mm BBH B BEBR FACULTY WORKING PAPER NO. 90-1650 Non-Existence and Inefficiency of Equilibria with American Options and Convertible Bonds Charles M. Kahn Stefan Krasa ■yd 5 1990 College of Commerce and Business Administration Bureau of Economic and Business Research University of Illinois Urbana-Champaign BEBR FACULTY WORKING PAPER NO. 90-1650 College of Commerce and Business Administration University of Illinois at Urb ana-Champaign April 1990 Non-Existence and Inefficiency of Equilibria with American Options and Convertible Bonds Charles M. Kahn Department of Economics University of Illinois, Urb ana-Champaign and Stefan Krasa Department of Economics University of Illinois, Urb ana-Champaign and University of Vienna Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/nonexistenceinef1650kahn ABSTRACT We analyze three different examples of economies with incomplete financial markets, in the first model we consider a bond and a convertible bond and a convertible bond, and in the second model a stock and an American put option on the stock. Although there is only one commodity and asset payoffs therefore do not depend on spot prices, we derive robust non-existence of equilibria in both cases. In the last example we consider American call options with nominal striking prices. We show that in equilibrium the assets can never span. The Arrow-Debreu allocation cannot be implemented and the equilibrium in inefficient. This example is also robust. 1 Introduction The general equilibrium model with incomplete markets has been used to analyze a variety of financial contracts: real assets such as stocks and com- modity forward contracts, nominal assets such as bonds — and some types of derivative assets, most notably European options. This paper investi- gates the existence and efficiency of equilibrium for two kinds of assets which have not yet been analyzed: Convertible bonds and American options. We provide a robust example of non-existence of equilibrium for each of these assets, even in a one-commodity economy. We also show that even if there are potentially enough options (with nominal striking prices) to implement the complete market equilibrium, efficient equilibria need not exist. Both of these results are in sharp contrast to the results for other types of assets. The literature on existence of equilibrium in incomplete markets is vast. Although it has been shown that in models with purely financial assets com- petitive equilibria always exist, 1 Hart (1974) shows that equilibria do not always exist in models with real assets. However, Hart's example is not robust with respect to perturbations of the endowments of the consumers; Magill and Shafer (1985) show that if there are sufficient assets to potentially span the market then equilibria exist for a generic economy. The equilibrium allocations coincide with the equilibrium allocations of the complete Arrow- Debreu Market. Duffie and Shafer (1985) show that generic existence also holds if the market is effectively incomplete. In these models the non-existence is caused by a discontinuity of the excess demand function. This discontinuity is due to a change of the rank of the payoff matrix at prices at which some of the assets become redundant. The generic existence argument relies on the fact that with real assets the full rank of the payoff matrix can be restored by small perturbations of the endowments and of the asset structure. 1 see, for example, Duffie (1987), Werner (1985), and Cass (1984) This insight does not apply to models with European options. Polemar- chakis and Ku (1986) derive a robust non-existence example for an economy with a European put and a European call option. Like Hart's example, the Polemarchakis and Ku example depends on an economy in which there are multiple commodities. It is crucial that the value of the option varies with the relative value of prices on future spot markets: Geanakoplos and Pole- marchakis (1986) show that if assets pay off in the same commodity bundle in all states then equilibria always exist. The first example in this paper involves a convertible bond, and the second example involves an American put option. In contrast to Geanakoplos and Polemarchakis, we derive robust non-existence of equilibria, even though both examples are one-commodity economies. In both examples, the single commodity also serves as a numeraire. The assumption that there exists a numeraire commodity is essential for all exam- ples of non-existence of equilibria with options. As Krasa and Werner (1989) show, when European options have nominal striking prices, equilibria exist generically. The third example in this paper considers an economy with American call options with a nominal striking price. 2 There are three time periods and six states of nature. Given the structure of uncertainty there would be enough assets to dynamically complete the market. If the options where European options Krasa and Werner (1989) would show that we can in fact implement the Arrow-Debreu equilibrium. We show that this is not the case for American options. The only equilibria which exist in our example are equilibria where some of the options are redundant. The equilibrium allocation cannot coincide with the equilibrium allocation of the complete 2 This is again a one-commodity economy. Admittedly, it is odd to consider fiat money in a one-commodity world, but it simplifies the analysis, and demonstrates that the resul- tant inefficiency is fundamentally different from the inefficiency in the Hart example. The example can be generalized to a multicommodity example. Arrow-Debreu market and is therefore inefficient. Again this example is robust. Thus the result contrasts with earlier results by Ross (1976) and Mc- Manus (1984) showing that it is always possible to complete the market if there as many European options as there are states of nature. The differ- ence arises because the assets we consider allow a consumer who is long to decide between alternative payoffs before all the uncertainty of the model is resolved. With convertible bonds and American options this conversion or early exercise decision affects the actual span of the assets and therefore also the state prices. The actual decision of the consumers in turn depends on the price of the assets and consequently on the state prices which as we just argued depend on the decisions of the consumers. With European options a consumer who is long also makes a decision about exercising the option, but the decision does not affect the span. This difference drives all three exam- ples and therefore serves as a new source of non-existence and inefficiency of competitive equilibria with incomplete financial markets. 2 Non-existence with convertible bonds In the following section we give an example of a one-commodity economy with a convertible bond in which there is no equilibrium. This non-existence result is robust in the sense that small perturbations of the endowments and of the asset structure do not lead to the existence of an equilibrium. We proceed by showing this non-existence for a specific example and then give the argument that non-existence also has to hold for slightly perturbed economies by continuity. There are two time periods t — 0,1. At t = 1 there are two states of nature, denoted a, 6. The state of nature is not known at t = 0. There is a single consumption commodity in each state of nature. There is no consumption at t = 0; however there are two assets available for trade. The first asset is a riskless bond which pays of 1 unit of account independent of the state. The second asset is a convertible bond. A holder of a unit of the convertible bond can, at his choice, turn his convertible bond into a unit of the riskless bond or into a unit of risky asset. We assume a unit of risky asset pays 1.9 units of the commodity in state a and in state 6. 3 If the consumer chooses the risky payment we say that he converts the bond. The decision to convert the bond is made at t = 0. Agents are permitted to convert any fraction of their convertible bond holdings. There are two consumers /, J. The preferences of both consumers can be described by the Cobb-Douglas utility function u(x a ,x 6 ) = x a Xb. Let w 1 — (9, 11), and w J = (11,9) be the initial endowment of the two consumers. In the definition of a competitive equilibrium which we present below we also have to incorporate the decision of the consumers regarding conversion. Such a decision can only be made by an agent who is long in the convertible bond. Agents who are short have to expect a ratio of conversion which has to equal to true rate of conversion in equilibrium. We denote this expected rate by f , and the rate of conversion chosen by the agent i who is long by r\ We introduce the following notation: Let 9* be the portfolio choice of consumer z, an element of R 2 . The first component denotes holdings of the riskless bond and the second component denotes holdings of the con- vertible bond. (Negative numbers of course indicate short holdings). Let + = ((#i) + , (#2) + )> Le. we take the positive part of every component of 0. In a similar way we define 9~ to be the negative part of 9 taken for every component. Hence, 9 = 9 + + 0~. Let V(t) denote the payoff* matrix of all 3 For the robustness of the result we allow perturbations of the convertible bond of the following kind: The amount of bond the agent can get can be perturbed as well as the risky payoffs. This basically excludes the possibility that the payment of the bond and of the unconverted convertible bond are changed independent of each other. assets given the conversion rate r. Hence ™-(I TV)- An equilibrium is then a consumption-portfolio allocation (x\ 6 l ) € #5- x i? 2 , i = /, J; a system of asset prices 7r £ i? 2 , a total rate of conversion of the bond f , and conversion decisions t* of the consumers satisfying: (i) for every i, (x\ 6\ r') maximizes u'(:r) subject to the budget constraints 7T0 <0 x ', £,=/,j *' = 0, £ t =/,j7-«(^)+ = f E,=/.j(^) + . The last equation is the requirement that the actual rate of conversion equals the expected rate of conversion. To prove that no competitive equilibrium exists for the economy we have described, we proceed as follows: We first assume that there exists an equilibrium where a part of the bond is converted, i.e. where f ^ 0. In such a case the payoff matrix has full rank and consequently the set of equilibrium allocations has to coincide with the equilibrium allocations of the complete Arrow-Debreu economy. We then show that in an Arrow Debreu equilibrium it is better for the agent who is long in the convertible bond not to convert, i.e. to choose r* = 0. Therefore this cannot be an equilibrium. Thus, if there exists an equilibrium then the bond cannot be converted. The convertible bond is therefore redundant. However, it turns out that if there were such an equilibrium, one of the agents would be better of if he held a positive amount of bond converted into stock. Hence this cannot be an equilibrium either. We now show the first step. Clearly the only Arrow-Debreu equilibrium allocation is given by x 1 = x J = (10, 10) with equilibrium prices p a = p b = 1. Hence, consumer / has to finance the net trade (1,-1). This requires holding 6 1 = — 1 units of bond, 4 and 6 2 — |§ units of bond converted into stock. If consumer / now decides not to convert then his net trade is given by (^, —). Since consumer / will not convert; hence the Arrow-Debreu allocation cannot be an equilibrium (See figure la). ft*.«*J 0. At w 1 = (9,11) the marginal rate of substitution is ~. Since ^ > 09 this i m ph es that there exists a A > such that u 7 ((9, 11) + A(0.9, —1)) > u 7 (9, 11). Hence consumer / is going to convert bond into stock, and we have a contradiction to the assumption that an equilibrium exists (See figure lb). Since the preferences fulfill the gross substitution property the equilib- rium allocation is a continuous function of the endowments and of the asset structure. All the arguments still go through for slightly perturbed prices, asset payoffs, and endowments. Therefore our non-existence example is ro- bust. 3 Non-existence with American Options In this section we give a robust example of non-existence of competitive equilibria for an economy with an American put option. We prove this by showing that in both cases, i.e. if the option is exercised in t = or in t = 1, there would have to be an arbitrage. Hence an equilibrium cannot exist. As before, there are two time periods t = 0,1, and two states of nature a, b at t = 1. There is one consumption commodity in each state; however, now there is also a single consumption good in period t = 0. At t = there are two assets available for trade: a stock and an American put option on the stock. The stock pays 1 unit of the commodity in state a and 2 units in state b. The American option allows the holder to sell 1 unit of the stock at a given price k. Our example assumes a striking price k — 3. The option can be exercised in t = 0, or the decision to exercise can be delayed until t = 1. If the option is exercised in t = then the payoff of one unit of the option in the first period is A:, and and the payoff in the second period is — 1 in state a and —2 in state 6. If the decision is delayed, the payoff will be 8 (k — 1) + = max{A: — 1,0} in state a and (k — 2) + = ma.x{k — 2,0} in state b. As in the previous example we allow an agent to exercise any fraction of his option holdings. The decision whether to exercise can only be made by agents who are long. Agents who are short in the option have to form expectations as to the percentage to be exercised in each period. Let f denote this expected rate of early exercise by agents who are short, and let r* be the percentage of the option portfolio which is exercised early by an agent who is long. Again, there are two consumers /, J. The preferences of both consumers can be described by the Cobb- Douglas utility function i/(x ,x a ,Xk) = x x a xi,. Let w 1 = (25,3,600), and w J = (5,300,6) be the initial endowment of the consumers. The definition of an equilibrium of this model is similar to the definition of section 2; we only have to chance the payoff matrix and allow for consumption in period t = 0. Let x x = (x x ,x x aJ x x b ) denote agent i's consumption vector and let X = (0\, 6 X 2 ) denote his portfolio allocation. The first component of 6 l is the holding of stock; the second component is the holding of options. As in the previous section we define + and 0~ to be the vectors where we take the positive (respectively negative) part of every component of 0. Given the rate of early exercise r, the payoff matrix for this economy is /0 kr \ V(t)= 1 -lr + (l-r)(Jb-l) + . \2 -2r + (1 - r)(k - 2) + ) An equilibrium is therefore a consumption-portfolio allocation (x\ l ) 6 R z + x R 2 , i = /, J; a system of asset prices n £ R 2 , a total rate of expected early exercise f, and early exercise decisions r* of the consumers satisfying: (i) for every i, (x 1 , % , r') maximizes u x (x) subject to the budget constraints x + 7T0 < w x ' + V o (f)0- + V (r x )9 + Xs < < + V M (f)0- + V a {r i )0 + ; s = a, 6 xeiJj, ^Ei? 2 , r* e [0,1] (ii) £, = /,J *'' = £,=/,J w\ E,=r,j * = 0, £, = /,jr'(^)+ =fE, = u W) + . where V 5 (t) denotes the s th row of V a . We now show the non-existence of equilibria in two steps. We first con- sider the case where f < 1, so that not all of the option is exercised in period t = 0. In this case, since the option is in the money in every state, the assets span the market and the equilibrium allocations would have to be the equilibrium allocations of the complete market. We now compute the state prices associated with those equilibria and show that there exists an arbitrage portfolio given those state prices. Let p = (po,p a ,Pb) De the state prices in the Arrow-Debreu equilibrium. Given the Cobb- Douglas utility functions, the demand 7* of consumer i in the complete Arrow-Debreu market is given by ( pw l pw x pw x \ 1 iP0,Pa,Pb) = 3po 3p a 3pb J By setting p = 1 and solving the system of excess demand equations we get w l a + w J a w l b + w J h Hence, in our case, Pa = j§i and pi = ~. Clearly, p a and pb are exactly the state prices for our economy. Therefore the price ir\ of the real asset has to t> e Pa + 2/?6 = yjjy, and the price -k' 2 of one unit of the option not exercised in t = Ohas tobe(/:-l) + p a -f(A:-2) + p 6 = 2p a +p b = ||. Iff is the percentage of options being exercised in t = then ir 2 = r(k — 7Ti) + (1 — t)tt 2 . Consider now a portfolio consisting of 1 unit of the option and 1 unit of the stock. If the agent decides to exercise all of the option in t — then is payoff in both states s — a, b is 0. However, in t = his payoff is k — 7r x — ir 2 = 10 k — (r k + ( 1 — t)(7Ti+xJ) = (1 — t)(& — 7^ — tt' 2 ). Therefore k — ir\ — 7r 2 > for every r < 1 which means that the portfolio is an arbitrage portfolio. Hence there cannot exist an equilibrium where the two assets span the market. This proves the first step. In the second step we assume that all of the American option is exercised in t = 0. Hence f = 1 and the asset payoff matrix V(r) is redundant. Therefore all equilibria of the above economy would have to coincide with the equilibria of an economy where the stock is the only asset available for trade in t = 0. We now proceed by computing the equilibria. Let m, n be the payoff of the stock in the states t = a, 6, and let p be the price of the stock. With only the stock to trade consumer i solves ma.x(w x — sp)(w x a + sm)(wl + sn), subject to the constraint that the consumption in every state is positive. The first and second order conditions are necessary and sufficient for a maximum. The first order condition is given by — 3mnps 2 + 2{w x mn — w x a np — w\mp)s + w x Q w x a n -f w x w x b m — w x a w\p = 0. (1) The second order condition is — %mnps + 2(u;Q77in — w x a np — w x b rnp) < 0. (2) Solving the quadratic equation (1) for 5 we get the demand function of con- sumer i. t - w x Q mn — w x a np — w x b mp 3mnp + (w l Q mn - w\np - w x b mp) 2 w^wjn + w'owjm - w x a wlp \ (3mnp) 2 3mnp (The second order condition guarantees that the solution with the positive root is the unique maximum.) We now show that a x is a strictly decreasing function. 11 Denote the expression inside the radical by R. The derivative of demand with respect to price is dp 3 P 2 1 \2wi 2p 2 y/R 3 1 w l Q mn — w x a np — wlmp\ WqW^u + w l w l b m Zmnp J 3mn 2(wq) 2 w x w x a n + w x w\m + <0. 3p 2 2p 2 y/R [ 9p 9mn In other words a 1 and cr J are continuous, strictly decreasing functions for p> 0. We now compute the equilibrium price for the above economy. Inserting the parameters in (3) we get "'(?) =? - 101 + 3p * J (p) =f " 101 + 3p \ /25 \ 2 2525 -101 + 300, Up \ + 505 300. Since the demand functions are strictly monotonic there can exist only one equilibrium price. One can check that cr 7 (5) = 1.027 and cr J (5) = —0.993; whereas <7 7 (5.1) = 0.997 and cr J (b.l) = —1.003. Therefore the only possible equilibrium price p lies between 5 and 5.1. We now show that there exists an arbitrage also in this case. Since the option is always exercised in t = the only possible equilibrium price for the option is q = k — p. Otherwise there would immediately be an arbitrage since the second period payoff of the stock and of the option are the same. Given our choice of k and the equilibrium price p we have computed above, we get q < 0. If an agent holds 1 unit of the American option and decides to exercise it in t = 1 then his payoff is —q > in t = 0, and (k — 1) + , (k - 2) + > in s = a, 6. Consequently the agent has an arbitrage. Therefore this cannot be an equilibrium either. 12 It now remains to show that this counterexample is robust with respect to perturbations of the endowments and of the asset structure. The argument is straightforward. Because of (1) both demand functions a* are continu- ous, monotonic function of the endowments and of the payoffs of the assets. Small perturbations of the parameters will only chance the equilibrium price slightly. All inequalities which we need to establish the non-existence of equilibria will therefore still be fulfilled. Hence our example is robust. 4 Inefficiency of Equilibria In the previous example we assumed that the consumption commodity serves at the same time as a numeraire. In this section we relax this assumption. Specifically we consider a model in which there are American call options with nominal striking prices. In Krasa and Werner (1989) European options with nominal striking prices are analyzed in an incomplete market framework. It is shown that (a) equilibria exist generically, and that (b) the allocation of the complete Arrow-Debreu market can be implemented if there are enough assets to span the market. In the following example we are going to show that (b) breaks, down for American options. Again this phenomenon is robust with respect to perturbations of the asset structure. The example works in a way similar to the previous one. We compute the equilibrium of the complete Arrow-Debreu market and show that there would always exist an arbitrage by exercising the option early. However, in contrast to the example of section 2 we now have to determine whether such an arbitrage exists for all possible price normalizations in the different states. In the previous examples, by choosing sufficiently low absolute price levels in the states s = a, 6 it will always be better to exercise the American put option in t = 1. Hence the matrix of asset payoff has full rank and it is possible to implement the Arrow-Debreu allocation. This argument applies to all 13 models with American put options with nominal striking prices. However, as the following example shows it does not work for American call options in a three-period economy. We consider an economy with uncertainty described by six states of nature S = {aa, ab, ac, 6a, 66, 6c}. There are three time periods t — 0,1,2. In t = 1 a part of the uncertainty is resolved. Consumers receive the signal a if s 6 {aa,a6,ac}; they receive the signal 6 if s € {6a, 66, 6c}, where 3 is the underlying state of the economy. Uncertainty is fully resolved in t = 2. There is one consumption commodity in each state and time period. There are six agents 5 whose preferences are described by the utility function u(x ,x 1 (s) i x 2 (s)) = x Yl xi(s) Y[x 2 (s). 3=a,b a^S Let w* denote the endowment of the consumers. As in the previous sections we first give an example for a specific choice of parameters and then we argue that our result is robust. We assume that all consumers have 2 units of the commodity as initial endowment in t = and in each of the two states in t = 1. Endowments only differ in t = 2. We assume that for each of the states there exists a consumer who has an initial endowment of 7 units of the commodity in this state, and an endowment of 1 unit otherwise. In this example, the commodity does not serve as a numeraire; instead there is a separate unit of account. There are two assets available for trade in t = 0: a stock d and a European call option e. The stock pays dividends only in t = 2; it pays 1 unit of the commodity in s = a6, ac^ 66, 6c, and 2 units of the commodity in the remaining states. The European call option 6 traded in t = gives the holder 5 We need at least six agents since the net trades of the agents in the complete Arrow- Debreu market have to span the full market space. Otherwise the fact that the assets cannot span does not prove that the Arrow-Debreu equilibrium cannot be implemented. 6 Our inefficiency result would also work if the option where an American call option, however, it is easier to describe the maximization problem of the consumers this way. 14 the right to buy 1 unit of the the stock in t = 1 for the strike price k = 6 which is now measured in units of account. In t — 1 there are two American call options available for trade. The options can be exercised in t = 1 or t = 2. Assume that one of the American options has the striking prices ki = 0.5, and that the other option has the striking price k 2 = 0.4. Given this asset structure, a portfolio plan of consumer i is given by the vector 0* = (Oodi 0Oe,Q ad, 0a\i0a2i0bdi Oblate), where the first subscript denotes the date and state of purchase, and the second subscript denotes the asset purchased. We will denote the vector of assets purchased in one state by # 3 , where s == 0,a,6. Furthermore we use the notation 0+, and 6~ as in the previous sections to denote the positive (respectively negative) part of every component of 6 S . Given the rates of early exercise r = (rj tS ), j = 1,2, s = a, 6 for the American option j in state s. the matrix of payoffs of all assets traded in z = a, b is given by ( -hT\,z -k 2 T 2tZ \ $Pza 2p 2a r li2 + (l-r 1)2 )(2p 2a -A: 1 ) + 2p 2a r 2)2 + (1 - r 2 , 2 )(2p 2a - k 2 ) + Pzb PzbT\, z + (1 - Ti iZ )(p zb - ki) + PzbT 2 ,z + (1 ~ T 2 , 2 )(p zb ~ k 2 ) + \ Pzc PzcT\,z + (1 - T 1>2 )(p 2C - fcx)* Pzc^z + (1 - T2,z){Pzc ~ h)* ) where p 2a , p 2 &, p zc is the price of the commodity in the states za, zb, zc. We denote the first row of this matrix V z (r, p), the second row V 2a (r,p), the third row K 2 t(r,p), and the fourth row V zc (t, p). In other words, V s (r, p) concisely denotes the vector of asset payoffs in date 1 for s = a, b or in date 2 for s G S. An equilibrium is then a consumption-portfolio allocation (x\ 9 l ) G R\ x R 8 , i — 1,..., /; a system of asset prices x G R 8 , total rates of expected early exercise fj <3 for option j in state s = a, 6, and early exercise decisions Tj s of consumer i for option j in state s = a, 6 satisfying: (i) for every i, (x*, 0', r') maximizes u'(x) subject to the budget constraints Po^o + ^o^o < Po^o 15 p s x s + v s 9, < p a w a + e od TT ad + M*** - k) + + V 5 (f ,p)0; + V s (r\p)0+, for 5 = a, 6 p 5 z, < p s ™, + V 3 (f,p)6j + V a (r'',p)^ + , for s 3 1 \ l < 3 l \ *- 5 A \ TT 3 d = 2A 2 , 3a T 1" <*2,a6 7 1" ^2,5cT = 4A liS . ^2,aa *2,sb *2,sc In order to implement the Arrow-Debreu equilibrium the assets have to span the full market space in both periods. Consequently the call option has to be in-the-money at least in one of the states. That means (7r ad — k) + > or (7r w — k) + > 0. Given our choice of k 16 this implies A lf0 > 1.5 or A lf6 > 1.5. (4) Without loss of generality we can assume that the first inequality holds. The argument for the other case is similar. In order to span the complete market at least a part (1 — f Jt ,) > of both options would have to be exercised in t = 2. We now compute the price of the options in t = 1 and show that there would have to exist an arbitrage by exercising one of the options early. The price q a of an option in s = a which is not exercised early is given by 9i = A La Hence the price of the option is given by f{-K ad — k{) + (1 — f)q a . In order to implement the Arrow-Debreu equilibrium the options and the stock have to span. This is only possible if there is a state in t = 2 where the options with the striking price k l is out-of-the-money. 7 Hence we get qi < 3Ai,„. (5) We now show that a portfolio consisting of 1 unit of option 1 and —1 units of the stock allows for an arbitrage if all of the option is exercised early. In any of the states in t = 2 this portfolio has zero payoff. In s = a the payoff is given by m a = 7T ad - f(ir ad - k x ) - (1 - f)q\ - k = (1 - f)(ir ad - ki - q\). (6) (4), (5) and (6) now imply m« > (1 - f)(4A lia - 0.5 - 3A 1>a ) > (1 - f) > 0. 7 At this point we clearly need that there are 2 call options on the real asset. If there were only one call option then it could be in-the-money in all states and still span. 17 Hence this is an arbitrage portfolio. Therefore there cannot exist an equilib- rium which implements the complete market equilibrium. Again the example is robust since the equilibrium prices of the complete market are a continuous function of the endowment, and all the necessary inequalities still hold for slightly perturbed asset structures and prices. Finally we note that there exist inefficient equilibria. Consider the econ- omy where no options are available for trade in t = and where two European call options with the striking prices k\ and k% are traded in t = 1 instead of the American options. Such an economy has an equilibrium because of Krasa and Werner (1989) or Polemarchakis and Ku (1986) as we noted above. We then have to choose sufficiently low prices levels A 1)2 such that in the original economy the option traded in t = is always out of the money and such that T ac i — k — q\ < 0. Equation (6) then implies that it is never optimal to exercise the American options in t = 1. Therefore we have an equilibrium for the economy with the original asset structure which is, however, inefficient. 5 Conclusion The models we have analyzed demonstrate that the properties of European options in general equilibrium are quite special. Any financial instrument which offers the possibility of early exercise, will carry with it the possibil- ity for non-existence or inefficiency of equilibria, even in a single commod- ity economy. We have demonstrated this by reference to American options and convertible bonds, but we conjecture that the results will hold for most derivative financial instruments. 18 References: D. Cass (1984), "Competitive Equilibria in Incomplete Financial Markets", Work- ing Paper 84-09, University of Pennsylvania, CARESS. D. DUFFIE (1987), "Stochastic Equilibria with Incomplete Financial Markets". Journal of Economic Theory 41, 405-416 D. Duffie and W. Shafer (1985), "Equilibrium in Incomplete Markets: I. A Basic Model of Generic Existence", Journal of Mathematical Economics 14, 285-300. G. Geanakoplos and H. Polemarchakis (1986), "Existence, Regularity, and Constrained Suboptimality of Competitive Allocations when the Asset Market is Incomplete", In W. Heller and D. Starrett, Essays in Honor of Kenneth J. Arrow, Volume III, Cambridge University Press. O. Hart (1974), "On the Optimality of Equilibrium When the Market Structure is Incomplete", Journal of Economic Theory 11, 418-443. S. 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