MATH5320: Discrete Time Finance

Lecture 3

G. Aivaliotis, c University of Leeds

February 2021

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 In this lecture

• Brief Recap – Video 1

• Risk-neutral measures – Video 2
• Fundamental Theorem of Asset Pricing – Videos 3 and 4.
• Contingent Claims, Attainability, Completeness – Videos 5
and 6.

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 General single period market model

Ω := {ω1 , ..., ωk }, P

Assumption: P(ω) > 0 for all ω ∈ Ω.

Asset prices:

S1i : Ω → R, i = 1, . . . , n.

Money market account:

B1 = B0 (1 + r).

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 Risk neutral probability measure

A measure P̃ on Ω is called a risk neutral probability measure if

1 P̃(ω) > 0 for all ω ∈ Ω,

2 EP̃ (∆Ŝ i ) = 0 for i = 1, . . . , n.

Another way of formulating 2. is:

 
1
EP̃ S1i = S0i .
1+r

 
V1 (x, φ)
If (x, φ) is a trading strategy, then EP̃ = x.
1+r

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 Fundamental Theorem of Asset Pricing

No arbitrage if and only if there exists a risk neutral mea-

sure.

M = the set of all risk neutral measures

• empty
• 1 element
• more than 1 element
Cardinality of M is crucial for pricing...

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 Definitions
We are working on a probability space Ω = (ω1 , . . . , ωk ) with k ≥ 2.

W := {X ∈ Rk : X = Ĝ(x, φ) for some strategy (x, φ)}

W⊥ := {Y ∈ Rk : hX, Y i = 0 forall X ∈ W}
A := {X ∈ Rk : X ≥ 0, X 6= 0}
k
X
+ k
P := {Z ∈ R : Zi = 1, Zi > 0}
i=1

Lemma 1.2.9. The set of all risk-neutral measures coincides with

P + W⊥ =: M.
T

In particular, no risk-neutal measures means M = ∅.

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 Separating hyperplane theorem

This result will be used in the proof of the Theorem 1.2.8. Recall
that a set A is called convex iff x, y ∈ A implies that
αx + (1 − α)y ∈ A for any 0 ≤ α ≤ 1.

Theorem Let A ⊂ Rk be convex and compact, H 3 0 a

hyperplane in Rk , and A H = ∅. Then there exists a vector
T

v ∈ Rk such that v ⊥ H and min(hv, zi : z ∈ A) > 0.

A standard notation for v ⊥ H is

v ∈ H⊥ .

For the proof see, e.g., [Elliott, Kopp, 2005, section 3.1].

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 Proof of the Theorem 1.2.8

I. Assume first that the no arbitrage condition holds. Let

A+ = {X ∈ A|hX, Pi = 1}

be a closed, bounded and convex subset of Rk . Since A+ ⊂ A it

follows from the Lemma 1.2.9 that

no arbitrage ⇒ W ∩ A+ = ∅.

By the separating hyperplane theorem, there exists a vector

Y ∈ W⊥ , s.t.
hX, Y i > 0 for all X ∈ A+ . (1)

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 Proof, ctd2

Let us now define Q by

Y (ωi )
Q(ωi ) = .
Y (ω1 ) + ... + Y (ωk )

Clearly, Q ∈ P + . Since Q is just a scalar multiple of Y , and W⊥ is

a linear vector space, and hX, Y i = 0, ∀X ∈ W (i,e, Y ∈ W⊥ ), it
follows that Q ∈ W⊥ . Therefore

Q ∈ P + ∩ W⊥ .

By virtue of the Lemma 1.2.9 we have that Q is a risk neutral

measure on Ω. Hence, the condition of no arbitrage implies the
existence of a risk neutral measure.

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 Proof, ctd3

II. Let us now show the converse. We assume there exists a risk
neutral measure Q. Let (x, φ) be an arbitrary trading strategy. As
in the proof of the Lemma 1.2.9,
k k
!
X X  
i i
EQ (Ĝ(x, φ)) = EQ φ ∆Ŝ = φi EQ ∆Ŝ i = 0.
i=1 i=1

If we assume that Ĝ(x, φ) ≥ 0 then the last equation clearly implies

that Ĝ(x, φ)(ω) = 0 for all ω ∈ Ω since otherwise EQ Ĝ(x, φ)
would be positive: recall that Q(ω) > 0 for each ω. Hence by the
Proposition 1.2.6 about no arbitrage condition in terms of Ĝ, there
cannot be any trading strategy that is an arbitrage. 

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 Contingent claim

Proposition: Let X be a contingent claim in our general single

period market model, and let (x, φ) be a hedging strategy for X,
i.e. a trading strategy which satisfies V1 (x, φ) = X. The only price
of X which complies with the no arbitrage principle is V0 (x, φ),
which by definition is equal to x.

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 Replication principle

A replicating strategy for the contingent claim X is a trading

strategy (x, φ) which satisfies V1 (x, φ)(ω) = X(ω) for every ω ∈ Ω.

Financial institution ←→ customer

replicating strategy ←→ contingent claim
(x, φ) ←→ X

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 Attainable claims

A contingent claim X is called attainable, if there exists a trading

strategy (x, φ) which replicates X, i.e. satisfies V1 (x, φ) = X.

Proposition: Let X be an attainable contingent claim and P̃ be an

arbitrary risk neutral measure. Then the price x of X at time t = 0
defined via a replicating strategy can be computed by the formula

 
1
x = EP̃ 1+r X .

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 Complete market model

A financial market model is called complete if all

contingent claims are attainable.

Assume the model with n risky assets is arbitrage free. This model is complete
if and only if the k × (n + 1) matrix A given by

S11 (ω1 ) · · · S1n (ω1 )

 
1+r
 1+r S11 (ω2 ) · · · S1n (ω2 ) 
 · · · 
A= 
 · · · 
· · ·
 
1+r S11 (ωk ) · · · S1n (ωk )
has the rank k, i.e. rank(A) = k. Recall that k = |Ω|.

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 Stochastic volatility model

There are two assets: the money market account Bt and one stock St (t = 0, 1). There is also a random
quantity called the volatility v which determines the size of stock price jumps. Our state space consists of 4 states:

Ω := {ω1 , ω2 , ω3 , ω4 }

and the volatility is given by n

h, if ω = ω1 , ω2
v(ω) :=
l, if ω = ω3 , ω4

The stock price S1 is then modeled by

n
(1 + v(ω))S0 , if ω = ω1 , ω3
S1 (ω) :=
(1 − v(ω))S0 , if ω = ω2 , ω4

where S0 = the initial stock price. The money market account: B0 = 1, B1 = 1 + r.

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 Digital call

Let us now consider a digital call in this model, i.e.

(
1, if S1 > K
X=
0 otherwise

Assume that the strike price K satisfies

(1 + l)S0 < K < (1 + h)S0 .

Try to find a replicating strategy.

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