 2, · · · , m
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Characterization of Conditional Expectation
Therefore, we can state equivalently
Remark
Definition
We see that
Let X be a given r.v. . For any partition P = {A1 , A2 , · · · , Am }
(equivalently, associate filtration A(P)), its conditional expectation
E [X |A(P)] is the only A(P)-measurable r.v. Y satisfying
E [Y 1Ai ] = E [X 1Ai ] ⇐⇒ E [Y |Ai ] = E [X |Ai ]
for all
i = 1, 2, · · · , m
E [Y 1A ] = E [X 1A ]
Remark
⇐⇒
Since A(P) consists of union of atoms of P (plus ∅) we see that
E [Y 1Ai ] = E [X 1Ai ] for all
Z
A
E [X |A(P)] dP =
for all
X dP for all
A ∈ A(P)
A
i = 1, 2, · · · , m
Remark
is equivalent to
E [X |A(P)] is the unique r.v. that satisfies
E [Y 1A ] = E [X 1A ]
for all
(1) Measurability: E [X |A(P)] is A(P)-measurable,
A ∈ A(P).
(2) Partial averaging:
R
R
A E [X |A(P)] dP = A X dP
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A ∈ A(P),
Z
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for all A ∈ A(P)
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Characterization of Conditional Expectation
Characterization of Conditional Expectation
Remark
We have
In particular
Z
Definition
C
Z
P(A|C) dP =
C
1A dP = P(A ∩ C ) for all
C ∈ C(P)
Let an algebra C ⊂ F, and an event A ∈ F, we define
Example
P(A|C) := E [1A |C]
For C = {∅, Ω, B, B c } with 0 < P(B) < 1, we have
the conditional probability of A given C.

 P(A∩B) =: P(A|B),
P(B)
P(A|C)(ω) = P(A∩B
c)

=: P(A|B c ),
c
P(B )
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Consequence
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ω ∈ Bc
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Important Properties of the Conditional Expectation
Let a partition P = {B1 , B2 , · · · , Bm } of Ω be given and its
associated filtration A(P).
Corollary
Let a r.v. X and a partition P = {A1 , A2 , · · · , Am } be given.
Then
E [E [X |A(P)]] = E [X ] .
Proposition (Linearity of conditional expectation)
The operation of taking conditional expectations is linear, i.e.
E [X + λY |A(P)] = E [X |A(P)] + λ E [Y |A(P)]
Proof.
holds for r.v.s X and Y and any scalar λ ∈ R .
E [E [X |A(P)]] = E [E [X |A1 ] 1A1 + · · · + E [X |Am ] 1Am ]
= E [X |A1 ] E [1A1 ] + · · · + E [X |Am ] E [1Am ]
Proposition
= P(A1 ) E [X |A1 ] + · · · + P(Am ) E [X |Am ]
If X is P-measurable (equivalently A(P)-measurable), then
= E [X 1A1 + · · · + X 1Am ]
E [X |A(P)] = X
= E [X ] .
on arrival of P we know the exact value of X and do not
need to take any expectations.
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Important Properties of the Conditional Expectation
Important Properties of the Conditional Expectation
Proposition (Taking out what is known)
Proposition (Tower Property)
If Z is any P-measurable (equivalently A(P)-measurable), we have
If Q is a partition finer than P, i.e. A(P) ⊆ A(Q), then
E [Z X |A(P)] = Z E [X |A(P)],
E [E [X |A(Q)]|A(P)] = E [X |A(P)]
i.e. we may “take” out what is “known”.
this is sometimes called the tower property.
Proposition
Remark
Taking conditional expectations preserves positivity, i.e. if X
is a positive r.v., than E [X |A(P)] is also positive. More precisely,
if X ≥ 0
For A(P) ⊆ A(Q), We also trivially have
E [E [X |A(P)]|A(Q)] = E [X |A(P)]
then E [X |A(P)] ≥ 0
and as P(ω) > 0 ∀ω ∈ Ω,
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X > 0 =⇒ E [X |A(P)] > 0
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Important Properties of the Conditional Expectation
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Martingale
Proposition
Let (Ω, F, P, (Ft )t=0,1,··· ,T ) a a filtered (finite) probability space
Let a and b be real numbers, X and Y be r.v. on Ω, and F ⊆ G
be algebras/filtration on Ω. The following relations hold:
Definition (Martingale)
A stochastic process X = (Xt )t=0,1,··· ,T is a P-martingale w.r.t to
the filtration F = {(Ft )t=0,1,··· ,T } if
1. E [X |P(Ω)] = X and E [X |{Ω, ∅}] = E[X ]
2. E [a X + b Y |F] = a E [X |F] + b E [Y |F] and E [1Ω |F] = 1
E [Xt |Fs ] = Xs
3.
if X ≥ 0 then E [X |F] ≥ 0
and as P(ω) > 0
∀ω ∈ Ω,
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for all
0≤s≤t≤T.
Remark
X > 0 =⇒ E [X |F] > 0
I
Since for s = t, we have E [Xt |Ft ] = Xt , (Xt )t=0,1,··· ,T is
adapted.
4. Y ∈ F =⇒ E [X Y |F] = Y E [X |F]
I
for 0 ≤ s < t ≤ T , Xs is the best prediction of Xt given Fs
5. F ⊆ G =⇒ E [E [X |G]|F] = E [X |F]
I
The sum of two martingales is a martingale (useful property).
6. If F is independent of σ(X ) then E [X |F] = E [X ]
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Martingale
Martingale: Fair Game
Remark (Fair Game)
Definition (Martingale)
The notion of a martingale is, intuitively, a probabilistic model of
a fair game: the r.v. Xt of the process X can be interpreted as
the fortune at time t of a gambler playing a game. In this context
a martingale can be thought of as a “fair game” (a game which is
neither in your favour nor your opponent’s), in the sense that the
gambler’s future fortune is expected to be the same as the
gambler’s current fortune, given all the past and present
information on the game.
Let (Ω, F, P, (Ft )t=0,1,··· ,T ) a a filtered (finite) probability
space. A stochastic process X = (Xt )t=0,1,··· ,T is a P-martingale
w.r.t to the filtration F = {(Ft )t=0,1,··· ,T } if
1. X is F-adapted, i.e. Xt ∈ Ft
2. for all s, t = 0, 1, · · · , T with 0 ≤ s < t ≤ T we have
E [Xt |Fs ] = Xs
E [Xt − Xs |Fs ] = 0
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Martingale: Fair Game
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for all
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0≤s≤t≤T.
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Tutorial: Conditional Expectation
Exercise
Remark
The fundamental property of a martingale process is that its
future variations are completely unpredictable given the current
information Ft . In other words, the directions of the future
movements in martingales are impossible to forecast
∀u > 0, E [Xu+t − Xt |Ft ] = E [Xu+t |Ft ] − Xt = 0
In particular, for all 0 ≤ s ≤ t ≤ T
t 7→ E [Xt ] = E [Xs ] = E [X0 ](= X0 )
Let 0 < p < 1 be the probability of going up on each period.
Compute E [S2 |F1 ]? Compute E [S1 |σ(S2 )]?
i.e. its expectation function is constant.
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Tutorial: Conditional Expectation
Tutorial: Conditional Expectation
Solution
E [S2 |F1 ] is constant on the atoms of F1 : i.e. on
Au = {uu, ud}, Ad = {du, dd} and must satisfy the partial
averaging property on these sets:
Z
Au
E [S2 |F1 ]dP =
Z
Z
S2 dP,
Au
Ad
E [S2 |F1 ]dP =
Therefore, ∀ω ∈ {uu, ud}
E [S2 |F1 ](ω) = pS0 u 2 + qudS0 = (pu + qd)uS0 = (pu + qd)S1 (ω),
Z
S2 dP
Ad
Similarly, we can show that (check this)
On Au we have
Z
Au
E [S2 |F1 ](ω) = (pu + qd)S1 (ω),
E [S2 |F1 ]dP = P(Au ) E [S2 |F1 ](ω) why?
∀ω ∈ {du, dd}
Conclusion
= p E [S2 |F1 ] ∀ω ∈ Au
E [S2 |F1 ](ω) = (pu + qd)S1 (ω),
∀ω ∈ Ω
since E [S2 |F1 ] is constant over Au . On the other hand with
q =1−p
Z
S2 dP = p2 S0 u 2 + pqudS0 .
Au
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