4: single-period market models [.03in] · pdf file4: single-period market models marek...

4: SINGLE-PERIOD MARKET MODELS

Marek RutkowskiSchool of Mathematics and Statistics

University of Sydney

Semester 2, 2016

M. Rutkowski (USydney) Slides 4: Single-Period Market Models 1 / 87

General Single-Period Market Models

The main differences between the elementary and general singleperiod market models are:

The investor is allowed to invest in several risky securities instead ofonly one.The sample set is bigger, that is, there are more possible states ofthe 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 stockprices are represented by the probability measure P(ωi) = pi > 0 fori = 1, 2, . . . , k.

The savings account B equals B0 = 1 and B1 = 1 + r for someconstant r > −1.

The price of the jth stock at t = 1 is a random variable on Ω.It is denoted by Sj

t for t = 0, 1 and j = 1, . . . , n.

A contingent claim X = (X(ω1), . . . , X(ωk)) is a random variableon the probability space (Ω,F ,P).


Questions

1 Under which conditions a general single-period market modelM = (B,S1, . . . , Sn) is arbitrage-free?

2 How to define a risk-neutral probability measure for a model?3 How to use a risk-neutral probability measure to analyse general

single-period market models?4 Under which conditions a general single-period market model is

complete?5 Is completeness of a market model related to the uniqueness of a

risk-neutral probability measure?6 How to define and compute the arbitrage price of an attainable claim?7 Can we still apply the risk-neutral valuation formula to compute the

arbitrage price of an attainable claim?8 How to deal with contingent claims that are not attainable?9 How to use the class of risk-neutral probability measures to value

non-attainable claims?


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


PART 1

TRADING STRATEGIES AND ARBITRAGE-FREE MODELS


Trading Strategy

Definition (Trading Strategy)

A trading strategy (or a portfolio) in a general single-period marketmodel is defined as the vector

(x, φ1, . . . , φn) ∈ Rn+1

where x is the initial wealth of an investor and φj stands for the numberof shares of the jth stock purchased at time t = 0.

If an investor adopts the trading strategy (x, φ1, . . . , φn) at timet = 0 then the cash value of his portfolio at time t = 1 equals

V1(x, φ1, . . . , φn) :=

(x−

n∑j=1

φjSj0

)(1 + r) +

n∑j=1

φjSj1.


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 cashvalue of the portfolio at time t = 1

V1(x, φ1, . . . , φn) :=

(x−

n∑j=1

φjSj0

)(1 + r) +

n∑j=1

φjSj1.


Gains (Profits and Losses) Process

The profits or losses an investor obtains from the investment canbe 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)

=

(x−

n∑j=1

φjSj0

)r +

n∑j=1

φj∆Sj1

where the random variable ∆Sj1 = Sj

1 − Sj0 represents the nominal change

in the price of the jth stock.


Discounted Stock Price and Value Process

To understand whether the jth stock appreciates in real terms, weconsider the discounted stock prices of the jth stock

Sj0 := Sj

0 =Sj

0

B0, Sj

1 :=Sj

1

1 + r=Sj

1

B1.

Similarly, we define the discounted wealth process as

V0(x, φ1, . . . , φn) := x, V1(x, φ1, . . . , φn) :=V1(x, φ1, . . . , φn)

B1.

It is easy to see that

V1(x, φ1, . . . , φn) =

(x−

∑nj=1 φ

jSj0

)+∑n

j=1 φjSj

1

= x+∑n

j=1 φj(Sj

1 − Sj0).


Discounted Gains Process

Definition (Discounted Gains Process)

The discounted gains process for the investor is defined as

G0(x, φ1, . . . , φn) = 0

and

G1(x, φ1, . . . , φn) := V1(x, φ1, . . . , φn)− V0(x, φ1, . . . , φn)

=

n∑j=1

φj∆Sj1

where ∆Sj1 = Sj

1 − Sj0 is the change in the discounted price of the jth

stock.


Arbitrage: Definition

The concept of an arbitrage in a general single-period market modelis essentially the same as in the elementary market model. It is worthnoting that the real-world probability P can be replaced here by anyequivalent probability measure Q.

Definition (Arbitrage)

A trading strategy (x, φ1, . . . , φn) in a general single-period market modelis 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. EPV1(x, φ1, . . . , φn)

> 0, that is,

k∑i=1

V1(x, φ1, . . . , φn)(ωi)P(ωi) > 0.


Arbitrage: Equivalent Conditions

The following condition is equivalent to A.3.

A.3′. There exists ω ∈ Ω such that V1(x, φ1, . . . , φn)(ω) > 0.

The definition of arbitrage can be formulated using the discounted valueand gains processes. This is sometimes very helpful.

Proposition (4.1)

A trading strategy (x, φ1, . . . , φn) in a general single-period market modelis an arbitrage opportunity if and only if one of the following conditionsholds:

1 Assumptions A.1-A.3 in the definition of arbitrage hold withV (x, φ1, . . . , φn) instead of V (x, φ1, . . . , φn).

2 x = 0 and A.2-A.3 in the definition of arbitrage are satisfied withG1(x, φ1, . . . , φn) instead of V1(x, φ1, . . . , φn).


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 areequivalent.

In Proposition 4.1, condition 1 is equivalent to condition 2.

To prove the first statement, we use the relationships betweenV (x, φ1, . . . , φn) and V (x, φ1, . . . , φn), that is,

V0(x, φ1, . . . , φn) = V0(x, φ1, . . . , φn) = x,

V1(x, φ1, . . . , φn) =1

1 + rV1(x, φ1, . . . , φn),

EPV1(x, φ1, . . . , φn)

=

1

1 + rEPV1(x, φ1, . . . , φn)

.

This shows that the first statement holds.



Proof of Proposition 4.1: Second step.

To prove the second statement, we recall the relation betweenV (x, φ1, . . . , φn) and G1(x, φ1, . . . , φn)

G1(x, φ1, . . . , φn) = V1(x, φ1, . . . , φn)− V0(x, φ1, . . . , φn)

= V1(x, φ1, . . . , φn)− x.

It is now clear that for x = 0 we have

G1(x, φ1, . . . , φn) = V1(x, φ1, . . . , φn).

Hence the second statement is true as well.

In fact, one can also observe that G1(x, φ1, . . . , φn) does not dependon x at all, since G1(x, φ1, . . . , φn) =

∑nj=1 φ

j∆Sj1.


Verification of the Arbitrage-Free Property

It can be sometimes hard to check directly whether arbitrageopportunities exist in a given market model, especially when dealingwith several risky assets or in the multi-period setup.

We have introduced the risk-neutral probability measure in theelementary market model and we noticed that it can be used tocompute the arbitrage price of any contingent claim.

We will show that the concept of a risk-neutral probability measure isalso a convenient tool for checking whether a general single-periodmarket model is arbitrage-free or not.

In addition, we will argue that a risk-neutral probability measure canalso be used for the purpose of valuation of a contingent claim (eitherattainable or not).


Risk-Neutral Probability Measure

Definition (Risk-Neutral Probability Measure)

A probability measure Q on Ω is called a risk-neutral probabilitymeasure for a general single-period market model M if:

R.1. Q (ωi) > 0 for all ωi ∈ Ω,

R.2. EQ(∆Sj

1

)= 0 for j = 1, 2, . . . , n.

We denote by M the class of all risk-neutral probability measures forthe market model M.

Condition R.1 means that Q and P are equivalent probabilitymeasures. A risk-neutral probability measure is also known as anequivalent martingale measure.

Note that condition R.2 is equivalent to EQ(Sj

1

)= Sj

0 or, moreexplicitly,

EQ(Sj

1

)= (1 + r)Sj

0.


Example: Stock Prices

Example (4.1)

We consider the following model featuring two stocks S1 and S2

on the sample space Ω = ω1, ω2, ω3.The interest rate r = 1

10 so that B0 = 1 and B1 = 1 + 110 .

We deal here with the market model M = (B,S1, S2).

The stock prices at t = 0 are given by S10 = 2 and S2

0 = 3.

The stock prices at t = 1 are represented in the table:

ω1 ω2 ω3

S11 1 5 3S2

1 3 1 6


Example: Value Process

Example (4.1 Continued)

For any trading strategy (x, φ1, φ2) ∈ R3, we have

V1(x, φ1, φ2) =(x− 2φ1 − 3φ2

)(1 +

1

10

)+ φ1S1

1 + φ2S21 .

We set φ0 := x− 2φ1 − 3φ2. Then V1(x, φ1, φ2) equals

V1(x, φ1, φ2)(ω1) = φ0

(1 +

1

10

)+ φ1 + 3φ2,

V1(x, φ1, φ2)(ω2) = φ0

(1 +

1

10

)+ 5φ1 + φ2,

V1(x, φ1, φ2)(ω3) = φ0

(1 +

1

10

)+ 3φ1 + 6φ2.


Example: Gains Process


The increments ∆Sj1 are represented by the following table

ω1 ω2 ω3

∆S11 -1 3 1

∆S21 0 -2 3

The gains G1(x, φ1, φ2) are thus given by

G1(x, φ1, φ2)(ω1) =1

10φ0 − φ1 + 0φ2,

G1(x, φ1, φ2)(ω2) =1

10φ0 + 3φ1 − 2φ2,

G1(x, φ1, φ2)(ω3) =1

10φ0 + φ1 + 3φ2.


Example: Discounted Stock Prices


Out next goal is to compute the discounted wealth processV (x, φ1, φ2) and the discounted gains process G1(x, φ1, φ2).

To this end, we first compute the discounted stock prices.

Of course, Sj0 = Sj

0 for j = 1, 2.

The following table represents the discounted stock prices Sj1 for

j = 1, 2 at time t = 1

ω1 ω2 ω3

S11

1011

5011

3011

S21

3011

1011

6011


Example: Discounted Value Process


The discounted value process V (x, φ1, φ2) is thus given by

V0(x, φ1, φ2) = V0(x, φ1, φ2) = x

and

V1(x, φ1, φ2)(ω1) = φ0 +10

11φ1 +

30

11φ2,

V1(x, φ1, φ2)(ω2) = φ0 +50

11φ1 +

10

11φ2,

V1(x, φ1, φ2)(ω3) = φ0 +30

11φ1 +

60

11φ2,

where φ0 = x− 2φ1 − 3φ2 is the amount of cash invested in B attime 0 (as opposed to the initial wealth given by x).


Example: Discounted Gains Process


The increments of the discounted stock prices equal

ω1 ω2 ω3

∆S11 −12

112811

811

∆S21 − 3

11 −2311

2711

Hence the discounted gains G1(x, φ1, φ2) are given by

G1(x, φ1, φ2)(ω1) = −12

11φ1 − 3

11φ2,

G1(x, φ1, φ2)(ω2) =28

11φ1 − 23

11φ2,

G1(x, φ1, φ2)(ω3) =8

11φ1 +

27

11φ2.


Example: Arbitrage-Free Property


The condition G1(x, φ1, φ2) ≥ 0 is equivalent to

−12φ1 − 3φ2 ≥ 0

28φ1 − 23φ2 ≥ 0

8φ1 + 27φ2 ≥ 0

Can we find (φ1, φ2) ∈ R2 such that all inequalities are valid andat least one of them is strict?

It appears that the answer is negative, since the unique vectorsatisfying all inequalities above is (φ1, φ2) = (0, 0).

Therefore, the single-period market model M = (B,S1, S2) isarbitrage-free.


Example: Risk-Neutral Probability Measure


We will now show that this market model admits a unique risk-neutralprobability measure on Ω = ω1, ω2, ω3.Let us denote qi = Q(ωi) for i = 1, 2, 3. From the definition of arisk-neutral probability measure, we obtain the following linear system

−12

11q1 +

28

11q2 +

8

11q3 = 0

− 3

11q1 −

23

11q2 +

27

11q3 = 0

q1 + q2 + q3 = 1

The unique solution equals Q = (q1, q2, q3) =(

4780 ,

1580 ,

1880

).


PART 2

FUNDAMENTAL THEOREM OF ASSET PRICING


Fundamental Theorem of Asset Pricing (FTAP)

In Example 4.1, we have checked directly that the market modelM = (B,S1, S2) is arbitrage-free.

In addition, we have shown that the unique risk-neutral probabilitymeasure exists in this model.

Is there any relation between no arbitrage property of a market modeland the existence of a risk-neutral probability measure?

The following important result, known as the FTAP, gives a completeanswer to this question within the present setup.

The FTAP was first established by Harrison and Pliska (1981) and itwas later extended to continuous-time market models.

Theorem (FTAP)

A general single-period model M = (B,S1, . . . , Sn) is arbitrage-free ifand only if there exists a risk-neutral probability measure for M, that is,M 6= ∅.


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 Qexists.

Let (0, φ) = (0, φ1, . . . , φn) be any trading strategy with null initialendowment. Then for any Q ∈M

EQ(V1(0, φ)

)= EQ

( n∑j=1

φj∆Sj1

)=

n∑j=1

φj EQ(∆Sj

1

)︸︷︷︸=0

= 0.

If we assume that V1(0, φ) ≥ 0 then the last equation implies that theequality V1(0, φ)(ω) = 0 must hold for all ω ∈ Ω.

Hence no trading strategy satisfying all conditions of an arbitrageopportunity may exist.


Geometric Interpretation of X and Q

The proof of the implication (⇒) in the FTAP needs somepreparation, 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 avector in Rk

Q = (Q(ω1), . . . ,Q(ωk)) = (q1, . . . , qk) ∈ Rk.

We note that

EQ (X) =

k∑i=1

X(ωi)Q(ωi) =

k∑i=1

xiqi = 〈X,Q〉

where 〈·, ·〉 denotes the inner product of two vectors in Rk.


Auxiliary Subsets of Rk

We define the following classes:

W =X ∈ Rk

∣∣∣X = V1(0, φ1, . . . , φn) for some φ1, . . . , φn

W⊥ =Z ∈ Rk | 〈X,Z〉 = 0 for all X ∈W

The set W is the image of the map V1(0, ·, . . . , ·) : Rn → Rk.

We note that W represents all discounted values at t = 1 of tradingstrategies 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:

A =X ∈ Rk

∣∣X 6= 0, xi ≥ 0 for i = 1, . . . , k

P+ =

Q ∈ Rk

∣∣∣ k∑i=1

qi = 1, qi > 0


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 V1(0, ·, . . . , ·) : Rn → Rk is linear.

In other words, for any trading strategies (0, η1, . . . , ηn) and(0, κ1, . . . , κn) and arbitrary real numbers α, β

(0, φ1, . . . , φn) = α(0, η1, . . . , ηn) + β(0, κ1, . . . , κn)

is also a trading strategy. Hence W is a vector subspace of Rk. Inparticular, the zero vector (0, 0, . . . , 0) belongs to W.

It us easy to check that W⊥, that is, the orthogonal complement ofW is a vector subspace as well.


Risk-Neutral Probability Measures

Lemma (4.1)

A single-period market model M = (B,S1, . . . , Sn) is arbitrage free ifand 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 asingle-period market model M = (B,S1, . . . , Sn) if and only ifQ ∈W⊥ ∩ P+.

Hence the set M of all risk-neutral probability measures for the modelM satisfies M = W⊥ ∩ P+ and thus

M 6= ∅ ⇔ W⊥ ∩ P+ 6= ∅.M. Rutkowski (USydney) Slides 4: Single-Period Market Models 31 / 87

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 = V1(0, φ) ∈W

〈X,Q〉 = EQ(V1(0, φ)

)= EQ

( n∑j=1

φj∆Sj1

)=

n∑j=1

φj EQ(∆Sj

1

)︸︷︷︸=0

= 0.

We conclude that Q belongs to W⊥ as well.

Consequently, Q ∈W⊥ ∩ P+ as was required to show.


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 satisfyingcondition 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 tradingstrategy (0, φ1, . . . , φn) with

(φ1, . . . , φn) = (0, . . . , 0, 1, 0, . . . , 0) = ej .

This trading strategy only invests in the savings account and the jthasset.

The discounted wealth of this strategy is V1(0, ej) = ∆Sj1.


Proof of Lemma 4.2

Proof of (⇐) in Lemma 4.2 (Continued).

SinceV1(0, ej) ∈W and Q ∈W⊥

we obtain

0 = 〈V1(0, ej),Q〉 = 〈∆Sj1,Q〉 = EQ

(∆Sj

1

).

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 geometricreformulation of the FTAP:

W ∩ A = ∅ ⇔ W⊥ ∩ P+ 6= ∅.


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

〈b− a, y〉 < 0 for all b ∈ B

and〈c− a, y〉 > 0 for all c ∈ C.

Proof of the Separating Hyperplane Theorem.

The proof can be found in any textbook of convex analysis or functionalanalysis. It is sketched in the course notes.


Separating Hyperplane Theorem: Interpretation

Let the vectors a, y ∈ Rk be as in the statement of the SeparatingHyperplane Theorem

It is clear that y ∈ Rk is never a zero vector.

We define the (k − 1)-dimensional hyperplane H ⊂ Rk by setting

H = a+x ∈ Rk | 〈x, y〉 = 0

= a+ y⊥.

Then we say that the hyperplane H strictly separates the convexsets B and C.

Intuitively, the sets B and C lie on different sides of the hyperplaneH and thus they can be seen as geometrically separated by H.

Note that the compactness of at least one of the sets is a necessarycondition for the strict separation of B and C.


Separating Hyperplane Theorem: Corollary

The following corollary is a consequence of the separating hyperplanetheorem.

It is more suitable for our purposes: it will be later applied to B = Wand C = A+ := X ∈ A | 〈X,P〉 = 1 ⊂ A.

Corollary (4.1)

Assume that B ⊂ Rk is a vector subspace and set C is a compact convexset such that B ∩ C = ∅. Then there exists a vector y ∈ Rk such that

〈b, y〉 = 0 for all b ∈ B

that is, y ∈ B⊥, and

〈c, y〉 > 0 for all c ∈ C.


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 suchthat the inequality

〈b, y〉 < 〈a, y〉

is satisfied for all vectors b ∈ B.

Since B is a vector subspace, the vector λb belongs to B for anyλ ∈ R. Hence for any b ∈ B and λ ∈ R we have

〈λb, y〉 = λ〈b, y〉 < 〈a, y〉.

This in turn implies that 〈b, y〉 = 0 for any vector b ∈ B, meaningthat y ∈ B⊥. Also, we have that 〈a, y〉 > 0.


Proof of Corollary 4.1

Proof of Corollary 4.1: Second step.

To establish the second inequality, we observe that from the separatinghyperplane theorem, we obtain

〈c, y〉 > 〈a, y〉 for all c ∈ C.

Consequently, for any c ∈ C

〈c, y〉 > 〈a, y〉 > 0.

We conclude that 〈c, y〉 > 0 for all c ∈ C.

We now are ready to establish the implication (⇒ ) in the FTAP:

W ∩ A = ∅ ⇒ W⊥ ∩ P+ 6= ∅.


Proof of (⇒ ) in FTAP: 1

Proof of (⇒ ) in FTAP: First step.

We assume that the model is arbitrage-free. From Lemma 4.1, this isequivalent 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 | 〈X,P〉 = 1.Observe that A+ is a closed, bounded (hence compact) and convexsubset of Rk. Since A+ ⊂ A, it is clear that

W ∩ A = ∅ ⇒ W ∩ A+ = ∅.

Hence in the next step we may assume that W ∩ A+ = ∅.



Proof of (⇒ ) in FTAP: Second step.

By applying Corollary 4.1 to B = W and C = A+, we see that thereexists a vector Y ∈W⊥ such that

〈X,Y 〉 > 0 for all X ∈ A+. (1)

Our goal is to show that Y can be used to define a risk-neutralprobability 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) = 〈Xi,P〉 = 1.



Proof of (⇒ ) in FTAP: Third step.

Let yi be the ith component of Y . It follows from (1) that

0 < 〈Xi, Y 〉 = (P(ωi))−1 yi

and thus yi > 0 for all i = 1, 2, . . . , k. We set Q(ωi) = qi where

qi :=yi

y1 + · · ·+ yk= cyi > 0

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 thatW⊥ ∩ P+ 6= ∅.From Lemma 4.2, Q is a risk-neutral probability and thus M 6= ∅.


PART 3

EXAMPLES OF MARKET MODELS


Example: Arbitrage-Free Market Model


We consider the market model M = (B,S1, S2) introduced inExample 4.1.

The interest rate r = 110 so that B0 = 1 and B1 = 1 + 1

10 .


0 = 3.

We have shown that the increments of the discounted stock prices S1

and S2 equal

ω1 ω2 ω3

∆S11 −12

112811

811

∆S21 − 3

11 −2311

2711




The vector spaces W and W⊥ are given by

W =

α −12

288

+ β

−3−23

27

∣∣∣∣α, β ∈ R

and

W⊥ =

γ 47

1518

∣∣∣∣ γ ∈ R

.

We first show the model is arbitrage-free using Lemma 4.1.

It thus suffices to check that W ∩ A = ∅




If there exists a vector X ∈W ∩ A then the following threeinequalities 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 thusW ∩ A = ∅. This is left as an easy exercise.

In view of Lemma 4.1, we conclude that the market model isarbitrage-free.

In the next step, our goal is to show that M is non-empty.




Lemma 4.2 tells us that M = W⊥ ∩ P+. If Q ∈W⊥ then

Q = γ

471518

for some γ ∈ R.

If Q ∈ P+ then 47γ + 15γ + 18γ = 1 so that γ = 180 > 0.

We conclude that the unique risk-neutral probability measure Q isgiven by

Q =1

80

471518

.

The FTAP confirms that the market model is arbitrage-free.


Example: Market Model with Arbitrage

Example (4.2)

We consider the following model featuring two stocks S1 and S2 onthe sample space Ω = ω1, ω2, ω3.The interest rate r = 1

10 so that B0 = 1 and B1 = 1 + 110 .


0 = 2 and thestock prices at t = 1 are represented in the table:

ω1 ω2 ω3

S11 1 1

2 3S2

152 4 1

10

Does this market model admit an arbitrage opportunity?




Once again, we will analyse this problem using Lemma 4.1, Lemma4.2 and the FTAP.

To tell whether a model is arbitrage-free it suffices to know theincrements of discounted stock prices.

The increments of discounted stock prices are represented in thefollowing table

ω1 ω2 ω3

∆S11 − 1

11 − 611

2011

∆S21

311

1811 −21

11




Recall thatG1(x, φ1, φ2) = φ1∆S1

1 + φ2∆S21

Hence, by the definition of W, we have

W =

α −1−620

+ β

318−21

∣∣∣∣α, β ∈ R

.

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 marketmodel is not arbitrage-free.




We note that

W⊥ =

γ −6

10

∣∣∣ γ ∈ R

.

If there exists a risk-neutral probability measure Q thenQ ∈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, norisk-neutral probability measure exists, that is, M = ∅.Hence the FTAP confirms that the model is not arbitrage-free.


PART 4

RISK-NEUTRAL VALUATION OF CONTINGENT CLAIMS


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 ina general single-period market model?

In a general single-period market model, the idea of pricing Europeanoptions can be extended to any contingent claim.

Definition (Contingent Claim)

A contingent claim is a random variable X defined on Ω andrepresenting the payoff at the maturity date.

Derivatives nowadays are usually quite complicated and thus it makessense to analyse valuation and hedging of a general contingent claim,and not only European call and put options.


No-Arbitrage Principle

Definition (Replication and Arbitrage Price)

A trading strategy (x, φ1, . . . , φn) is called a replicating strategy (ahedging strategy) for a claim X when V1(x, φ1, . . . , φn) = X. Then theinitial 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 tradingstrategy (x, φ1, . . . , φn). Then the unique price of X at 0 consistent withno-arbitrage principle equals V0(x, φ1, . . . , φn) = x.

Proof.

If the price of X is higher (lower) than x, one can short sell (buy) Xand buy (short sell) the replicating portfolio. This will yield an arbitrageopportunity in the extended market in which X is traded at time t = 0.


Example: Stochastic Volatility Model

In the elementary market model, a replicating strategy for anycontingent claim always exists. However, in a general single-periodmarket model, a replicating strategy may fail to exist for some claims.

For instance, when there are more sources of randomness than thereare stocks available for investment then replicating strategies do notexist for some claims.

Example (4.3)

Consider a market model consisting of bond B, stock S, and arandom variable v called the volatility.

The volatility determines whether the stock price can make eithera big or a small jump.

This is a simple example of a stochastic volatility model.




The sample space is given by

Ω = ω1, ω2, ω3, ω4

and the volatility is defined as

v(ωi) =

h for i = 1, 4,l for i = 2, 3.

We furthermore assume that 0 < l < h < 1. The stock price S1 isgiven by

S1(ωi) =

(1 + v(ωi))S0 for i = 1, 2,(1− v(ωi))S0 for i = 3, 4.




Unlike in examples we considered earlier, the amount by which thestock 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 doesnot exist. In that case, we say that a claim is not attainable.

To justify this claim, we consider the digital call option X with thepayoff

X =

1 if S1 > K,0 otherwise,

where K > 0 is the strike price.




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 for i = 1,0 otherwise.

Suppose that (x, φ) is a replicating strategy for X. EqualityV1(x, φ) = X becomes

(x− φS0)

1 + r1 + r1 + r1 + r

+ φ

(1 + h)S0

(1 + l)S0

(1− l)S0

(1− h)S0

=

1000




Upon setting β = φS0 and α = (1 + r)x− φS0r, we see that theexistence of a solution (x, φ) to this system is equivalent to theexistence of a solution (α, β) to the system

α

1111

+ β

hl−l−h

=

1000

It is easy to see that the above system of equations has no solutionand thus a digital call is not an attainable contingent claim withinthe framework of the stochastic volatility model.

Intuitively, the randomness generated by the volatility cannot behedged, since the volatility is not a traded asset.


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 areplicating strategy for X.

Let us summarise the known properties of attainable claims:

It is clear how to price attainable contingent claims by the replicatingprinciple.

There might be more than one replicating strategy, but no arbitrageprinciple leads the initial wealth x to be unique.

In the two-state single-period market model, one can use therisk-neutral probability measure to price contingent claims.


Risk-Neutral Valuation Formula

Our next objective is to extend the risk-neutral valuation formulato any attainable contingent claim within the framework of a generalsingle-period market model.

Proposition (4.2)

Let X be an attainable contingent claim and let Q ∈M be any risk-neutralprobability measure. Then the arbitrage price of X at t = 0 equals

π0(X) = EQ

(X

1 + r

).

Proof of Proposition 4.2.

Recall that a trading strategy (x, φ1, . . . , φn) is a replicating strategy forX whenever V1(x, φ1, . . . , φn) = X


Proof of the Risk-Neutral Valuation Formula


We divide both sides by 1 + r, to obtain

X

1 + r=V1(x, φ1, . . . , φn)

1 + r= V1(x, φ1, . . . , φn).

Hence

1

1 + rEQ (X) = EQ

V1(x, φ1, . . . , φn)

= EQ

x+ G1(x, φ1, . . . , φn)

= x+ EQ

n∑j=1

φj∆Sj1

= x+n∑

j=1

φj EQ(∆Sj

1

)= x. (from R.2.)




Proposition 4.2 shows that risk-neutral probability measures can beused to price attainable contingent claims.

Consider the market model introduced in Example 4.3 with theinterest rate r = 0.

Recall that in this case the model is arbitrage-free since1− h < 1 + r = 1 < 1 + h.

The increments of the discounted stock price S are represented inthe following table

ω1 ω2 ω3 ω4

∆S1 hS0 lS0 −lS0 −hS0




By the definition of the linear subspace W ⊂ R4, we have

W =

γ

hl−l−h

∣∣∣∣∣ γ ∈ R

.

The orthogonal complement of W is thus the three-dimensionalsubspace of R4 given by

W⊥ =

z1z2z3z4

∈ R4

∣∣∣∣∣ 〈

z1z2z3z4

,

hl−l−h

〉 = 0

.




Recall that a vector (q1, q2, q3, q4)> belongs to P+ if and only ifthe 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 byM = W⊥ ∩ P+, we find that

q1q2q3q4

∈M ⇔

(q1, q2, q3, q4)>∣∣ qi > 0,

4∑i=1

qi = 1

and h(q1 − q4) + l(q2 − q3) = 0.




The class M of all risk-neutral probability measures in our stochasticvolatility model is therefore given by

M =

q1q2q3

1− q1 − q2 − q3

∣∣∣∣∣ q1 > 0, q2 > 0, q3 > 0,

q1 + q2 + q3 < 1,l(q2 − q3) = h(1− 2q1 − q2 − q3)

.

This set appears to be non-empty and thus we conclude that ourstochastic volatility model is arbitrage-free.

Recall that we have already shown that the digital call option is notattainable if

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




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

2− q1.

We apply the risk-neutral valuation formula to the digital callX = (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, 12) can be achieved.


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 theno-arbitrage principle if the extended model, which consists of B, theoriginal stocks S1, . . . , Sn, as well as an additional asset Sn+1 satisfyingSn+1

0 = π0(X) and Sn+11 = X, is arbitrage-free.

The interpretation of Definition 4.1 is as follows:

We assume that the model M = (B,S1, . . . , Sn) is arbitrage-free.

We regard the additional asset as a tradable risky asset in theextended market model M = (B,S1, . . . , Sn+1).

We postulate its price at time 0 should be selected in such a way thatthe extended market model M is still arbitrage-free.


Valuation of Non-Attainable Claims

We already know that the risk-neutral valuation formula returns thearbitrage price for any attainable claim.

The next result shows that it also yields a price consistent with theno-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 arbitraryrisk-neutral probability measure. Then π0(X) given by

π0(X) := EQ

(X

1 + r

)(2)

defines a price at t = 0 for the claim X that is consistent with theno-arbitrage principle.




Let Q ∈M be an arbitrary risk-neutral probability for M.

We will show that Q is also a risk-neutral probability measure for theextended model M = (B,S1, . . . , Sn+1) in which Sn+1

0 = π0(X) andSn+1

1 = X.

For this purpose, we check that

EQ

(∆Sn+1

1

)= EQ

(X

1 + r− π0(X)

)= 0

and thus Q ∈ M is indeed a risk-neutral probability in the extendedmarket model.

By the FTAP, the extended model M is arbitrage-free. Hence theprice π0(X) given by (2) complies with the no-arbitrage principle.


PART 5

COMPLETENESS OF MARKET MODELS


Complete and Incomplete Models

The non-uniqueness of arbitrage prices is a serious theoreticalproblem, which is still not completely resolved.

We categorise market models into two classes: complete andincomplete models.

Definition (Completeness)

A financial market model is called complete if for any contingent claim Xthere exists a replicating strategy (x, φ) ∈ Rn+1. A model is incompletewhen there exists a claim X for which a replicating strategy does not exist.

Given an arbitrage-free and complete model, the issue of pricing allcontingent claims is completely solved.

How can we tell whether a given model is complete?


Algebraic Criterion for Market Completeness

Proposition (4.4)

Assume that a single-period market model M = (B,S1, . . . , Sn) definedon the sample space Ω = ω1, . . . , ωk is arbitrage-free. Then M iscomplete if and only if the k × (n+ 1) matrix A

A =

1 + r S1

1(ω1) · · · Sn1 (ω1)

1 + r S11(ω2) · · · Sn

1 (ω2)· · · ·· · · ·· · · ·

1 + r S11(ωk) · · · Sn

1 (ωk)

= (A0, A1, . . . , An)

has a full row rank, that is, rank (A) = k. Equivalently, M is completewhenever the linear subspace spanned by the vectors A0, A1, . . . , An

coincides with the full space Rk.




By the linear algebra, A has a full row rank if and only if for everyX ∈ Rk the equation AZ = X has a solution Z ∈ Rn+1.

If we set φ0 = x−∑n

j=1 φjSj

0 then we have1 + r S1

1(ω1) · · · Sn1 (ω1)

1 + r S11(ω2) · · · Sn

1 (ω2)· · · ·· · · ·· · · ·

1 + r S11(ωk) · · · Sn

1 (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 equivalentto solving the equation AZ = X.


Example: Incomplete Model


Consider the stochastic volatility model from Example 4.3.

We already know that this model is incomplete, since the digital callis not an attainable claim.

The matrix A is given by

A =

1 + r S1

1(ω1)1 + r S1

1(ω2)1 + r S1

1(ω3)1 + r S1

1(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 isincomplete.


Probabilistic Criterion for Attainability

Proposition 4.4 yields a method for determining whether a marketmodel 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 isnon-empty.

Proposition (4.5)

Assume that a single-period model M = (B,S1, . . . , Sn) is arbitrage-free.Then a contingent claim X is attainable if and only if the expected value

EQ

(X

1 + r

)has the same value for all risk-neutral probability measures Q ∈M.


Proof of (⇒) in Proposition 4.5: 1


(⇒) It is easy to deduce from Proposition 4.2 that if a contingent claimX is attainable then the expected value

EQ((1 + r)−1X

)has the same value for all Q ∈M. To this end, it is possible to argue bycontradiction. We leave the details as an exercise.

(⇐) (MATH3975) We prove this implication by contrapositive. Let usthus assume that the contingent claim X is not attainable. Our goal isto find two risk-neutral probabilities, say Q and Q, for which

EQ((1 + r)−1X

)6= EQ

((1 + r)−1X

). (3)


Proof of (⇐) in Proposition 4.5: 2


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 systemAZ = 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 isconvex and compact. Moreover, B ∩ C = ∅.




In view of Corollary 4.1, there exists a non-zero vectorY = (y1, . . . , yk) ∈ Rk such that

〈b, Y 〉 = 0 for all b ∈ B,〈c, Y 〉 > 0 for all c ∈ C.

In view of the definition of B and C, this means that for everyj = 0, 1, . . . , n

〈Aj , Y 〉 = 0 and 〈X,Y 〉 > 0 (4)

where Aj is the jth column of the matrix A.

It is worth noting that the vector Y depends on X.




We assumed that the market model is arbitrage-free and thus, by theFTAP, 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 toensure that for every i = 1, 2, . . . , k

Q(ωi) := Q(ωi) + λ(1 + r)yi > 0. (5)

In the next step, our next goal is to show that Q is also a risk-neutralprobability measure and it is different from Q.

In the last step, we will show that inequality (3) is valid.




From the definition of A in Proposition 4.4 and the first equality in(4) with j = 0, we obtain

k∑i=1

λ(1 + r)yi = λ〈A0, Y 〉 = 0.

It then follows from (5) that

k∑i=1

Q(ωi) =

k∑i=1

Q(ωi) +

k∑i=1

λ(1 + r)yi = 1

and thus Q is a probability measure on the space Ω.

In view of (5), it is clear that Q satisfies condition R.1.




It remains to check that Q satisfies also condition R.2.

We examine the behaviour under Q of the discounted stock price Sj1.

For every j = 1, 2, . . . , n, we have

EQ(Sj

1

)=

k∑i=1

Q(ωi)Sj1(ωi)

=k∑

i=1

Q(ωi)Sj1(ωi) + λ

k∑i=1

Sj1(ωi)(1 + r)yi

= EQ(Sj

1

)+ λ 〈Aj , Y 〉︸︷︷︸

=0

(in view of (4))

= Sj0 (since Q ∈M)




We conclude that EQ(∆Sj

1

)= 0 and thus Q ∈M, that is, Q is a

risk-neutral probability measure for the market model M.

From (5), it is clear that Q 6= Q. We have thus proven that if Mis arbitrage-free and incomplete, then there exists more than onerisk-neutral probability measure.

To complete the proof, it remains to show that inequality (3) issatisfied for the claim X.

Recall that X was a fixed non-attainable contingent claim and weconstructed a risk-neutral probability measure Q corresponding to X.




We observe that

EQ

(X

1 + r

)=

k∑i=1

Q(ωi)X(ωi)

1 + r

=

k∑i=1

Q(ωi)X(ωi)

1 + r+ λ

k∑i=1

yiX(ωi)︸︷︷︸>0

>

k∑i=1

Q(ωi)X(ωi)

1 + r= EQ

(X

1 + r

)since the inequalities 〈X,Y 〉 > 0 and λ > 0 imply that the bracedexpression is strictly positive.


Probabilistic Criterion for Market Completeness

Theorem (4.1)

Assume that a single-period model M = (B,S1, . . . , Sn) is arbitrage-free.Then M is complete if and only if the class M consists of a singleelement, 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 thatthere exists at least one risk-neutral probability measure, that is, the classM 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.


Proof of (⇒) in Theorem 4.1

Proof of (⇒) Theorem 4.1.

(⇒) Assume M is complete and consider any two risk-neutral probabilitymeasures Q and Q from M. For a fixed, but arbitrary, i = 1, 2, . . . , k, letthe contingent claim Xi be given by

Xi(ω) =

1 + r if ω = ωi,

0 otherwise.

Since M is now assumed to be complete, the contingent claim Xi isattainable. From Proposition 4.2, it thus follows that

Q(ωi) = EQ

(Xi

1 + r

)= π0(Xi) = EQ

(Xi

1 + r

)= Q(ωi).

Since i was arbitrary, we see that the equality Q = Q holds.


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 admitsat least one risk-neutral probability measure.

2 An arbitrage-free single-period market model is complete if and only ifthe risk-neutral probability measure is unique.

3 Under the assumption that the model is arbitrage-free:

An arbitrary attainable contingent claim X (that is, any claim thatcan be replicated by means of some trading strategy) has the uniquearbitrage price π0(X).The arbitrage price π0(X) of any attainable contingent claim X can becomputed from the risk-neutral valuation formula using any risk-neutralprobability Q.If X is not attainable then we may define a price of X consistent withthe no-arbitrage principle. It can be computed using the risk-neutralvaluation formula, but it will depend on the choice of a risk-neutralprobability Q.


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