Unformatted text preview: 5
NoArbitrage Pricing
Theory
5.1 INTRODUCTION
In this chapter, we study the fundamental concept of arbitrage and examine its
implications for the pricing of cash ﬂow streams. Only the ﬁnite discretetime and
discretestate theory is considered. A more advanced treatment of the topics in this
chapter is given in Chapter 11.
The main result of this chapter, presented in various versions, is the Fundamental
Theorem of Asset Pricing, which says that “absence of arbitrage is equivalent to the
existence of a strictly positive linear pricing rule.” Separating hyperplane arguments
underly this result.
With some effort, all the results of this chapter could be expressed (and proved)
without using probability concepts. However, employing probability has the advantages of (i) signiﬁcantly shortening the proofs, (ii) introducing concepts which
are intuitively appealing, and (iii) preparing the reader for the continuoustime case,
where probability cannot be avoided. Relevant probabilistic concepts will be introduced as needed. 5.2 SINGLEPERIOD MODEL
5.2.1 Description of the Model
We begin with a ﬁnite number Æ of securities or assets Ë ½ Ë¾
ËÆ . Only their
values at times ¼ and ½ are considered in this section. At time 0, the investors know
the time0 values but the time1 values are random variables. These random variables
159 160 NOARBITRAGE PRICING THEORY are deﬁned with respect to a sample space ª
½¾
Å consisting of a
ﬁnite number Å of “states of nature” or possible outcomes. At time 0, the investors
know the list of these possible outcomes but they do not know until time 1 which
one will occur. There is also a probability measure È satisfying È ´ µ ¼ ¾ ª,
although this will play a minor role throughout this section.
The price or value of security , in state of nature , at time , will be denoted as
Ë ´ µ. We assume that these values are always nonnegative; in other words, they
are limited liability securities, because their owner’s ﬁnancial liability is limited to
the price paid for the security. Although we are using the symbol Ë , these assets need
not be stocks. They could be bonds, call options, or any of the other traded securities
discussed in Chapter 2.
The time0 prices of the securities are assumed to be strictly positive. Since
Ë ´¼ µ is the same for all ¾ ª, we simply denote this common value as Ë ´¼µ
and consider the row vector Ë ´¼µ Ë½ ´¼µ Ë¾ ´¼µ ¡ ¡ ¡ ËÆ ´¼µ Meanwhile, it is convenient to organize the time1 prices as the matrix ¾ ½
¾ Ë½ ´½ Ë ´½ ªµ Ë½ ´½
Ë½ ´½ Å µ Ë¾ ´½ Å µ .
.
. µ
µ Ë¾ ´½
Ë¾ ´½ ½
¾ .
.
. µ
µ ¡¡¡
¡¡¡
.
.
.
¡¡¡ ËÆ ´½
ËÆ ´½ ½
¾ ËÆ ´½ ¿ Åµ .
.
. µ
µ (5.2.1) The evolution of the market in the model is illustrated in Figure 5.1.
Investors select a portfolio of the assets at time ¼. The number of units of asset
held from time 0 to time 1 will be denoted by the numbers
½¾
Æ.
If is positive,
units of security are purchased. If is negative,
units of
security are sold short. We sometimes refer to the column vector ¾ ½
¾ ¿ (5.2.2) .
.
. Æ
as a trading strategy. Then the value of the corresponding portfolio at time 0 is Ë ´¼µ ½ Ë½ ´¼µ · ¾ Ë¾ ´¼µ · ¡ ¡ ¡ · Æ ËÆ ´¼µ (5.2.3) The time1 value of this same portfolio will depend upon the state of nature; if state
occurs, then the time1 value is
½ Ë½ ´½ µ · ¾ ´½ µ · ¡ ¡ ¡ · Æ ËÆ ´½ µ SINGLEPERIOD MODEL 161 ½
¾
¿ time 0 time 1
Fig. 5.1 Evolution of the Market which is the th component of the column vector ¾ Ë ´½ ªµ ½ Ë½ ´½
½ Ë½ ´½
½ Ë½ ´½ ½µ · ¾ Ë¾ ´½
¾µ · ¾ Ë¾ ´½ ½µ · ¡ ¡ ¡ ·
¾µ · ¡ ¡ ¡ · .
.
. Å µ · ¾ Ë¾ ´½ Å µ · ¡¡¡· Æ ËÆ ´½
Æ ËÆ ´½ ¿ ½µ
¾µ (5.2.4) Æ ËÆ ´½ Å µ Thus from the time0 perspective of the investors, the time1 value of the portfolio
corresponding to the trading strategy is the random variable whose outcomes are
the components of the vector Ë ´½ ªµ . Examples of singleperiod security models
will be found in Subsections 5.2.2 and 5.2.6.
Sometimes it is useful to designate security 1 as the bank account and stipulate
that it is of the form Ë½ ´¼µ ½ and Ë½ ´½ µ ½· for all ¾ ª where the nonnegative number is interpreted as the oneperiod interest rate or the
short rate. Here Ë½ ´½ µ is constant with respect to , because in this case the short
rate is fully known to the investors at time 0.
Subject to the assumptions that there are only one period and a ﬁnite number of
states of nature, is this model reasonable from the economic point of view? Not 162 NOARBITRAGE PRICING THEORY necessarily, for in the next subsection, it will be seen that there can exist arbitrage
opportunities.
5.2.2 Arbitrage and the Fundamental Theorem of Asset Pricing
The Random House Dictionary deﬁnes arbitrage as “the simultaneous purchase and
sale of the same securities, commodities, or foreign exchange in different markets to
proﬁt from unequal prices.” Our deﬁnition of arbitrage expresses this idea mathematically.
Deﬁnition 5.2.1 An Ö ØÖ ÓÔÔÓÖØÙÒ ØÝ is a trading strategy Ë ´¼µ ¼ and We say that a securities market model is
opportunities. Ë ´½ ¼ ªµ Ö ØÖ  Ö such that 1
(5.2.5) if there are no arbitrage Intuitively, a model admits an arbitrage opportunity if an investor can select a
portfolio that costs nothing now, pays something to the investor at the end of the
period in at least one state of nature, but never ends up with the investor having an
obligation to pay.
Clearly arbitrage opportunities are unreasonable from an economic standpoint,
for investors seeing them in real world would probably want to establish such large
positions that the prices would be affected in such a way that the arbitrage opportunities would be quickly eliminated. In other words, if investors were to ﬁnd a
trading strategy that started with no money and, without any risk of losing money,
had the potential of a positive return, then their buy and sell orders, together with the
principle of supply and demand, would cause the market prices to quickly change
and the arbitrage opportunity to disappear. Thus for our singleperiod model to have
a measure of economic realism, it will be necessary to assume that it is arbitragefree.
Example 5.2.1 Suppose Å ¾ Æ Ë ´½ ªµ Ë ´¼µ
½· ½½ and Ù ½· where Ù and are numbers such that
½
Ù. Thus Ë½ represents a bank account
with short interest rate
¼ whereas Ë ¾ represents a stock whose ﬁnal value is either
Ì
Ù (“up”) or
(“down”). Now consider whether the trading strategy
½ ¾ is
Let Ü Ý ¾ Ò denote Òdimensional vectors with Ü and Ý being
the th component of Ü and Ý . If Ü
¼,
½¾
Ò we write Ü
¼ and say that Ü is nonnegative.
If Ü is nonnegative and Ü
¼ for some , we write Ü ¼ and say that Ü is positive. If Ü is positive
¼ for all we write Ü
¼ and say that Ü is strictly positive. We write Ü Ý if Ü Ý ¼,
and Ü
Ü Ý if Ü Ý ¼, and Ü Ý if Ü Ý ¼. Corresponding deﬁnitions are made for inequalities
Ò Ü ¼ and Ò
, and . We also deﬁne Ò
Ü¾ Ò Ü ¼
· Ü¾
··
1 Notation and inequalities for vectors: SINGLEPERIOD MODEL an arbitrage opportunity. Firstly, we must have Ë ´¼µ
½· ¾ Secondly, we must have Ë ´½
´½ · µ ½ · Ù that is, ¼ ¼ that is ªµ
¼ ¾ ¼ 163 ´½ · µ ½ · and ¾ ¼ Finally, in order to have Ë ´½ ªµ
¼ at least one of these last two inequalities
must be strictly positive.
If ¾ ¼ and these three inequalities all hold, then since ½ ·
, it follows that
¼ and
½ ´½ · ¾ This means that, in fact, ½
On the other hand, if ¾ µ½ ½ ¾ ¼ that is, cannot be an arbitrage opportunity.
and all three inequalities hold, then ½ ¼ and ¾
¼ Moreover, since ´½ · µ ½ ¾ Ù Ù ½ This implies ½ ·
Ù
Ù¾ ¾ it follows that the third
inequality is strict, which means is actually an arbitrage opportunity.
Conversely, suppose ½ ·
Ù Then choose ½
½ and ¾ ´½ · µ Ù ½
so that the ﬁrst two inequalities will hold. Since Ù
the third inequality will hold
in a strict fashion, and thus will be an arbitrage opportunity.
In summary, an arbitrage opportunity will exist for this model if and only if
½·
Ù, that is, if and only if the bank account’s return is certain to be at least as
great as the stock’s return.
¾
Example 5.2.2 Suppose in Example 5.2.1, a stock with price 1.00 will increase to
1.20 or decrease to 0.80 after one period. If the oneperiod rate of interest is 10%,
then an investment of 1.00 in the bank account and 1.00 in the stock has current price
Ë ´¼µ ½ ½ and Ë ´½ ªµ ½ ½¼ ½ ¾¼ ½ ½¼ ¼ ¼ ¾ As seen by the preceding examples, the existence of arbitrage opportunities for the
singleperiod model depends on some algebraic calculations. Not surprisingly, these
calculations can be organized into a concise, algebraic statement. What is surprising,
however, is the fact that, as explained in Subsection 5.2.4, the resulting statement has
enormous implications for ﬁnancial theory and practice. The theorem to follow is
often called the fundamental theorem of asset pricing because it says that the absence
of arbitrage is equivalent to the existence of a state price vector that prices all traded
assets.
Deﬁnition 5.2.2 A ×Ø Ø ÔÖ Ú ØÓÖ is a strictly positive row vector 164
´ NOARBITRAGE PRICING THEORY ½µ ´ ¾µ ´ Å µ for which Ë ´¼µ
Equivalently, Ë ´½ (5.2.6) ªµ is a random variable for which Ë ´¼µ ¾ª ´ Ë µ ´½ µ ½ Æ A state price vector expresses the current prices of traded assets as a positive linear
combination of their uncertain cash ﬂows at time ½. If there are fewer than Å assets
with linearly independent payoff vectors at time ½, then Ë ´½ ªµ is not of full rank.
Hence there may exist more than one solution of (5.2.6).
Before presenting the theorem, it should be mentioned that the proof involves
securities which pay one unit in one state of nature and nothing in all others. These
are called ArrowDebreu securities and have already been discussed in Chapter 4.
For Ñ ½ ¾
Å , the ArrowDebreu security for outcome Ñ is a security that
at time 1 pays ½ if outcome Ñ occurs, and ¼ otherwise. In other words, the ArrowÌ
Debreu security for outcome Ñ has payoff vector Ñ
¼
¼½¼
¼
with the “1" in the Ñth position.
Given an arbitrary ArrowDebreu security with payoff vector Ñ at time 1, consider whether there exists a trading strategy such that Ë ´½ ªµ
Ñ . If so, then
we say and the corresponding portfolio replicate the payoff Ñ . Now in a particular market model, none, some, or all ArrowDebreu securities may be obtained from
linear combinations of the assets Ë ½ Ë¾
ËÆ . If all ArrowDebreu securities can
be replicated in this way, then any time1 payoff vector may also be replicated by
some trading strategy. This is because the set of vectors ½
Å forms a basis
of Å . These ideas will be utilized in the proof of the theorem.
Theorem 5.2.3 (Fundamental Theorem of Asset Pricing) The singleperiod securities market model is arbitragefree if and only if there exists a state price vector.
Proof Suppose that there exists a state price vector . If Ë ´½ ªµ
¼, then by
(5.2.6), Ë ´¼µ
Ë ´½ ªµ ¼. Hence, all portfolios with positive payoff vectors
at time 1 have positive prices at time 0, and the model is arbitragefree.
Conversely, suppose that the model is arbitragefree and that there are Å assets
with linearly independent payoff vectors at time 1 (a proof for the case where fewer
than Å of the assets have linearly independent payoff vectors will be deferred until
Subsection 5.2.7). Thus Ë ´½ ªµ has full rank and Æ
Å . There is no loss of
generality in assuming that the ﬁrst Å assets are the linearly independent ones. We
partition the vector Ë ´¼µ as follows: Ë ´¼µ
where Ä´µ . Ä´¼µ .. Ê´¼µ Ë½ ´¼µ Ë¾ ´¼µ ¡¡¡ ËÅ ´¼µ SINGLEPERIOD MODEL and Ê´¼µ ËÅ ·½ ´¼µ ËÅ ·¾ ´¼µ Correspondingly, we partition the matrix Ë´½ ªµ
¾ where Ä´½µ Ë½ ´½ ½µ
Ë½ ´½ ¾µ
Ë½ ´½ ¾ and Ê´½µ .
.
. ËÅ ·½ ´½ ½µ
ËÅ ·½ ´½ ¾µ
.
.
. ËÅ ·½ ´½ Ä´½µ ... Ê´½µ Ë¾ ´½ .
.
. ¡¡¡ .. Åµ
.
.
. ËÅ ·¾ ´½ .
.
. . ËÅ ´½ ¡¡¡ Åµ ¿ ËÅ ´½ ½µ
ËÅ ´½ ¾µ ¡¡¡ ËÅ ·¾ ´½ ½µ
ËÅ ·¾ ´½ ¾µ Åµ ËÆ ´¼µ ¡¡¡ Ë´½ ªµ as Ë¾ ´½ ½µ
Ë¾ ´½ ¾µ Åµ 165 ËÆ ´½ ½µ
ËÆ ´½ ¾µ ¡¡¡
¡¡¡ .. Åµ
.
.
. . ËÆ ´½ ¡¡¡ ¿ Åµ Ä´½µ is invertible, we can deﬁne the Å dimensional row vector
Ä´¼µÄ´½µ ½.
Now for an arbitrary ArrowDebreu security having time1 payoff Ñ , consider
the trading strategy
´Ä´½µ ½ Ñ µÌ ¼
¼ Ì , which calls for a nontrivial Because position in the ﬁrst Å securities but no position in any of the last Æ
Note that Ë´½ ªµ Ä´½µÄ´½µ ½ Ñ Å securities. Ñ so replicates the payoff Ñ . Since Ñ ¼ and there are no arbitrage opportunities,
it follows that this replicating portfolio’s time0 price Ë´¼µ Ä´¼µÄ´½µ ½ Ñ Ñ is strictly positive. Hence the Ñth component of the vector is strictly positive.
Since Ñ is arbitrary, we conclude that every component of is strictly positive.
Now, Ë´½ ªµ Ä´¼µÄ´½µ ½ Ä´½µ ... Ê´½µ Ä´¼µ ... Ä´¼µÄ´½µ ½Ê´½µ
If Æ
Å Ë ´¼µ Ä´¼µ (and there is no Ê´½µ); hence we have found a state price
vector. For Æ
Å , since the columns of Ä´½µ form a basis of Å , there exists
an Å ´Æ Å µ matrix Ã such that Ê´½µ
Ä´½µÃ . Because the model is
¢ arbitragefree, the prices of the Æ Å redundant securities must equal the cost of
the unique portfolio of the linearly independent assets that produce the same payoffs
at time ½. Algebraically, this is to say Ê´¼µ Ä´¼µÃ Ä´½µÃ Ê´½µ 166 NOARBITRAGE PRICING THEORY We ﬁnally get Ë ´½ . . Ä´½µ .. Ê´½µ
.
Ä´¼µ .. Ê´¼µ ªµ Ä´½µ .. Ê´½µ
Ë ´¼µ
¾ For a general singleperiod model, it is difﬁcult to provide intuition for why the
absence of arbitrage opportunities should imply the existence of a state price vector.
Easier to understand is the important special case of the following subsection. Before
turning to that, however, it is worthwhile to reconsider Example 5.2.1.
Example 5.2.1 (continued) Consider whether there exists a state price vector, that
is, a strictly positive solution ´ ½ ¾µ to the system (writing
for ´ µ)
½ ´½ · µ ½ · ´½ · µ ¾ ½ Ù ½· ¾ The solution is easily found to be
½ ½· ´½ · µ ´ Ù µ ¾ Ù ´½ · µ µ ´½ · µ´Ù Since we have already assumed that
¼ and Ù
½
¼, we conclude that
there exists a state price vector if and only if Ù ½ · . Note that this is precisely
the condition which we earlier found to be equivalent to the absence of arbitrage. ¾
Example 5.2.2 (continued) In Example 5.2.1, the state price vector has elements
½
¾ ¼ ¼ ¼ ¼µ
½ ¾¼ ½ ½¼
½ ½´½ ¾¼ ¼ ¼µ
½ ½¼ ¿¼ ½ ½´½ ¾¼ ½¼ ¾ 5.2.3 RiskNeutral Probability Measures
Throughout this subsection, assume security Ë ½ is a bank account, as introduced in
subsection 5.2.1. In other words, assume Ë½ ´¼µ
½ and Ë½ ´½ µ
½ · for all
¾ ª, where the number ¼ is the oneperiod interest rate.
Now given a state price vector (see Deﬁnition 5.2.2), we can always deﬁne the
quantities
É´ µ
´½ · µ ´ µ
¾ª
(5.2.7) SINGLEPERIOD MODEL 167 and note all these quantities are strictly positive. Thus (5.2.6) becomes Ë
In particular, for É´ µË ´¼µ µ for ½· ¾ª
½ ´½ ½ Æ (5.2.8) we have Ë½ ´¼µ ½ ¾ª É´ µ (5.2.9) Hence it is convenient to interpret É´ µ as a probability associated with the state
of nature , in which case É should be interpreted as a probability measure on the
sample space ª.
Another thing to notice is that the quantity Ë ´½ µ ´½ · µ can be interpreted as
the discounted (to time 0) time1 price of security if the state of the world is .
Hence the righthand side of (5.2.8) can be interpreted as the expected discounted
price of security , only where the expectation is computed with respect to the
probability measure É, not the original probability measure È .
In summary, starting with a state price vector, we have constructed what is called
a riskneutral probability measure.
Deﬁnition 5.2.4 A Ö × Ò ÙØÖ Ð ÔÖÓ
sure É on ª such that
a. É´ µ ¼ for all Ð ØÝ Ñ ×ÙÖ is a probability mea ¾ª
¾ b. Equation (5.2.8) holds for Æ. Thus a riskneutral probability measure is a strictly positive probability measure
under which the expected discounted price of any security equals the initial price of
the same security.
What about the converse? Starting with a riskneutral probability measure É, can
one construct a state price vector ? The answer is yes, as will now be explained. By
deﬁning with (5.2.7), the requirement that the probabilities sum to one becomes
½ ¾ª É´ µ ¾ª ´ µ´½ · µ ¾ª ´ µË½ ´½ µ which is the ﬁrst row of (5.2.6). Furthermore, equation (5.2.7) for
¾ becomes
the th row of (5.2.6). Hence when there is a bank account, the existence of a state
price vector is equivalent to the existence of a riskneutral probability measure.
Our conclusions can be summarized as follows:
Theorem 5.2.5 Suppose security 1 is a bank account. Then the following are equivalent:
a. The singleperiod model is arbitragefree.
b. There exists a state price vector. 168 NOARBITRAGE PRICING THEORY c. There exists a riskneutral probability measure.
Example 5.2.1 (continued) It is readily seen that security 1 is a bank account . By
(5.2.7) and the earlier results we thus have (denoting É´ Ñ µ ÕÑ ) Õ½ Ù ½· and Õ¾ Ù ´½ ·
Ù µ Note that Õ½ · Õ¾ ½. Moreover, both quantities are strictly positive if and only if
Ù ½ · , the condition we already obtained for the absence of arbitrage. Finally,
note that for the expected discounted price of security 2 we have Õ½Ë¾ ´½ ½· ½ µ · Õ¾Ë¾ ´½ ¾µ
½· which is (5.2.8) for ½ · Ù · Ù ´½ · µ
Ù ½·
Ù ½·
Ù ½ Ë¾ ´¼µ
Ù ¾. ¾ Example 5.2.2 (continued) In Example 5.2.1, the riskneutral probabilities are ½ ½¼ ¼ ¼ ¿
½ ¾¼ ¼ ¼
½ ¾¼ ½ ½¼ ½
½ ¾¼ ¼ ¼ Õ½
Õ¾ From this it can be seen that the discounted expected value of the stock cash ﬂows is ½¿
½
½ ½¼ ½ ¾¼ · ¼ ¼ ½ ¼¼
¾ which is the current price of the stock.
5.2.4 Valuation of Cash Flows A principal purpose of securities market models is to derive the time0 value of future
uncertain cash ﬂows. In the case of singleperiod models, such a cash ﬂow can be
simply described by a random variable representing the time1 payment. In other
words, ´ µ is the time1 payment that occurs if ¾ ª turns out to be the state of
the world. Sometimes is called a contingent claim, and it can readily be used to
model European options, as will be seen in Chapter 6. The aim of this subsection
is to consider the time0 value of the cash ﬂow
in the context of arbitragefree
singleperiod models.
The ﬁrst important principle is the idea of cash ﬂow replication. We say that the
trading strategy replicates the cash ﬂow vector
´ ½µ
´ Å µ Ì if Ë ´½ ªµ SINGLEPERIOD MODEL 169 By considering the components of this vector equation, it is equivalent to say that the
trading strategy replicates the cash ﬂow random variable if Ë ´½ µ ´µ all ¾ª We also say that the cash ﬂow is attainable if it can be replicated by some trading
strategy. Thus if a cash ﬂow is attainable, then there exists a trading strategy such
that the time1 value of the corresponding portfolio coincides with the cash ﬂow
in every state of nature ¾ ª. Note that this concept was introduced in Subsection
5.2.2, where the cash ﬂows were ArrowDebreu payoffs Ñ .
The second important principle is the idea of arbitrage pricing: if the cash ﬂow
is attainable, then its time0 value must equal the time0 value of its replicating
portfolio. This is because the introduction of a cash ﬂow is like adding to the model a
new security having time1 price and some time0 price, say . If is not equal to
the time0 value of the replicating portfolio, then the augmented singleperiod model
will not be arbitragefree.
To see this, suppose
Ë ´¼µ , that is, you can buy or sell at time 0 a position
in the cash ﬂow at a price that is greater than the initial value of the replicating
portfolio. In this case the arbitrageur would sell the cash ﬂow at time 0, thereby
collecting dollars and promising to deliver ´ µ dollars at time 1 if state ¾ ª
occurs. The arbitrageur would also take a long position in the replicating portfolio,
costing Ë´¼µ dollars but leaving a net proﬁt of Ë ´¼µ
¼ dollars. At time
1, the arbitrageur will have the obligation to pay , but this will be precisely equal
to Ë ´½ ªµ , the time1 value of his portfolio, so the net obligation will be precisely
zero. The arbitrageur will thus have made a proﬁt of Ë ´¼µ
¼ without needing
initial funds and without any risk of losing money. Thus the condition
Ë ´¼µ
is unreasonable from the economic point of view. Similarly (this is left to the reader
for veriﬁcation), the condition
Ë ´¼µ is unreasonable from the economic point
of view. Only when
Ë ´¼µ will the augmented singleperiod model be free of
arbitrage opportunities.
These results are summarized as follows:
Theorem 5.2.6 In an arbitragefree, singleperiod model, the time0 value of an
attainable cash ﬂow is equal to the time0 value of the portfolio which replicates
.
But there is more, a remarkable consequence of the Fundamental Theorem of Asset
Pricing. Consider the state price vector . If Ë ´½ ªµ
, then Ë ´½ ªµ
However, we also have Ë ´¼µ
Ë ´½ ªµ from (5.2.6), hence Ë ´¼µ
This says that the time0 price of the replicating portfolio, and thus the time0 value
of the cash ﬂow , is equal to
, a quantity that can be computed by knowing
the state price vector without any knowledge of the replicating trading strategy
itself. 170 NOARBITRAGE PRICING THEORY ¼, then by (5.2.7) the If security 1 is a bank account with short interest rate
time0 value of the cash ﬂow is ¾ª É´ ´µ ´µ ¾ª µ ´µ
½· This says that the time0 value of
is equal to the expected discounted value of
, with the expectation computed with the riskneutral probability measure. This
principle, ﬁrst encountered in Section 3.7, is called riskneutral valuation and has
enormous implications for the valuation of options, as will be seen in Chapter 6.
We conclude this subsection with a summarizing theorem, another look at our
familiar example, and some ﬁnal remarks. Theorem 5.2.7 In an arbitragefree, singleperiod model, the time0 value of an
attainable cash ﬂow
is equal to
, where
is the state price vector. If, in
addition, security 1 is a bank account with short interest rate
¼, then
É´ ¾ª µ ´µ
½· where É is the riskneutral probability measure.
Example 5.2.1 (continued) Suppose the model is arbitragefree (i.e. Ù ½ · ) and
consider an arbitrary cash ﬂow . Easy algebraic equations will verify that Ë ´½ ªµ ´ ¾µ Ì ´ ½µ will always have the solution
Ë ´½ ªµ ½ , so every possible cash ﬂow will
be attainable in this example. Since Ë ´¼µ
½ ½ , the time0 value of will be
½ ½ Ë´½ ªµ ½
In view of Theorem 5.2.7, this will also equal ½· ´½ · µ´Ù µ
½·
Ù ´ ½µ ½ ½· Õ ´ ½µ · ´ ½µ
·
½·
· Õ¾ Ù ´ ¾µ
½· ´½ · µ
´½ · µ´Ù µ
Ù ´½ ·
Ù µ ´ ¾µ ´ ¾µ
½· ¾ Example 5.2.2 (continued) In Example 5.2.1, if we wish to price a call option with
a strike price of 1.10, then the cash ﬂows are ´½µ ¼ ½¼
´¾µ
¼ ¼¼ and the
option price is
¿ ¼ ½¼ ½ ¼ ¼¼
¿
·
½ ½¼
½ ½¼ SINGLEPERIOD MODEL Note that Ë ´½ ªµ 171 is ½ ½¼ ½ ¾¼
½ ½¼ ¼ ¼ ½
¾ ¼ ½¼
¼ ¼¼ yielding ½ ¾ ½½ and ¾ ½ This means that the option is replicated by
borrowing ¾ ½½ and investing ½ in the stock. This results in a net investment of
¿
the price of the option. At the end of the period ´¾ ½½µ½ ½¼ ¼ ¾¼ is owed. If
the stock increases in value to ´½ µ½ ¾¼ ¼ ¿¼ After repayment of the borrowing
the net cash ﬂow is 0.10. Similarly, if the stock goes down, the ½ in stock is worth
only ¼ ¾¼, with a net cash ﬂow of 0. This demonstrates that the combination of
borrowing and investment in the stock replicates the call option on the stock.
¾
The reader should bear in mind that the preceding theory for valuing cash ﬂows
applies only for cash ﬂows that are attainable. If a cash ﬂow cannot be replicated
by some trading strategy, then arbitrage arguments cannot be used to determine the
cash ﬂow’s value. Instead, the equilibrium pricing methods of Chapter 4 must be
employed.
How does one check whether a cash ﬂow is attainable? There is rarely a shortcut;
usually, one must investigate whether Ë ´½ ªµ
has a solution . However, as
illustrated with the preceding example, sometimes a singleperiod model is such that
every cash ﬂow is attainable. This is the subject of the following subsection.
5.2.5 Completeness in the SinglePeriod Model
Deﬁnition 5.2.8 An arbitragefree market model is said to be complete if for every
¾ Å there exists some trading strategy ¾ Æ such that
cash ﬂow vector Ë ´½ ªµ Thus in complete markets one can construct a portfolio that replicates any prescribed
contingent cash ﬂow, in which case every cash ﬂow can be priced by arbitrage. The
following theorem provides a useful necessary and sufﬁcient condition for the model
to be complete.
Theorem 5.2.9 Suppose a singleperiod model is arbitragefree. Then the model is
complete if and only if there is a unique state price vector.
Proof Suppose the model is complete but there are two distinct state price vectors,
say and . Since every cash ﬂow is attainable, there must exist some attainable
cash ﬂow such that
. But this contradicts Theorem 5.2.6, for cannot
have two distinct time0 values in an arbitragefree model.
Conversely, suppose that the model is not complete. Then there is some
¾Å
Æ such that Ë ´½ ªµ
for which there is no ¾
. Thus the rank of Ë ´½ ªµ
is strictly smaller than Å . This in turn means that the Å rows of that matrix are
linearly dependent, so there is a nonzero ¾ Å such that Ë ´½ ªµ ¼. Since 172 NOARBITRAGE PRICING THEORY the model is arbitragefree, there is at least one state price vector, say
number
¼ so small that ·
¼, we get ´ · µË ´½ ªµ Ë ´½ ªµ · Ë ´½ ªµ Ë ´½ ªµ . For any real Ë ´¼µ
¾ showing that the number of state price vectors is inﬁnite. The proof of this theorem also shows that, when the securities market model is
complete, the price of the ArrowDebreu security for outcome is equal to ´ µ.
Combining the Fundamental Theorem of Asset Pricing with this theorem gives the
result: A singleperiod model is arbitragefree and complete if and only if there exists
a unique state price vector.
If, in addition, security 1 is a bank account, then we have the following result, the
proof of which is left to the reader.
Corollary 5.2.10 Suppose security 1 in a singleperiod model is a bank account.
Then this model is arbitragefree and complete if and only if the riskneutral probability measure is unique.
We see that in a complete arbitragefree market each contingent cash ﬂow has a
unique price. The next section gives some examples with incomplete markets.
5.2.6 Applications of the Fundamental Theorem of Asset Pricing
For a given securities market model one may compute all state price vectors by
determining all strictly positive solutions of Ë ´¼µ Ë ´½ ªµ As long as the securities market model is arbitragefree, the Fundamental Theorem
of Asset Pricing assures us that at least one strictly positive solution of this equation
exists. We can try to ﬁnd all possible state price vectors.
Example 5.2.3 Let us compute all state price vectors for the securities market model ¾ Ë ´¼µ ½½ Ë ´½ ªµ ¾¼
¾¼
¾
¾ ¿ This market model consists of two assets, each with initial price 1. There are four
possible state of nature, one corresponding to each row of Ë ´½ ªµ. A solution
of the linear equation Ë ´¼µ
Ë ´½ ªµ is equivalent to
½¾¿
solving ¾ ½·¾ ¾·¾ ¿·¾
¿· ½
½ SINGLEPERIOD MODEL 173 As this is an underdetermined system, its solution will involve two parameters. One
parameterization of the solutions is as follows: Ø
¿ ½ Ø ¾ × ½ ½ × The requirement that a state price vector be strictly positive imposes the reﬁnements
that ¼ Ø ½ and ¼ × ½ . Thus the collection of all state price vectors for
this model is the set À of all vectors of the form ½ × × ½ Ø Ø for × and
Ø satisfying ¼ × ½ and ¼ Ø ½ that is ¬
½ × × ½ Ø Ø ¬ ´× Øµ ¾ Ï
¬
¬ À
where Ï ¬
¬
´× Øµ ¬ ¼ × ½ ¼ Ø ½
¬
¾ is the parameter region for the model. Example 5.2.4 Consider the following securities market model, which differs only
slightly from the preceding example: ¾ Ë ´¼µ ½½ Ë ´½ ªµ ¾¼
¾¼
¾
¾¿ ¿ We may compute all strictly positive solutions of the equation Ë ´¼µ
Ë ´½ ªµ
to characterize the state price vectors in this model. Proceeding as in the preceding
example, we ﬁnd that the set of all state price vectors for this model is À
where Ï ¬
½ × Ø × ½ ¿Ø Ø ¬ ´× Øµ ¾ Ï
¬
¬ ¬
¬
´× Øµ ¬ ¼ × ¼ Ø ½ × · Ø
¬
¿ ½
¾ Suppose that a new security is to be introduced into an existing incomplete securities market model. If the resulting securities market model is to remain arbitragefree,
we know from the Fundamental Theorem of Asset Pricing that there must still exist
a state price vector. Such a state price vector must be consistent with the original 174 NOARBITRAGE PRICING THEORY securities market model, so a state price vector for the augmented securities market
model must be chosen from the collection, denoted , of state price vectors for the
original model. Consequently, a new security having time1 price vector may be
introduced in an arbitragefree fashion if and only if its time0 price equals
for
some ¾ .
For instance, over what range of time0 prices would the security with time1 price
vector
½¼¼½Ì À À be introduced into the securities market model of Example 5.2.4 such that the resulting
model is arbitragefree? The answer is the set of current prices
that is compatible
with the set ; that is, ¾ . Since À À ½ ½ × ½ Ø · ¼× · ¼ ½ ¿ Ø · ½Ø
½ × · ¿Ø the range of possible time0 prices for this new security is then ¬
½ × · ¿ Ø¬ ´× Øµ ¾ Ï
¬
¬ Some simple calculations then show that ¬
½ × · ¿ Ø ¬ ´× Øµ ¾ Ï
¬
¬ ¼½
¾ Therefore, the new security with time1 price vector may be introduced at any initial
price in ´¼ ½ ¾µ, and the resulting securities market model will be arbitragefree.
5.2.7 Proof of Theorem 5.2.3 (completed)
The proof of the “only if" part of Theorem 5.2.3 given in Subsection 5.2.2 assumed
that there are
assets with linearly independent payoff vectors at time 1. Here we
will provide a proof for the other case, namely, where fewer than
of the assets
have linearly independent payoffs.
One difﬁculty in proving this result lies in constructing a state price vector when
the securities market model is arbitragefree. We now discuss this construction. It
is based on adding assets to the model to obtain an extended arbitragefree securities
market model which will ultimately have full rank, thereby placing us back in the
domain of the earlier proof.
Recall that arbitrage means there exists a trading strategy or portfolio such that Å Å Ë ´½µ Ë ´¼µ ¼ ...
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 Spring '08
 Tang,Q
 Linear Algebra, Arbitrage, bank account, state price vector

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