4: SINGLE-PERIOD MARKET MODELS
Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
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General Single-Period Market Models
The main differences between the elementary and general single
period market models are:
The investor is allowed to invest in several risky securities instead of
only one.
The sample set is bigger, that is, there are more possible states of
the world at time t = 1.
The sample space is Ω = {ω1 , ω2 , . . . , ωk } with F = 2Ω .
An investor’s personal beliefs about the future behaviour of stock
prices are represented by the probability measure P(ωi ) = pi > 0 for
i = 1, 2, . . . , k.
The savings account B equals B0 = 1 and B1 = 1 + r for some
constant r > −1.
The price of the jth stock at t = 1 is a random variable on Ω.
It is denoted by Stj for t = 0, 1 and j = 1, . . . , n.
A contingent claim X = (X(ω1 ), . . . , X(ωk )) is a random variable
on the probability space (Ω, F, P).
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Questions
1
2
3
4
5
6
7
8
9
Under which conditions a general single-period market model
M = (B, S 1 , . . . , S n ) is arbitrage-free?
How to define a risk-neutral probability measure for a model?
How to use a risk-neutral probability measure to analyse general
single-period market models?
Under which conditions a general single-period market model is
complete?
Is completeness of a market model related to the uniqueness of a
risk-neutral probability measure?
How to define and compute the arbitrage price of an attainable claim?
Can we still apply the risk-neutral valuation formula to compute the
arbitrage price of an attainable claim?
How to deal with contingent claims that are not attainable?
How to use the class of risk-neutral probability measures to value
non-attainable claims?
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Outline
We will examine the following issues:
1
Trading Strategies and Arbitrage-Free Models
2
Fundamental Theorem of Asset Pricing
3
Examples of Market Models
4
Risk-Neutral Valuation of Contingent Claims
5
Completeness of Market Models
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PART 1
TRADING STRATEGIES AND ARBITRAGE-FREE MODELS
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Trading Strategy
Definition (Trading Strategy)
A trading strategy (or a portfolio) in a general single-period market
model is defined as the vector
(x, φ1 , . . . , φn ) ∈ Rn+1
where x is the initial wealth of an investor and φj stands for the number
of shares of the jth stock purchased at time t = 0.
If an investor adopts the trading strategy (x, φ1 , . . . , φn ) at time
t = 0 then the cash value of his portfolio at time t = 1 equals
n
n
X
X
j j
V1 (x, φ , . . . , φ ) := x −
φ S0 (1 + r) +
φj S1j .
1
n
j=1
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Wealth Process of a Trading Strategy
Definition (Wealth Process)
The wealth process (or the value process) of a trading strategy
(x, φ1 , . . . , φn ) is the pair
(V0 (x, φ1 , . . . , φn ), V1 (x, φ1 , . . . , φn )).
The real number V0 (x, φ1 , . . . , φn ) is the initial endowment
V0 (x, φ1 , . . . , φn ) := x
and the real-valued random variable V1 (x, φ1 , . . . , φn ) represents the cash
value of the portfolio at time t = 1
1
n
V1 (x, φ , . . . , φ ) :=
x−
n
X
φ
j
S0j
n
X
(1 + r) +
φj S1j .
j=1
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Gains (Profits and Losses) Process
The profits or losses an investor obtains from the investment can
be calculated by subtracting V0 (·) from V1 (·). This is called the
(undiscounted) gains process.
The ‘gain’ can be negative; hence it may also represent a loss.
Definition (Gains Process)
The gains process is defined as G0 (x, φ1 , . . . , φn ) = 0 and
G1 (x, φ1 , . . . , φn ) := V1 (x, φ1 , . . . , φn ) − V0 (x, φ1 , . . . , φn )
n
n
X
X
j j
= x−
φ S0 r +
φj ∆S1j
j=1
j=1
where the random variable ∆S1j = S1j − S0j represents the nominal change
in the price of the jth stock.
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Discounted Stock Price and Value Process
To understand whether the jth stock appreciates in real terms, we
consider the discounted stock prices of the jth stock
Sj
Sb0j := S0j = 0 ,
B0
S1j
Sj
Sb1j :=
= 1.
1+r
B1
Similarly, we define the discounted wealth process as
Vb0 (x, φ1 , . . . , φn ) := x,
V1 (x, φ1 , . . . , φn )
Vb1 (x, φ1 , . . . , φn ) :=
.
B1
It is easy to see that
Vb1 (x, φ1 , . . . , φn ) =
Pn
P
j
j
x − j=1 φ S0 + nj=1 φj Sb1j
=x+
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Pn
j=1 φ
Slides 4: Single-Period Market Models
j (S
bj
1
− Sb0j ).
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Discounted Gains Process
Definition (Discounted Gains Process)
The discounted gains process for the investor is defined as
b 0 (x, φ1 , . . . , φn ) = 0
G
and
b 1 (x, φ1 , . . . , φn ) := Vb1 (x, φ1 , . . . , φn ) − Vb0 (x, φ1 , . . . , φn )
G
n
X
=
φj ∆Sb1j
j=1
where ∆Sb1j = Sb1j − Sb0j is the change in the discounted price of the jth
stock.
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Arbitrage: Definition
The concept of an arbitrage in a general single-period market model
is essentially the same as in the elementary market model. It is worth
noting that the real-world probability P can be replaced here by any
equivalent probability measure Q.
Definition (Arbitrage)
A trading strategy (x, φ1 , . . . , φn ) in a general single-period market model
is called an arbitrage opportunity if
A.1. V0 (x, φ1 , . . . , φn ) = 0,
A.2. V1 (x, φ1 , . . . , φn )(ωi ) ≥ 0 for i = 1, 2, . . . , k,
A.3. EP V1 (x, φ1 , . . . , φn ) > 0, that is,
k
X
V1 (x, φ1 , . . . , φn )(ωi ) P(ωi ) > 0.
i=1
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Arbitrage: Equivalent Conditions
The following condition is equivalent to A.3.
A.30 . There exists ω ∈ Ω such that V1 (x, φ1 , . . . , φn )(ω) > 0.
The definition of arbitrage can be formulated using the discounted value
and gains processes. This is sometimes very helpful.
Proposition (4.1)
A trading strategy (x, φ1 , . . . , φn ) in a general single-period market model
is an arbitrage opportunity if and only if one of the following conditions
holds:
1
Assumptions A.1-A.3 in the definition of arbitrage hold with
Vb (x, φ1 , . . . , φn ) instead of V (x, φ1 , . . . , φn ).
2
x = 0 and A.2-A.3 in the definition of arbitrage are satisfied with
b 1 (x, φ1 , . . . , φn ) instead of V1 (x, φ1 , . . . , φn ).
G
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Proof of Proposition 4.1
Proof of Proposition 4.1: First step.
We will show that the following two statements are true:
The definition of arbitrage and condition 1 in Proposition 4.1 are
equivalent.
In Proposition 4.1, condition 1 is equivalent to condition 2.
To prove the first statement, we use the relationships between
V (x, φ1 , . . . , φn ) and Vb (x, φ1 , . . . , φn ), that is,
Vb0 (x, φ1 , . . . , φn ) = V0 (x, φ1 , . . . , φn ) = x,
1
V1 (x, φ1 , . . . , φn ),
Vb1 (x, φ1 , . . . , φn ) =
1+r
1
EP Vb1 (x, φ1 , . . . , φn ) =
EP V1 (x, φ1 , . . . , φn ) .
1+r
This shows that the first statement holds.
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Proof of Proposition 4.1
Proof of Proposition 4.1: Second step.
To prove the second statement, we recall the relation between
b 1 (x, φ1 , . . . , φn )
Vb (x, φ1 , . . . , φn ) and G
b 1 (x, φ1 , . . . , φn ) = Vb1 (x, φ1 , . . . , φn ) − Vb0 (x, φ1 , . . . , φn )
G
= Vb1 (x, φ1 , . . . , φn ) − x.
It is now clear that for x = 0 we have
b 1 (x, φ1 , . . . , φn ) = Vb1 (x, φ1 , . . . , φn ).
G
Hence the second statement is true as well.
b 1 (x, φ1 , . . . , φn ) does not depend
In fact, one can also observe that G
P
1
n
b
on x at all, since G1 (x, φ , . . . , φ ) = nj=1 φj ∆Sb1j .
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Verification of the Arbitrage-Free Property
It can be sometimes hard to check directly whether arbitrage
opportunities exist in a given market model, especially when dealing
with several risky assets or in the multi-period setup.
We have introduced the risk-neutral probability measure in the
elementary market model and we noticed that it can be used to
compute the arbitrage price of any contingent claim.
We will show that the concept of a risk-neutral probability measure is
also a convenient tool for checking whether a general single-period
market model is arbitrage-free or not.
In addition, we will argue that a risk-neutral probability measure can
also be used for the purpose of valuation of a contingent claim (either
attainable or not).
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Risk-Neutral Probability Measure
Definition (Risk-Neutral Probability Measure)
A probability measure Q on Ω is called a risk-neutral probability
measure for a general single-period market model M if:
R.1. Q (ωi ) > 0 for all ωi ∈ Ω,
R.2. EQ ∆Sb1j = 0 for j = 1, 2, . . . , n.
We denote by M the class of all risk-neutral probability measures for
the market model M.
Condition R.1 means that Q and P are equivalent probability
measures. A risk-neutral probability measure is also known as an
equivalent martingale measure.
Note that condition R.2 is equivalent to EQ Sb1j = Sb0j or, more
explicitly,
EQ S1j = (1 + r)S0j .
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Example: Stock Prices
Example (4.1)
We consider the following model featuring two stocks S 1 and S 2
on the sample space Ω = {ω1 , ω2 , ω3 }.
The interest rate r =
1
10
so that B0 = 1 and B1 = 1 +
1
10 .
We deal here with the market model M = (B, S 1 , S 2 ).
The stock prices at t = 0 are given by S01 = 2 and S02 = 3.
The stock prices at t = 1 are represented in the table:
S11
S12
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ω1
1
3
ω2
5
1
ω3
3
6
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Example: Value Process
Example (4.1 Continued)
For any trading strategy (x, φ1 , φ2 ) ∈ R3 , we have
1
1 2
1
2
+ φ1 S11 + φ2 S12 .
V1 (x, φ , φ ) = x − 2φ − 3φ
1+
10
We set φ0 := x − 2φ1 − 3φ2 . Then V1 (x, φ1 , φ2 ) equals
1
1 2
0
V1 (x, φ , φ )(ω1 ) = φ 1 +
+ φ1 + 3φ2 ,
10
1
1 2
0
V1 (x, φ , φ )(ω2 ) = φ 1 +
+ 5φ1 + φ2 ,
10
1
1 2
0
V1 (x, φ , φ )(ω3 ) = φ 1 +
+ 3φ1 + 6φ2 .
10
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Example: Gains Process
Example (4.1 Continued)
The increments ∆S1j are represented by the following table
∆S11
∆S12
ω1
-1
0
ω2
3
-2
ω3
1
3
The gains G1 (x, φ1 , φ2 ) are thus given by
1 0
φ − φ1 + 0φ2 ,
10
1 0
G1 (x, φ1 , φ2 )(ω2 ) =
φ + 3φ1 − 2φ2 ,
10
1 0
φ + φ1 + 3φ2 .
G1 (x, φ1 , φ2 )(ω3 ) =
10
G1 (x, φ1 , φ2 )(ω1 ) =
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Example: Discounted Stock Prices
Example (4.1 Continued)
Out next goal is to compute the discounted wealth process
b 1 (x, φ1 , φ2 ).
Vb (x, φ1 , φ2 ) and the discounted gains process G
To this end, we first compute the discounted stock prices.
Of course, Sbj = S j for j = 1, 2.
0
0
The following table represents the discounted stock prices Sb1j for
j = 1, 2 at time t = 1
Sb11
Sb12
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ω1
ω2
ω3
10
11
30
11
50
11
10
11
30
11
60
11
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Example: Discounted Value Process
Example (4.1 Continued)
The discounted value process Vb (x, φ1 , φ2 ) is thus given by
Vb0 (x, φ1 , φ2 ) = V0 (x, φ1 , φ2 ) = x
and
10
Vb1 (x, φ1 , φ2 )(ω1 ) = φ0 + φ1 +
11
50
1
2
0
Vb1 (x, φ , φ )(ω2 ) = φ + φ1 +
11
30
Vb1 (x, φ1 , φ2 )(ω3 ) = φ0 + φ1 +
11
30 2
φ ,
11
10 2
φ ,
11
60 2
φ ,
11
where φ0 = x − 2φ1 − 3φ2 is the amount of cash invested in B at
time 0 (as opposed to the initial wealth given by x).
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Example: Discounted Gains Process
Example (4.1 Continued)
The increments of the discounted stock prices equal
∆Sb11
∆Sb2
1
ω1
− 12
11
3
− 11
ω2
ω3
28
11
− 23
11
8
11
27
11
b 1 (x, φ1 , φ2 ) are given by
Hence the discounted gains G
b 1 (x, φ1 , φ2 )(ω1 ) = − 12 φ1 − 3 φ2 ,
G
11
11
23
28
1
2
1
b 1 (x, φ , φ )(ω2 ) = φ − φ2 ,
G
11
11
8
b 1 (x, φ1 , φ2 )(ω3 ) = φ1 + 27 φ2 .
G
11
11
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Example: Arbitrage-Free Property
Example (4.1 Continued)
b 1 (x, φ1 , φ2 ) ≥ 0 is equivalent to
The condition G
−12φ1 − 3φ2 ≥ 0
28φ1 − 23φ2 ≥ 0
8φ1 + 27φ2 ≥ 0
Can we find (φ1 , φ2 ) ∈ R2 such that all inequalities are valid and
at least one of them is strict?
It appears that the answer is negative, since the unique vector
satisfying all inequalities above is (φ1 , φ2 ) = (0, 0).
Therefore, the single-period market model M = (B, S 1 , S 2 ) is
arbitrage-free.
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Example: Risk-Neutral Probability Measure
Example (4.1 Continued)
We will now show that this market model admits a unique risk-neutral
probability measure on Ω = {ω1 , ω2 , ω3 }.
Let us denote qi = Q(ωi ) for i = 1, 2, 3. From the definition of a
risk-neutral probability measure, we obtain the following linear system
12
q1 +
11
3
− q1 −
11
−
28
8
q2 + q3 = 0
11
11
23
27
q2 + q3 = 0
11
11
q1 + q2 + q3 = 1
The unique solution equals Q = (q1 , q2 , q3 ) =
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80 , 80 , 80
.
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PART 2
FUNDAMENTAL THEOREM OF ASSET PRICING
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Fundamental Theorem of Asset Pricing (FTAP)
In Example 4.1, we have checked directly that the market model
M = (B, S 1 , S 2 ) is arbitrage-free.
In addition, we have shown that the unique risk-neutral probability
measure exists in this model.
Is there any relation between no arbitrage property of a market model
and the existence of a risk-neutral probability measure?
The following important result, known as the FTAP, gives a complete
answer to this question within the present setup.
The FTAP was first established by Harrison and Pliska (1981) and it
was later extended to continuous-time market models.
Theorem (FTAP)
A general single-period model M = (B, S 1 , . . . , S n ) is arbitrage-free if
and only if there exists a risk-neutral probability measure for M, that is,
M 6= ∅.
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Proof of ( ⇐ ) in FTAP
Proof of ( ⇐ ) in FTAP.
(⇐) We first prove the ‘if’ part.
We assume that M 6= ∅, so that a risk-neutral probability measure Q
exists.
Let (0, φ) = (0, φ1 , . . . , φn ) be any trading strategy with null initial
endowment. Then for any Q ∈ M
n
n
X
X
EQ Vb1 (0, φ) = EQ
φj ∆Sb1j =
φj EQ ∆Sb1j = 0.
| {z }
j=1
j=1
=0
If we assume that Vb1 (0, φ) ≥ 0 then the last equation implies that the
equality Vb1 (0, φ)(ω) = 0 must hold for all ω ∈ Ω.
Hence no trading strategy satisfying all conditions of an arbitrage
opportunity may exist.
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Geometric Interpretation of X and Q
The proof of the implication (⇒) in the FTAP needs some
preparation, since it is based on geometric arguments.
Any random variable on Ω can be identified with a vector in Rk ,
specifically,
X = (X(ω1 ), . . . , X(ωk ))T = (x1 , . . . , xk )T ∈ Rk .
An arbitrary probability measure Q on Ω can also be interpreted as a
vector in Rk
Q = (Q(ω1 ), . . . , Q(ωk )) = (q1 , . . . , qk ) ∈ Rk .
We note that
EQ (X) =
k
X
i=1
X(ωi )Q(ωi ) =
k
X
xi qi = hX, Qi
i=1
where h·, ·i denotes the inner product of two vectors in Rk .
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Auxiliary Subsets of Rk
We define the following classes:
n
o
W = X ∈ Rk X = Vb1 (0, φ1 , . . . , φn ) for some φ1 , . . . , φn
n
o
W⊥ = Z ∈ Rk | hX, Zi = 0 for all X ∈ W
The set W is the image of the map Vb1 (0, ·, . . . , ·) : Rn → Rk .
We note that W represents all discounted values at t = 1 of trading
strategies with null initial endowment.
The set W⊥ is the set of all vectors in Rk orthogonal to W.
We introduce the following sets of k-dimensional vectors:
n
o
A = X ∈ Rk X 6= 0, xi ≥ 0 for i = 1, . . . , k
k
X
+
k
P = Q∈R qi = 1, qi > 0
i=1
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W and W⊥ as Vector Spaces
Corollary
The sets W and W⊥ are vector (linear) subspaces of Rk .
Proof.
It suffices to observe that the map Vb1 (0, ·, . . . , ·) : Rn → Rk is linear.
In other words, for any trading strategies (0, η 1 , . . . , η n ) and
(0, κ1 , . . . , κn ) and arbitrary real numbers α, β
(0, φ1 , . . . , φn ) = α(0, η 1 , . . . , η n ) + β(0, κ1 , . . . , κn )
is also a trading strategy. Hence W is a vector subspace of Rk . In
particular, the zero vector (0, 0, . . . , 0) belongs to W.
It us easy to check that W⊥ , that is, the orthogonal complement of
W is a vector subspace as well.
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Risk-Neutral Probability Measures
Lemma (4.1)
A single-period market model M = (B, S 1 , . . . , S n ) is arbitrage free if
and only if W ∩ A = ∅.
Proof.
The proof hinges on an application of Proposition 4.1.
Lemma (4.2)
A probability measure Q is a risk-neutral probability measure for a
single-period market model M = (B, S 1 , . . . , S n ) if and only if
Q ∈ W⊥ ∩ P + .
Hence the set M of all risk-neutral probability measures for the model
M satisfies M = W⊥ ∩ P + and thus
M 6= ∅
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⇔
W⊥ ∩ P + 6= ∅.
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Proof of Lemma 4.2
Proof of (⇒) in Lemma 4.2.
(⇒) We assume that Q is a risk-neutral probability measure.
By the property R.1, it is obvious that Q belongs to P + .
Using the property R.2, we obtain for any vector X = Vb1 (0, φ) ∈ W
hX, Qi = EQ
n
X
b
V1 (0, φ) = EQ
φj ∆Sb1j
j=1
=
n
X
φj EQ ∆Sb1j = 0.
| {z }
j=1
=0
We conclude that Q belongs to W⊥ as well.
Consequently, Q ∈ W⊥ ∩ P + as was required to show.
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Proof of Lemma 4.2
Proof of (⇐) in Lemma 4.2.
(⇐) We now assume that Q is an arbitrary vector in W⊥ ∩ P + .
Since Q ∈ P + , we see that Q defines a probability measure satisfying
condition R.1.
It remains to show that Q satisfies condition R.2 as well. To this end,
for a fixed (but arbitrary) j = 1, 2, . . . , n, we consider the trading
strategy (0, φ1 , . . . , φn ) with
(φ1 , . . . , φn ) = (0, . . . , 0, 1, 0, . . . , 0) = ej .
This trading strategy only invests in the savings account and the jth
asset.
The discounted wealth of this strategy is Vb1 (0, ej ) = ∆Sbj .
1
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Proof of Lemma 4.2
Proof of (⇐) in Lemma 4.2 (Continued).
Since
Vb1 (0, ej ) ∈ W and Q ∈ W⊥
we obtain
0 = hVb1 (0, ej ), Qi = h∆Sb1j , Qi = EQ ∆Sb1j .
Since j was arbitrary, we see that Q satisfies condition R.2.
Hence Q is a risk-neutral probability measure.
From Lemmas 4.1 and 4.2, we get the following purely geometric
reformulation of the FTAP:
W∩A=∅
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⇔
W⊥ ∩ P + 6= ∅.
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Separating Hyperplane Theorem: Statement
Theorem (Separating Hyperplane Theorem)
Let B, C ⊂ Rk be nonempty, closed, convex sets such that B ∩ C = ∅.
Assume, in addition, that at least one of these sets is compact (that is,
bounded and closed). Then there exist vectors a, y ∈ Rk such that
hb − a, yi < 0
for all
b∈B
hc − a, yi > 0
for all
c ∈ C.
and
Proof of the Separating Hyperplane Theorem.
The proof can be found in any textbook of convex analysis or functional
analysis. It is sketched in the course notes.
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Separating Hyperplane Theorem: Interpretation
Let the vectors a, y ∈ Rk be as in the statement of the Separating
Hyperplane Theorem
It is clear that y ∈ Rk is never a zero vector.
We define the (k − 1)-dimensional hyperplane H ⊂ Rk by setting
n
o
H = a + x ∈ Rk | hx, yi = 0 = a + {y}⊥ .
Then we say that the hyperplane H strictly separates the convex
sets B and C.
Intuitively, the sets B and C lie on different sides of the hyperplane
H and thus they can be seen as geometrically separated by H.
Note that the compactness of at least one of the sets is a necessary
condition for the strict separation of B and C.
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Separating Hyperplane Theorem: Corollary
The following corollary is a consequence of the separating hyperplane
theorem.
It is more suitable for our purposes: it will be later applied to B = W
and C = A+ := {X ∈ A | hX, Pi = 1} ⊂ A.
Corollary (4.1)
Assume that B ⊂ Rk is a vector subspace and set C is a compact convex
set such that B ∩ C = ∅. Then there exists a vector y ∈ Rk such that
hb, yi = 0 for all
b∈B
hc, yi > 0
c ∈ C.
that is, y ∈ B ⊥ , and
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for all
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Proof of Corollary 4.1
Proof of Corollary 4.1: First step.
We note that any vector subspace of Rk is a closed and convex set.
From the separating hyperplane theorem, there exist a, y ∈ Rk such
that the inequality
hb, yi < ha, yi
is satisfied for all vectors b ∈ B.
Since B is a vector subspace, the vector λb belongs to B for any
λ ∈ R. Hence for any b ∈ B and λ ∈ R we have
hλb, yi = λhb, yi < ha, yi.
This in turn implies that hb, yi = 0 for any vector b ∈ B, meaning
that y ∈ B ⊥ . Also, we have that ha, yi > 0.
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Proof of Corollary 4.1
Proof of Corollary 4.1: Second step.
To establish the second inequality, we observe that from the separating
hyperplane theorem, we obtain
hc, yi > ha, yi
for all
c ∈ C.
Consequently, for any c ∈ C
hc, yi > ha, yi > 0.
We conclude that hc, yi > 0 for all c ∈ C.
We now are ready to establish the implication ( ⇒ ) in the FTAP:
W∩A=∅
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⇒
W⊥ ∩ P + 6= ∅.
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Proof of ( ⇒ ) in FTAP: 1
Proof of ( ⇒ ) in FTAP: First step.
We assume that the model is arbitrage-free. From Lemma 4.1, this is
equivalent to the condition W ∩ A = ∅.
Our goal is to show that the class M is non-empty.
In view of Lemma 4.2, it thus suffices to show that
W∩A=∅
⇒
W⊥ ∩ P + 6= ∅.
We define an auxiliary set A+ = {X ∈ A | hX, Pi = 1}.
Observe that A+ is a closed, bounded (hence compact) and convex
subset of Rk . Since A+ ⊂ A, it is clear that
W ∩ A = ∅ ⇒ W ∩ A+ = ∅.
Hence in the next step we may assume that W ∩ A+ = ∅.
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Proof of ( ⇒ ) in FTAP: 2
Proof of ( ⇒ ) in FTAP: Second step.
By applying Corollary 4.1 to B = W and C = A+ , we see that there
exists a vector Y ∈ W⊥ such that
hX, Y i > 0
for all
X ∈ A+ .
(1)
Our goal is to show that Y can be used to define a risk-neutral
probability Q. We need first to show that yi > 0 for every i.
For this purpose, for any fixed i = 1, 2, . . . , k, we define
Xi = (P(ωi ))−1 (0, . . . , 0, 1, 0 . . . , 0) = (P(ωi ))−1 ei
so that Xi ∈ A+ since
EP (Xi ) = hXi , Pi = 1.
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Proof of ( ⇒ ) in FTAP: 3
Proof of ( ⇒ ) in FTAP: Third step.
Let yi be the ith component of Y . It follows from (1) that
0 < hXi , Y i = (P(ωi ))−1 yi
and thus yi > 0 for all i = 1, 2, . . . , k. We set Q(ωi ) = qi where
qi :=
yi
= cyi > 0
y1 + · · · + yk
It is clear that Q is a probability measure and Q ∈ P + .
Since Y ∈ W⊥ , Q = cY for some scalar c and W⊥ is a vector space,
we have that Q ∈ W⊥ . We conclude that Q ∈ W⊥ ∩ P + so that
W⊥ ∩ P + 6= ∅.
From Lemma 4.2, Q is a risk-neutral probability and thus M 6= ∅.
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PART 3
EXAMPLES OF MARKET MODELS
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Example: Arbitrage-Free Market Model
Example (4.1 Continued)
We consider the market model M = (B, S 1 , S 2 ) introduced in
Example 4.1.
The interest rate r =
1
10
so that B0 = 1 and B1 = 1 +
1
10 .
The stock prices at t = 0 are given by S01 = 2 and S02 = 3.
We have shown that the increments of the discounted stock prices Sb1
and Sb2 equal
∆Sb11
∆Sb2
1
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ω1
− 12
11
3
− 11
ω2
ω3
28
11
− 23
11
8
11
27
11
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Example: Arbitrage-Free Market Model
Example (4.1 Continued)
The vector spaces W and W⊥ are given by

 



−12
−3 

W = α  28  + β  −23  α, β ∈ R


8
27
and
 


47 

W⊥ = γ  15  γ ∈ R .


18
We first show the model is arbitrage-free using Lemma 4.1.
It thus suffices to check that W ∩ A = ∅
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Example: Arbitrage-Free Market Model
Example (4.1 Continued)
If there exists a vector X ∈ W ∩ A then the following three
inequalities are satisfied by a vector X = (x1 , x2 , x3 ) ∈ R3
x1 = x1 (α, β) = −12α − 3β ≥ 0
x2 = x2 (α, β) = 28α − 23β ≥ 0
x3 = x3 (α, β) = 8α + 27β ≥ 0
with at least one strict inequality, where α, β ∈ R are arbitrary.
It can be shown that such a vector X ∈ R3 does not exist and thus
W ∩ A = ∅. This is left as an easy exercise.
In view of Lemma 4.1, we conclude that the market model is
arbitrage-free.
In the next step, our goal is to show that M is non-empty.
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Example: Arbitrage-Free Market Model
Example (4.1 Continued)
Lemma 4.2 tells us that M = W⊥ ∩ P + . If Q ∈ W⊥ then


47
Q = γ  15  for some γ ∈ R.
18
If Q ∈ P + then 47γ + 15γ + 18γ = 1 so that γ =
1
80
> 0.
We conclude that the unique risk-neutral probability measure Q is
given by


47
1 
15  .
Q=
80
18
The FTAP confirms that the market model is arbitrage-free.
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Example: Market Model with Arbitrage
Example (4.2)
We consider the following model featuring two stocks S 1 and S 2 on
the sample space Ω = {ω1 , ω2 , ω3 }.
The interest rate r =
1
10
so that B0 = 1 and B1 = 1 +
1
10 .
The stock prices at t = 0 are given by S01 = 1 and S02 = 2 and the
stock prices at t = 1 are represented in the table:
S11
S12
ω1
1
ω2
5
2
4
1
2
ω3
3
1
10
Does this market model admit an arbitrage opportunity?
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Example: Market Model with Arbitrage
Example (4.2 Continued)
Once again, we will analyse this problem using Lemma 4.1, Lemma
4.2 and the FTAP.
To tell whether a model is arbitrage-free it suffices to know the
increments of discounted stock prices.
The increments of discounted stock prices are represented in the
following table
∆Sb11
∆Sb12
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ω1
1
− 11
3
11
ω2
6
− 11
18
11
ω3
20
11
− 21
11
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Example: Market Model with Arbitrage
Example (4.2 Continued)
Recall that
b 1 (x, φ1 , φ2 ) = φ1 ∆Sb11 + φ2 ∆Sb12
G
Hence, by the definition of W, we have
 




−1
3 

W = α  −6  + β  18  α, β ∈ R .


20
−21
Let us take α = 3 and β = 1. Then we obtain the vector (0, 0, 39)T ,
which manifestly belongs to A.
We conclude that W ∩ A 6= ∅ and thus, by Lemma 4.1, the market
model is not arbitrage-free.
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Example: Market Model with Arbitrage
Example (4.2 Continued)
We note that

 

−6 

W⊥ = γ  1  γ ∈ R .


0
If there exists a risk-neutral probability measure Q then
Q ∈ W⊥ ∩ P + .
Since Q ∈ W⊥ , we obtain Q(ω1 ) = −6Q(ω2 ).
However, Q ∈ P + implies that Q(ω) > 0 for all ω ∈ Ω.
We conclude that W⊥ ∩ P + = ∅ and thus, by Lemma 4.2, no
risk-neutral probability measure exists, that is, M = ∅.
Hence the FTAP confirms that the model is not arbitrage-free.
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PART 4
RISK-NEUTRAL VALUATION OF CONTINGENT CLAIMS
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Contingent Claims
We now know how to check whether a given model is arbitrage-free.
Hence the following question arises:
What should be the ‘fair’ price of a European call or put option in
a general single-period market model?
In a general single-period market model, the idea of pricing European
options can be extended to any contingent claim.
Definition (Contingent Claim)
A contingent claim is a random variable X defined on Ω and
representing the payoff at the maturity date.
Derivatives nowadays are usually quite complicated and thus it makes
sense to analyse valuation and hedging of a general contingent claim,
and not only European call and put options.
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No-Arbitrage Principle
Definition (Replication and Arbitrage Price)
A trading strategy (x, φ1 , . . . , φn ) is called a replicating strategy (a
hedging strategy) for a claim X when V1 (x, φ1 , . . . , φn ) = X. Then the
initial wealth is denoted as π0 (X) and it is called the arbitrage price of X.
Proposition (No-Arbitrage Principle)
Assume that a contingent claim X can be replicated by means of a trading
strategy (x, φ1 , . . . , φn ). Then the unique price of X at 0 consistent with
no-arbitrage principle equals V0 (x, φ1 , . . . , φn ) = x.
Proof.
If the price of X is higher (lower) than x, one can short sell (buy) X
and buy (short sell) the replicating portfolio. This will yield an arbitrage
opportunity in the extended market in which X is traded at time t = 0.
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Example: Stochastic Volatility Model
In the elementary market model, a replicating strategy for any
contingent claim always exists. However, in a general single-period
market model, a replicating strategy may fail to exist for some claims.
For instance, when there are more sources of randomness than there
are stocks available for investment then replicating strategies do not
exist for some claims.
Example (4.3)
Consider a market model consisting of bond B, stock S, and a
random variable v called the volatility.
The volatility determines whether the stock price can make either
a big or a small jump.
This is a simple example of a stochastic volatility model.
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Example: Stochastic Volatility Model
Example (4.3 Continued)
The sample space is given by
Ω = {ω1 , ω2 , ω3 , ω4 }
and the volatility is defined as
h for i = 1, 4,
v(ωi ) =
l for i = 2, 3.
We furthermore assume that 0 < l < h < 1. The stock price S1 is
given by
(1 + v(ωi ))S0 for i = 1, 2,
S1 (ωi ) =
(1 − v(ωi ))S0 for i = 3, 4.
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Example: Stochastic Volatility Model
Example (4.3 Continued)
Unlike in examples we considered earlier, the amount by which the
stock price in this market model jumps is random.
It is easy to check that the model is arbitrage-free whenever
1 − h < 1 + r < 1 + h.
We claim that for some contingent claims a replicating strategy does
not exist. In that case, we say that a claim is not attainable.
To justify this claim, we consider the digital call option X with the
payoff
1 if S1 > K,
X=
0 otherwise,
where K > 0 is the strike price.
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Example: Stochastic Volatility Model
Example (4.3 Continued)
We assume that (1 + l)S0 < K < (1 + h)S0 , so that
(1 − h)S0 < (1 − l)S0 < (1 + l)S0 < K < (1 + h)S0
and thus
X(ωi ) =
1
0
for i = 1,
otherwise.
Suppose that (x, φ) is a replicating strategy for X. Equality
V1 (x, φ) = X becomes



  
1+r
(1 + h)S0
1
 1+r 
 (1 + l)S0   0 


  
(x − φS0 ) 
 1 + r  + φ  (1 − l)S0  =  0 
1+r
(1 − h)S0
0
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Example: Stochastic Volatility Model
Example (4.3 Continued)
Upon setting β = φS0 and α = (1 + r)x − φS0 r, we see that the
existence of a solution (x, φ) to this system is equivalent to the
existence of a solution (α, β) to the system
 

  
1
h
1
 1 
 l   0 


  
α
 1  + β  −l  =  0 
1
−h
0
It is easy to see that the above system of equations has no solution
and thus a digital call is not an attainable contingent claim within
the framework of the stochastic volatility model.
Intuitively, the randomness generated by the volatility cannot be
hedged, since the volatility is not a traded asset.
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Valuation of Attainable Contingent Claims
We first recall the definition of attainability of a contingent claim.
Definition (Attainable Contingent Claim)
A contingent claim X is called to be attainable if there exists a
replicating strategy for X.
Let us summarise the known properties of attainable claims:
It is clear how to price attainable contingent claims by the replicating
principle.
There might be more than one replicating strategy, but no arbitrage
principle leads the initial wealth x to be unique.
In the two-state single-period market model, one can use the
risk-neutral probability measure to price contingent claims.
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Risk-Neutral Valuation Formula
Our next objective is to extend the risk-neutral valuation formula
to any attainable contingent claim within the framework of a general
single-period market model.
Proposition (4.2)
Let X be an attainable contingent claim and let Q ∈ M be any risk-neutral
probability measure. Then the arbitrage price of X at t = 0 equals
X
π0 (X) = EQ
.
1+r
Proof of Proposition 4.2.
Recall that a trading strategy (x, φ1 , . . . , φn ) is a replicating strategy for
X whenever V1 (x, φ1 , . . . , φn ) = X
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Proof of the Risk-Neutral Valuation Formula
Proof of Proposition 4.2.
We divide both sides by 1 + r, to obtain
V1 (x, φ1 , . . . , φn )
X
=
= Vb1 (x, φ1 , . . . , φn ).
1+r
1+r
Hence
n
o
1
EQ (X) = EQ Vb1 (x, φ1 , . . . , φn )
1+r
n
o
b 1 (x, φ1 , . . . , φn )
= EQ x + G
X
n
j
j
b
= x + EQ
φ ∆S1
j=1
=x+
n
X
φj EQ ∆Sb1j = x.
(from R.2.)
j=1
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Example: Stochastic Volatility Model
Example (4.3 Continued)
Proposition 4.2 shows that risk-neutral probability measures can be
used to price attainable contingent claims.
Consider the market model introduced in Example 4.3 with the
interest rate r = 0.
Recall that in this case the model is arbitrage-free since
1 − h < 1 + r = 1 < 1 + h.
The increments of the discounted stock price Sb are represented in
the following table
∆Sb1
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ω1
hS0
ω2
lS0
ω3
−lS0
ω4
−hS0
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Example: Stochastic Volatility Model
Example (4.3 Continued)
By the definition of the linear subspace W ⊂ R4 , we have
 
h


 
l
W= γ

−l



−h




  γ ∈ R .





The orthogonal complement of W is thus the three-dimensional
subspace of R4 given by


z1



z2 
4

W⊥ = 
 z3  ∈ R



z4
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
 
z1
h
 z2  
l
 
h
 z3  ,  −l
z4
−h
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




i = 0 .





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Example: Stochastic Volatility Model
Example (4.3 Continued)
Recall that aP
vector (q1 , q2 , q3 , q4 )> belongs to P + if and only if
the equality 4i=1 qi = 1 holds and qi > 0 for i = 1, 2, 3, 4.
Since the set of risk-neutral probability measures is given by
M = W⊥ ∩ P + , we find that


q1
 q2 


 q3  ∈ M
q4
⇔
4
n
X
(q1 , q2 , q3 , q4 )> qi > 0,
qi = 1
i=1
o
and h(q1 − q4 ) + l(q2 − q3 ) = 0 .
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Example: Stochastic Volatility Model
Example (4.3 Continued)
The class M of all risk-neutral probability measures in our stochastic
volatility model is therefore given by




M= 





q1
q1 > 0, q2 > 0, q3 > 0,
 q2

q1 + q2 + q3 < 1,

q3
l(q2 − q3 ) = h(1 − 2q1 − q2 − q3 )
1 − q1 − q2 − q3




.



This set appears to be non-empty and thus we conclude that our
stochastic volatility model is arbitrage-free.
Recall that we have already shown that the digital call option is not
attainable if
(1 + l)S0 < K < (1 + h)S0 .
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Example: Stochastic Volatility Model
Example (4.3 Continued)
It is not difficult to check that for every 0 < q1 < 12 there exists a
probability measure Q ∈ M such that Q(ω1 ) = q1 .
Indeed, it suffices to take q1 ∈ (0, 12 ) and to set
q4 = q1 ,
q2 = q3 =
1
− q1 .
2
We apply the risk-neutral valuation formula to the digital call
X = (1, 0, 0, 0)> . For Q = (q1 , q2 , q3 , q4 )> ∈ M, we obtain
EQ (X) = q1 · 1 + q2 · 0 + q3 · 0 + q4 · 0 = q1 .
Since q1 is any number from (0, 12 ), we see that every value from the
open interval (0, 21 ) can be achieved.
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Extended Market Model and No-Arbitrage Principle
We no longer assume that a contingent claim X is attainable.
Definition
A price π0 (X) of a contingent claim X is said to be consistent with the
no-arbitrage principle if the extended model, which consists of B, the
original stocks S 1 , . . . , S n , as well as an additional asset S n+1 satisfying
S0n+1 = π0 (X) and S1n+1 = X, is arbitrage-free.
The interpretation of Definition 4.1 is as follows:
We assume that the model M = (B, S 1 , . . . , S n ) is arbitrage-free.
We regard the additional asset as a tradable risky asset in the
f = (B, S 1 , . . . , S n+1 ).
extended market model M
We postulate its price at time 0 should be selected in such a way that
f is still arbitrage-free.
the extended market model M
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Valuation of Non-Attainable Claims
We already know that the risk-neutral valuation formula returns the
arbitrage price for any attainable claim.
The next result shows that it also yields a price consistent with the
no-arbitrage principle when it is applied to any non-attainable claim.
The price obtained in this way is not unique, however.
Proposition (4.3)
Let X be a possibly non-attainable contingent claim and Q is an arbitrary
risk-neutral probability measure. Then π0 (X) given by
X
π0 (X) := EQ
(2)
1+r
defines a price at t = 0 for the claim X that is consistent with the
no-arbitrage principle.
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Proof of Proposition 4.3
Proof of Proposition 4.3.
Let Q ∈ M be an arbitrary risk-neutral probability for M.
We will show that Q is also a risk-neutral probability measure for the
f = (B, S 1 , . . . , S n+1 ) in which S n+1 = π0 (X) and
extended model M
0
S1n+1 = X.
For this purpose, we check that
X
− π0 (X) = 0
EQ ∆Sb1n+1 = EQ
1+r
e is indeed a risk-neutral probability in the extended
and thus Q ∈ M
market model.
f is arbitrage-free. Hence the
By the FTAP, the extended model M
price π0 (X) given by (2) complies with the no-arbitrage principle.
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PART 5
COMPLETENESS OF MARKET MODELS
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Complete and Incomplete Models
The non-uniqueness of arbitrage prices is a serious theoretical
problem, which is still not completely resolved.
We categorise market models into two classes: complete and
incomplete models.
Definition (Completeness)
A financial market model is called complete if for any contingent claim X
there exists a replicating strategy (x, φ) ∈ Rn+1 . A model is incomplete
when there exists a claim X for which a replicating strategy does not exist.
Given an arbitrage-free and complete model, the issue of pricing all
contingent claims is completely solved.
How can we tell whether a given model is complete?
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Algebraic Criterion for Market Completeness
Proposition (4.4)
Assume that a single-period market model M = (B, S 1 , . . . , S n ) defined
on the sample space Ω = {ω1 , . . . , ωk } is arbitrage-free. Then M is
complete if and only if the k × (n + 1) matrix A




A=



1+r
1+r
·
·
·
1+r
S11 (ω1 ) · · · S1n (ω1 )
S11 (ω2 ) · · · S1n (ω2 )
·
·
·
·
·
·
·
·
·
S11 (ωk ) · · · S1n (ωk )




 = (A0 , A1 , . . . , An )



has a full row rank, that is, rank (A) = k. Equivalently, M is complete
whenever the linear subspace spanned by the vectors A0 , A1 , . . . , An
coincides with the full space Rk .
M. Rutkowski (USydney)
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Proof of Proposition 4.4
Proof of Proposition 4.4.
By the linear algebra, A has a full row rank if and only if for every
X ∈ Rk the equation AZ = X has a solution Z ∈ Rn+1 .
P
If we set φ0 = x − nj=1 φj S0j then we have








1+r
1+r
·
·
·
1+r
S11 (ω1 ) · · · S1n (ω1 )
S11 (ω2 ) · · · S1n (ω2 )
·
·
·
·
·
·
·
·
·
S11 (ωk ) · · · S1n (ωk )















φ0
φ1
·
·
·
φn


 
 
 
=
 
 
 
V1 (ω1 )
V1 (ω2 )
·
·
·
V1 (ωk )








where V1 (ωi ) = V1 (x, φ)(ωi ).
This shows that computing a replicating strategy for X is equivalent
to solving the equation AZ = X.
M. Rutkowski (USydney)
Slides 4: Single-Period Market Models
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Example: Incomplete Model
Example (4.3 Continued)
Consider the stochastic volatility model from Example 4.3.
We already know that this model is incomplete, since the digital call
is not an attainable claim.
The matrix A is given by

1+r
 1+r
A=
 1+r
1+r

S11 (ω1 )
S11 (ω2 ) 

S11 (ω3 ) 
S11 (ω4 )
The rank of A is 2, and thus it is not equal to k = 4.
In view of Proposition 4.4, this confirms that this market model is
incomplete.
M. Rutkowski (USydney)
Slides 4: Single-Period Market Models
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Probabilistic Criterion for Attainability
Proposition 4.4 yields a method for determining whether a market
model is complete.
Given an incomplete model, how to recognize an attainable claim?
Recall that if a model M is arbitrage-free then the class M is
non-empty.
Proposition (4.5)
Assume that a single-period model M = (B, S 1 , . . . , S n ) is arbitrage-free.
Then a contingent claim X is attainable if and only if the expected value
X
EQ
1+r
has the same value for all risk-neutral probability measures Q ∈ M.
M. Rutkowski (USydney)
Slides 4: Single-Period Market Models
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Proof of (⇒) in Proposition 4.5: 1
Proof of Proposition 4.5.
(⇒) It is easy to deduce from Proposition 4.2 that if a contingent claim
X is attainable then the expected value
EQ (1 + r)−1 X
has the same value for all Q ∈ M. To this end, it is possible to argue by
contradiction. We leave the details as an exercise.
(⇐) (MATH3975) We prove this implication by contrapositive. Let us
thus assume that the contingent claim X is not attainable. Our goal is
b for which
to find two risk-neutral probabilities, say Q and Q,
EQ (1 + r)−1 X 6= EQb (1 + r)−1 X .
(3)
M. Rutkowski (USydney)
Slides 4: Single-Period Market Models
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Proof of (⇐) in Proposition 4.5: 2
Proof of Proposition 4.5.
Consider the matrix A introduced in Proposition 4.4.
Since the claim X is not attainable, there is no solution Z ∈ Rn+1
to the linear system
AZ = X.
We define the following subsets of Rk
B = image (A) = AZ | Z ∈ Rn+1 ⊂ Rk
and C = {X}.
Then B is a proper subspace of Rk and, obviously, the set C is
convex and compact. Moreover, B ∩ C = ∅.
M. Rutkowski (USydney)
Slides 4: Single-Period Market Models
78 / 87
Proof of (⇐) in Proposition 4.5: 3
Proof of Proposition 4.5.
In view of Corollary 4.1, there exists a non-zero vector
Y = (y1 , . . . , yk ) ∈ Rk such that
hb, Y i = 0 for all b ∈ B,
hc, Y i > 0 for all c ∈ C.
In view of the definition of B and C, this means that for every
j = 0, 1, . . . , n
hAj , Y i = 0 and hX, Y i > 0
(4)
where Aj is the jth column of the matrix A.
It is worth noting that the vector Y depends on X.
M. Rutkowski (USydney)
Slides 4: Single-Period Market Models
79 / 87
Proof of (⇐) in Proposition 4.5: 4
Proof of Proposition 4.5.
We assumed that the market model is arbitrage-free and thus, by the
FTAP, the class M is non-empty.
Let Q ∈ M be an arbitrary risk-neutral probability measure.
We may choose a real number λ > 0 to be small enough in order to
ensure that for every i = 1, 2, . . . , k
b i ) := Q(ωi ) + λ(1 + r)yi > 0.
Q(ω
(5)
b is also a risk-neutral
In the next step, our next goal is to show that Q
probability measure and it is different from Q.
In the last step, we will show that inequality (3) is valid.
M. Rutkowski (USydney)
Slides 4: Single-Period Market Models
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Proof of (⇐) in Proposition 4.5: 5
Proof of Proposition 4.5.
From the definition of A in Proposition 4.4 and the first equality in
(4) with j = 0, we obtain
k
X
λ(1 + r)yi = λhA0 , Y i = 0.
i=1
It then follows from (5) that
k
X
i=1
b i) =
Q(ω
k
X
i=1
Q(ωi ) +
k
X
λ(1 + r)yi = 1
i=1
b is a probability measure on the space Ω.
and thus Q
b satisfies condition R.1.
In view of (5), it is clear that Q
M. Rutkowski (USydney)
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Proof of (⇐) in Proposition 4.5: 6
Proof of Proposition 4.5.
b satisfies also condition R.2.
It remains to check that Q
b of the discounted stock price Sbj .
We examine the behaviour under Q
1
For every j = 1, 2, . . . , n, we have
EQb Sb1j
=
=
k
X
i=1
k
X
b i )Sbj (ωi )
Q(ω
1
Q(ωi )Sb1j (ωi ) + λ
i=1
=
EQ Sb1j
k
X
Sb1j (ωi )(1 + r)yi
i=1
+ λ hAj , Y i
| {z }
(in view of (4))
=0
= Sb0j
M. Rutkowski (USydney)
Slides 4: Single-Period Market Models
(since Q ∈ M)
82 / 87
Proof of (⇐) in Proposition 4.5: 7
Proof of Proposition 4.5.
b ∈ M, that is, Q
b is a
We conclude that EQb ∆Sb1j = 0 and thus Q
risk-neutral probability measure for the market model M.
b We have thus proven that if M
From (5), it is clear that Q 6= Q.
is arbitrage-free and incomplete, then there exists more than one
risk-neutral probability measure.
To complete the proof, it remains to show that inequality (3) is
satisfied for the claim X.
Recall that X was a fixed non-attainable contingent claim and we
b corresponding to X.
constructed a risk-neutral probability measure Q
M. Rutkowski (USydney)
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Proof of (⇐) in Proposition 4.5: 8
Proof of Proposition 4.5.
We observe that
EQb
X
1+r
=
=
k
X
i=1
k
X
b i ) X(ωi )
Q(ω
1+r
k
X
X(ωi )
Q(ωi )
+λ
yi X(ωi )
1+r
i=1
i=1
|
{z
}
>0
>
k
X
i=1
X(ωi )
Q(ωi )
= EQ
1+r
X
1+r
since the inequalities hX, Y i > 0 and λ > 0 imply that the braced
expression is strictly positive.
M. Rutkowski (USydney)
Slides 4: Single-Period Market Models
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Probabilistic Criterion for Market Completeness
Theorem (4.1)
Assume that a single-period model M = (B, S 1 , . . . , S n ) is arbitrage-free.
Then M is complete if and only if the class M consists of a single
element, that is, there exists a unique risk-neutral probability measure.
Proof of (⇐) in Theorem 4.1.
Since M is assumed to be arbitrage-free, from the FTAP it follows that
there exists at least one risk-neutral probability measure, that is, the class
M is non-empty.
(⇐) Assume first that a risk-neutral probability measure for M is unique.
Then the condition of Proposition 4.5 is trivially satisfied for any claim X.
Hence any claim X is attainable and thus the model M is complete.
M. Rutkowski (USydney)
Slides 4: Single-Period Market Models
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Proof of (⇒) in Theorem 4.1
Proof of (⇒) Theorem 4.1.
(⇒) Assume M is complete and consider any two risk-neutral probability
b from M. For a fixed, but arbitrary, i = 1, 2, . . . , k, let
measures Q and Q
the contingent claim X i be given by
1 + r if ω = ωi ,
i
X (ω) =
0
otherwise.
Since M is now assumed to be complete, the contingent claim X i is
attainable. From Proposition 4.2, it thus follows that
Xi
Xi
i
b i ).
Q(ωi ) = EQ
= π0 (X ) = EQb
= Q(ω
1+r
1+r
b holds.
Since i was arbitrary, we see that the equality Q = Q
M. Rutkowski (USydney)
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Summary
Let us summarise the properties of single-period market models:
1
A single-period market model is arbitrage-free if and only if it admits
at least one risk-neutral probability measure.
2
An arbitrage-free single-period market model is complete if and only if
the risk-neutral probability measure is unique.
Under the assumption that the model is arbitrage-free:
3
An arbitrary attainable contingent claim X (that is, any claim that
can be replicated by means of some trading strategy) has the unique
arbitrage price π0 (X).
The arbitrage price π0 (X) of any attainable contingent claim X can be
computed from the risk-neutral valuation formula using any risk-neutral
probability Q.
If X is not attainable then we may define a price of X consistent with
the no-arbitrage principle. It can be computed using the risk-neutral
valuation formula, but it will depend on the choice of a risk-neutral
probability Q.
M. Rutkowski (USydney)
Slides 4: Single-Period Market Models
87 / 87