Example: the binary tree of Example 2.1.2: The outcomes are the 8 - - PowerPoint PPT Presentation

Martingales

• Mathematical background of random variables and stochastic

processes: – A sample space Ω with outcomes ω ∈ Ω; – A collection of subsets F; F is a σ-field; Ω ∈ F, and F is closed under complementation and countable unions; – A probability measure P defined on F; P is countably ad- ditive.

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• Axioms of probability:

– 0 ≤ P[A] ≤ 1 for all A ∈ F; – P[Ω] = 1; – P[A ∪ B] = P[A] + P[B] for any disjoint A, B ∈ F; – If An ∈ F for all n ∈ N and A1 ⊆ A2 ⊆ . . . , then P[An] ↑

P[

• n An] as n ↑ ∞.
• These imply countable additivity: if Bn ∈ F for all n ∈ N and

{Bn} are disjoint, then P[

• n Bn] =

n P[Bn].

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• Example: the binary tree of Example 2.1.2:
• The outcomes are the 8 possible paths from t = 0 to t = T.
• That is, Ω = {uuu, uud, udu, udd, duu, dud, ddu, ddd}, (or equiv-

alently {000, 001, 010, 011, 100, 101, 110, 111}).

• F is the power set 2Ω: the collection of all subsets of Ω.

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• Random variable: a real-valued function X on Ω that is F-

measurable: {ω ∈ Ω : X(ω) ≤ x} ∈ F for all x ∈ R; – that is, the probability

P[X ≤ x] = P[{ω ∈ Ω : X(ω) ≤ x}]

is defined.

• The cumulative distribution function of X is

F(x) = P[X ≤ x].

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• Example: the binary tree.
• Because Ω is discrete and F = 2Ω, every real-valued function
• n Ω is measurable.
• Define X1 and X2 on Ω by

X1(uuu) = X1(uud) = X1(udu) = X1(udd) = 120, X1(duu) = X1(dud) = X1(ddu) = X1(ddd) = 80, X2(uuu) = X2(uud) = 140, X2(udu) = X2(udd) = 100, X2(duu) = X2(dud) = 100, X2(ddu) = X2(ddd) = 60. and similarly X3.

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• Note that X1 is also measurable with respect to

F1 = {Ω, ∅, {uuu, uud, udu, udd}, {duu, dud, ddu, ddd}} = {Ω, ∅, {X1 = 120}, {X1 = 80}} ⊂ F.

• We can similarly construct F2, with F1 ⊂ F2 ⊂ F, such that

X2 is F2-measurable, as well as F-measurable.

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• A filtration is a σ-field F and a sequence of sub-fields {Fn}

with Fn ⊆ Fn+1 ⊆ · · · ⊆ F.

• A stochastic process is a sequence of random variables {Xn}n≥0
• n Ω.
• The stochastic process {Xn}n≥0 is adapted to the filtration

{Fn} if Xn is Fn-measurable for each n ≥ 0.

• Note that the definition of a stochastic process and its adap-

tation do not depend on the existence of any probability measure P.

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• Conditional expectation:

– After one step in the tree, the remaining possible paths form a subtree. – We can define risk-neutral probabilities on the subtree, and hence the conditional expectation of the claim X3:

E[X3|X1 = 120] and E[X3|X1 = 80].

• The conditional expectation is a function of X1, and therefore

a random variable, measurable with respect to F1.

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• The formal definition is less obvious:

– Suppose that X is F-measurable with E[|X|] < ∞ and G ⊆ F is a sub-σ-field. – The conditional expectation of X given G is a G-measurable random variable E[X|G] such that, for any A ∈ G,

• A E[X|G]dP =
• A XdP.
• The conditional expectation always exists and is essentially

unique.

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• The “tower property:” if Fi ⊆ Fj, then

E[E[X|Fj]|Fi] = E[X|Fi].

• That is, we can calculate conditional expectation in steps:

– First condition on detailed information (Fj); – Then take the expected value conditional on less informa- tion (Fi).

• “Taking out what is known:” if E[|X|] < ∞ and E[|XY |] < ∞,

and Y is Fn-measurable, then

E[XY |Fn] = Y E[X|Fn].

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• Martingale: Suppose that
• Ω, {Fn}n≥0, F, P
• is a filtered prob-

ability space. The stochastic process {Xn}n≥0 adapted to {Fn} is a martingale with respect to this space if, for any n ≥ 0,

E[|Xn|] < ∞

and

E[Xn+1|Fn] = Xn.

• On the binomial tree with zero interest rate, both Sn and

the price of the option are martingales.

• With positive interest, the martingales are the discounted

stock price ˜ Sn = e−rnδtSn and the discounted option price.

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• Conditional expectation of a claim: as before,
• Ω, {Fn}n≥0, F, P
• is a filtered probability space (not necessarily a binary tree).

If CN is FN-measurable with E[|CN|] < ∞, and Xn = E[CN|Fn] , then {Xn}0≤n≤N is a

• P, {Fn}0≤n≤N
• martingale.
• More generally, if

Vn = e−r(N−n)δtE[CN|Fn] is the expected value of the claim, discounted from t = N to t = n, and ˜ Vn = e−rnδtVn is Vn discounted from t = n to t = 0, then {˜ Vn}0≤n≤N is a

• P, {Fn}0≤n≤N
• martingale.

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• New martingales from old. In the binary tree model for pric-

ing a European option, we write (φn, ψn) for the amounts of stock and cash held for the time step [(n − 1)δt, nδt).

• If Vn is the value of the portfolio at t = nδt, then

Vn = φn+1Sn + ψn+1Bn and because the strategy is self-financing, φn+1Sn + ψn+1Bn = φnSn + ψnBn

• r

φn+1 ˜ Sn + ψn+1 = φn ˜ Sn + ψn.

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• So

˜ Vn+1 − ˜ Vn = φn+1

• ˜

Sn+1 − ˜ Sn

• and, summing,

˜ Vn − ˜ V0 =

n−1

• j=0

φj+1

• ˜

Sj+1 − ˜ Sj

• .
• We can verify directly that {˜

Vn} is a martingale, using the fact that φn is known at time (n − 1)δt (φn is Fn−1-measurable). Such a process {φn}n≥1 is called {Fn}n≥0-predictable or {Fn}n≥0- previsible.

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• In general, if {Xn}n≥0 is a (P, {Fn}n≥0)-martingale and {φn}n≥1

is {Fn}n≥0-previsible, then Zn = Z0 +

n−1

• j=0

φj+1

• Xj+1 − Xj
• ,

where Z0 is a constant, is also a (P, {Fn}n≥0)-martingale.

• We can view this sum as a discrete stochastic integral.

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• Key theorem: Consider a more general multi-period market

model with several assets.

• No arbitrage: in this market model, there is no arbitrage if

and only if there is a measure Q such that the discounted stock price vector is a Q-martingale.

• In that case, the market price at t = 0 of an attainable claim

CN at t = Nδt is unique and is given by EQ[ψ0CN] where ψ0 = N

1 ψ(i)

is the discount factor over N periods.

• Q is the equivalent martingale measure.

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