Chapter_5

# Chapter_5 - 5 No-Arbitrage Pricing Theory 5.1 INTRODUCTION...

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Unformatted text preview: 5 No-Arbitrage 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 discrete-time and discrete-state 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 continuous-time case, where probability cannot be avoided. Relevant probabilistic concepts will be introduced as needed. 5.2 SINGLE-PERIOD 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 time-0 values but the time-1 values are random variables. These random variables 159 160 NO-ARBITRAGE 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 time-0 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 time-1 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 time-1 value of this same portfolio will depend upon the state of nature; if state occurs, then the time-1 value is ½ Ë½ ´½ µ · ¾ ´½ µ · ¡ ¡ ¡ · Æ ËÆ ´½ µ SINGLE-PERIOD 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 time-0 perspective of the investors, the time-1 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 single-period 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 one-period 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 NO-ARBITRAGE 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 single-period model to have a measure of economic realism, it will be necessary to assume that it is arbitrage-free. 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 Ò-dimensional 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 non-negative. If Ü is non-negative 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: SINGLE-PERIOD 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 one-period 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 single-period 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 ´ NO-ARBITRAGE 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 Arrow-Debreu securities and have already been discussed in Chapter 4. For Ñ ½ ¾ Å , the Arrow-Debreu 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 Ñ-th position. Given an arbitrary Arrow-Debreu 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 Arrow-Debreu securities may be obtained from linear combinations of the assets Ë ½ Ë¾ ËÆ . If all Arrow-Debreu securities can be replicated in this way, then any time-1 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 single-period securities market model is arbitrage-free 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 arbitrage-free. Conversely, suppose that the model is arbitrage-free 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 SINGLE-PERIOD 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 Arrow-Debreu security having time-1 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 time-0 price Ë´¼µ Ä´¼µÄ´½µ ½ Ñ Ñ is strictly positive. Hence the Ñ-th 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 ¢ arbitrage-free, 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 NO-ARBITRAGE 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 single-period 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 Risk-Neutral 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 one-period 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) SINGLE-PERIOD 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) time-1 price of security if the state of the world is . Hence the right-hand 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 risk-neutral 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 risk-neutral 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 risk-neutral 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 risk-neutral 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 single-period model is arbitrage-free. b. There exists a state price vector. 168 NO-ARBITRAGE PRICING THEORY c. There exists a risk-neutral 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 risk-neutral 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 time-0 value of future uncertain cash ﬂows. In the case of single-period models, such a cash ﬂow can be simply described by a random variable representing the time-1 payment. In other words, ´ µ is the time-1 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 time-0 value of the cash ﬂow in the context of arbitrage-free single-period 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 SINGLE-PERIOD 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 time-1 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 Arrow-Debreu payoffs Ñ . The second important principle is the idea of arbitrage pricing: if the cash ﬂow is attainable, then its time-0 value must equal the time-0 value of its replicating portfolio. This is because the introduction of a cash ﬂow is like adding to the model a new security having time-1 price and some time-0 price, say . If is not equal to the time-0 value of the replicating portfolio, then the augmented single-period model will not be arbitrage-free. 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 time-1 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 single-period model be free of arbitrage opportunities. These results are summarized as follows: Theorem 5.2.6 In an arbitrage-free, single-period model, the time-0 value of an attainable cash ﬂow is equal to the time-0 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 time-0 price of the replicating portfolio, and thus the time-0 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 NO-ARBITRAGE PRICING THEORY ¼, then by (5.2.7) the If security 1 is a bank account with short interest rate time-0 value of the cash ﬂow is ¾ª É´ ´µ ´µ ¾ª µ ´µ ½· This says that the time-0 value of is equal to the expected discounted value of , with the expectation computed with the risk-neutral probability measure. This principle, ﬁrst encountered in Section 3.7, is called risk-neutral 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 arbitrage-free, single-period model, the time-0 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 risk-neutral probability measure. Example 5.2.1 (continued) Suppose the model is arbitrage-free (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 time-0 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 SINGLE-PERIOD 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 single-period model is such that every cash ﬂow is attainable. This is the subject of the following subsection. 5.2.5 Completeness in the Single-Period Model Deﬁnition 5.2.8 An arbitrage-free 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 single-period model is arbitrage-free. 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 time-0 values in an arbitrage-free 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 non-zero ¾ Å such that Ë ´½ ªµ ¼. Since 172 NO-ARBITRAGE PRICING THEORY the model is arbitrage-free, 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 Arrow-Debreu security for outcome is equal to ´ µ. Combining the Fundamental Theorem of Asset Pricing with this theorem gives the result: A single-period model is arbitrage-free 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 single-period model is a bank account. Then this model is arbitrage-free and complete if and only if the risk-neutral probability measure is unique. We see that in a complete arbitrage-free 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 arbitrage-free, 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 SINGLE-PERIOD 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 arbitrage-free, 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 NO-ARBITRAGE 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 time-1 price vector may be introduced in an arbitrage-free fashion if and only if its time-0 price equals for some ¾ . For instance, over what range of time-0 prices would the security with time-1 price vector ½¼¼½Ì À À be introduced into the securities market model of Example 5.2.4 such that the resulting model is arbitrage-free? 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 time-0 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 time-1 price vector may be introduced at any initial price in ´¼ ½ ¾µ, and the resulting securities market model will be arbitrage-free. 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 arbitrage-free. We now discuss this construction. It is based on adding assets to the model to obtain an extended arbitrage-free 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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