We use the same model as outlined in the previous chapter. There are two periods, tD0; 1. In the second period a finite number of states of the world,sD1; 2; : : : ;S can occur. The time-uncertainty structure is thus described by a tree as in Fig.4.3:
There arek D 0; 1; 2; : : : ;K assets with payoffs denoted byAks. We gather the structure of all assets’ payoffs in the states-asset-payoff matrix,
AD 0 B@
A01 AK1 ::: :::
A0S AKS 1 CAD
A0 AK D
0 B@
A1 :::
AS
1 CA
An arbitrage is a trading strategy that an investor would definitely like to exercise. Note that, as we mentioned above, this definition of arbitrage depends on the qualitative properties of the investor’s utility function. For strictly monotonic utility functions an arbitrage is a trading strategy that leads to positive payoffs without requiring any payments. For mean-variance utility functions an arbitrage is a trading strategy that offers the risk-free payoffs without requiring any payments.
We first formalize an arbitrage opportunity for strictly monotonic utility func- tions. Under this assumption, an arbitrage is a trading strategy 2RKC1such that
q0 A
>0:
Hence, the trading strategy never requires any payments and it delivers a non- negative and non-zero payoff. To give an example, let there be just two assets and two states. Say, the payoff matrix is
AWD 1 2
1 3
154 4 Two-Period Model: State-Preference Approach while the asset prices areq D .1; 4/0. Maybe you want to stop a moment and try to find an arbitrage opportunity, before reading on? In case you have not found it:
by selling one unit of the second asset and using the receipts (four units of wealth) to buy three units of the first asset, you are left with one unit of wealth today, and tomorrow you will be hedged if the second state occurs while you have an extra unit of wealth if the first state occurs. How can we erase arbitrage opportunities in this example? Obviously asset 2 is too expensive relative to asset 1. Suppose now the asset prices areq D .1; 2:5/0. Can you still find an arbitrage opportunity? We see that trying will not always be successful and is not helpful at all if there is no arbitrage opportunity. Instead we need a general result that tells us whether arbitrage opportunities exist. This is the content of the following theorem:
Theorem 4.2 (Fundamental Theorem of Asset Prices, FTAP) The following two statements are equivalent:
1. There exists no 2RKC1such that q0
A
>0:
2. There exists aD.1; : : : ; s; : : : ; S/02RSCCsuch that
qkD XS sD1
Akss; kD0; : : : ;K:
In the example above we see that for the state prices WD .0:5; 0:5/0 the two asset prices can be displayed as the weighted sum of their payoffs and therefore, applying the FTAP we know that there are no arbitrage opportunities at the asset pricesqD.1; 2:5/0. Hence, any effort to find an arbitrage must fail!
The proof of the FTAP has an easy and a tough part. It is straightforward to show that (2.) implies (1.): Suppose (2.) holds and consider a portfolio such thatA 0. Then by the strict positivity of state prices0A 0. But this implies q0 0 ruling out arbitrage opportunities.
In the following we first give a geometric proof of the more difficult part of the FTAP for the case of two assets and two states of the world. This will provide us with some intuitive understanding on the FTAP. Afterwards we give a proof of the general result which will be based on the Riesz representation theorem (TheoremA.1).
Proof of Theorem4.2(simple case) In the case of two assets and two states the payoffs of the assets in the two states s D 1; 2can be represented by the two dimensional vectorsA1andA2. To find the set of non-negative portfolio payoffs in a particular state, we first determine the set of assets where the asset payoff,As, is
Fig. 4.4 Finding arbitrage opportunities
q
A2
arbitrage A1
equal to0. This is a line orthogonal to the payoff vector.10Plotting these orthogonal lines for the vectorsA1andA2, we determine the set of non-negative payoffs in both states as the area of the intersection of two half planes as shown in Fig.4.4below.
To determine the set of arbitrage opportunities, we have then to find a strategy requiring no investments, i.e.q0 0orq00with a positive payoff in at least one of the states. To find the set of arbitrage portfolios we then plot the price vector qso that conditionsq0 0andAs0are satisfied. This is possible if and only if qdoes not belong to the cone ofA1andA2, i.e. if there are no constants1; 2> 0
such thatq0D1A1C2A2. ut
Proof of Theorem4.2(general case) The general argument is easy if markets are complete, in which case it follows from the Law of One Price. For any given payoff asset matrixA, consider the set of all payoffs that you can generate with alternative portfolios:
spanfAg D fy2RSC1WyDA for some 2RKC1g
Let q.y/ be the price associated with the payoff y in spanfAg. The function q W spanfAg ! Ris called the pricing functional on the set of attainable payoffs spanfAg. By the Law of One Priceqis linear, i.e., for ally;y02spanfAgand˛2R we have
(i) q.yCy0/Dq.y/Cq.y0/, (ii) q.˛y/D˛q.y/.
10The scalar product is positive (negative) if the angle is smaller (greater) than 90ı. The scalar product of orthogonal vectors is equal to 0 (see AppendixA.1).
156 4 Two-Period Model: State-Preference Approach Why is this true? – Since otherwise, one could find an arbitrage opportunity with hedged payoffs tomorrow and positive payoff today.
By the Riesz representation theorem (TheoremA.1) linear functionals can be represented asq.y/ D 0y for some vector of state prices 2 RSC1.11 From the various assumptions on the utility functions we get additional restrictions on the state prices. Suppose, for example, the utility function is increasing in the risk- free asset. Then the sum of the state prices must be positive because otherwise one could get the risk-free asset for free. If the utility function were strictly monotonic then each state price must be positive because otherwise the portfolio delivering a positive payoff in the state with zero or negative price would be an arbitrage opportunity for this type of investors. Note that this argument assumes that all portfolios can be built, i.e., that markets are complete. A proof for the general case can be found in the book by Magill and Quinzii [MQ02]. ut Let us now formulate the variant of the FTAP for mean-variance utilities, its proof is analogous to Theorem4.2:
Theorem 4.3 (FTAP for mean-variance utility functions) The following two conditions are equivalent:
1. There exists no 2RKC1such that
q00 and A Dv1; for somev > 0.
Note1D.1; : : : ; 1/0.
2. There exists a2RSwithPS
sD1s> 0such that qkD
XS sD1
Akss; kD0; : : : ;K:
To conclude this section we will give different equivalent formulations of the No- arbitrage Principle. These formulations are very useful to deepen the understanding of the main idea. Also it is important to study them because different fields of finance use different formulations of it.
The first reformulation of the No-arbitrage Principle displays asset prices as their discounted expected payoffs. This formulation follows from a normalization of state prices: applying the linear pricing rule,
qkD XS sD1
Akss; kD0; : : : ;K;
11It is obvious that a representation by state prices satisfies linearity. The converse is a bit harder to see (compare AppendixA.1).
to the risk-free asset,k D 0, we see that the risk-free rate is the reciprocal of the sum of the state prices:
q0D 1 Rf
D XS sD1
s: On defining the normalized state prices assWDs=.P
z/we get the discounted expected payoff formulation of asset prices12:
qkD 1 Rf
XS sD1
AkssD 1 Rf
E.Ak/:
There are similar results for situations where the no-arbitrage condition is disturbed by transaction costs or by short sale constraints [JK95b,JK95a].
The normalized state prices are called the martingale measure in financial mathematics and they are called therisk neutral probabilitiesin finance. The latter is a bit confusing since is actually accounting for the risk preferences of the agents, as we will see in Sect.4.3. “Risk neutral probabilities” therefore means:
probabilities that take the risk preferences already into account and can therefore be usedas if they were physical probabilities and the investor were risk-neutral.
Calling themrisk adjusted probabilitieswould be less confusing.
From this way of writing the pricing formula we also get an immediate formulation in terms of returns:
Rf D XS
sD1
Aks
qksDE.Rk/:
Hence, in the light of the normalized state prices all assets deliver the risk-free rate.
Indeed, a reformulation of the FTAP for returns does read like this:
Corollary 4.4 (FTAP for returns) The following two statements are equivalent:
1. There exists no2RKC1with
1 1 R
>0: (4.1)
12Note that we did not assume that all state prices are positive. For this formulation we only need that the sum of the state prices is positive, which holds, for example, with mean-variance utilities.
158 4 Two-Period Model: State-Preference Approach
2. There exists a2RSCCwith
Rf D XS sD1
Rkss; kD0; : : : ;K:
Finally, we would like to mention how the FTAP looks like in the case of the Ross APT. Recall that according to the arbitrage pricing theory of Ross, asset returns are thought of as being determined by several factors. The asset returns matrix is the product of the factor matrix and the matrix of factor loadings:RDFB. Accordingly asset payoffs can be written in terms of factors by noting that
ADA.q/1.q/DR.q/DFB.q/:
By defining zas the allocation of factor risks, i.e., z WD B.q/, we can write ADFzand the value of asset portfolios is
q0 Dq0A1FzDW Oq0Fz:
Hence, Corollary4.4can be rewritten as follows:
Corollary 4.5 (FTAP for Ross APT) The following two statements are equiva- lent:
1. There exists noz2RKC1with Oq0
F
z>0:
2. There exists aL 2RSCCwith
qOf D XS sD1
FsfLs; f D1; : : : ;F:
In other words: factors have a price that can be expressed as the weighted sum of their payoffs, weighted with some state prices. This concludes our remarks on the Fundamental Theorem of Asset Pricing. In the next section we introduce an important application of this theory: the pricing of derivatives.