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 FACULTY WORKING 
 PAPER NO. 957 
 
 Price Taking Behavior and Trading in Options 
 
 Leonard J. Mirrnan 
 Nairn Reisman 
 
 College of Commerce anc Business Aom in istr.nt Ion 
 Bureau of Economic and Business Researcn 
 University of Illinois, Urbara-Cnampaign 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 957 
 College of Commerce and Business Administration 
 University of Illinois at Urbana-Champaign 
 May 1983 
 
 Price Taking Behavior and Trading in Options 
 
 Leonard J. Mirman, Professor 
 Department of Economics 
 
 Haim Reisman 
 Tulane University 
 
 We would like to acknowledge the aid of an anonymous referee 
 for remarks helping to clarify the exposition of this paper. 
 
Abstract 
 Equilibrium prices of options are abritrage prices in economies in 
 which prices are determined endogenously and all agents are price takers. 
 We show that the price taking assumption in options' markets is unreason- 
 able because by not being a price taker a small agent can gain much. 
 
Black and Scholes (1973) considered the problem of option pricing in an 
 economy with two securities. In their model the prices of these securities are 
 given exogenously by 
 
 (1) B(t) = e rt , S(t) = e at + b w(t) 
 
 where {W(t); 0<_t<_T} is a standard Brownian Mortion. In this economy it is 
 assumed that traders can trade continously without transaction costs. 
 
 The Black and Scholes result implies that a generalized option (contingent 
 claim) that pays at date t = T, f(S(T)) dollars, has a unique, reasonable 
 price. One can show that a trader can form a portfolio of the two securities, 
 change this portfolio continously in the interval [0,T], in such a way that 
 buying an additional amount of one security is financed by selling an amount of 
 the other security of equal value (i.e., the trading strategy used is self 
 financing), so that the payoff at date t = T is f(S(T)) dollars with 
 probability one. The value of this portfolio at t = is called the arbitrage 
 price of the claim at date t = 0. 
 
 A trader in this economy will agree to sell any number of options at 
 prices which are a bit higher than their arbitrage price, knowing that a sure 
 profit can be made by duplicating the cash flow of the option at date t = T by 
 trading continously in the securities 3 and S. Therefore if options are traded 
 in this economy their arbitrage prices will be their equilibrium prices. 
 
 In the Black and Scholes model it is assumed implicity that the trading in 
 options does not affect the prices of the securities. Since, in this model, the 
 basic securities prices are given exogenously, this assumption is reasonable. 
 However to study the reasonableness as well as the implications of this 
 assumption, a model in which both option prices and security prices are 
 determined endogenously is needed. 
 
-2- 
 
 In this paper we examine the arbitrage method in an economy in which 
 securities prices are the result of the agents efforts to maximize their 
 utility. An example of this type of economy is described by Kreps (1979) or 
 Harrison and Kreps (1979). We shall consider in this paper an economy which 
 is a special case of the economy described by Kreps (1979) . 
 
 In Kreps (1979) and Harrison and Kreps (1979) an economy in which equi- 
 librium prices of the securities can be given by (1) was developed. In 
 their model all agents are assumed to be price takers. This assumption 
 implies that trading in options does not affect the prices of the securities 
 and also that the equilibrium prices of the options are their arbitrage 
 prices. The assumption that agents are price takers is very strong and 
 inconsistent since it is assumed that the traders are, in all other respects, 
 very sophisticated. Hence one would expect that if prices can be manipulated 
 by options trading these agents will in fact manipulate them. 
 
 In this paper we consider the problem of whether, in the Harrison- 
 Kreps economy (when a large number of options can be traded) , it is reasonable 
 to assume that agents are price takers believing that prices of the basic 
 securities are determined by (1) . In effect we study whether price taking 
 behavior is reasonable in the sense that by not accepting the "equilibrium" 
 security prices a small agent can gain a lot or even corner the market and 
 thus, in effect, make a pronounced change in the "equilibrium" securities 
 prices . 
 
 In order to study this question we assume that all agents but one 
 believe that no matter what options they sell, the prices of the basic 
 securities do not change. They are therefore willing to sell any option 
 at a little higher than its arbitrage price on the belief that they can 
 make arbitrage profits. 
 
-3- 
 
 It will be shown that there are certain arbitrarily inexpensive options 
 such that the one remaining agent, by buying one of these options at date t = 
 at its arbitrage price, (or a bit higher) , can gain a lot, to the detriment of 
 those who sold him this option, by not behaving as a price taker after date t = 
 0. If all agents anticipate this, no agent would sell options at their 
 arbitrage prices, and all agents would want to buy certain options at their 
 arbitrage price (or slightly higher). 
 
 The key to this argument is that a single agent, investing a small amount 
 of funds, can reap enormous profits, if this agent does not take prices as 
 given. Thus assuming price-taking in a "large-but-finite" economy of this sort 
 is unwarranted. (A similar sort of result is obtained by Hart (1979). Hart 
 shows that when short-selling is permitted in securities markets, firms that 
 are small but not infinites imally small can have large effects on the economy. 
 Hence "price taking" behavior is not appropriate in his model, as well.) Since 
 price taking is shown to be unreasonable, one wonders what is a reasonable way 
 to define equilibrium for such economies. This question is beyond the scope of 
 this paper and will not be discussed. 
 
 This might be an explanation for the fact that the option pricing model fails 
 to predict prices of options that are well out of the money in real markets. The 
 reason suggested in this paper for the occurrence of this phenomenon is that the 
 behavior of real world traders for well out of the money options is not approximated 
 well by the price taking assumption. 
 
 In Section 2 we describe an economy which is a special case of the economy 
 described by Kreps. In Section 3 we construct our example. In Section 4 some 
 concluding remarks are made. 
 
-4- 
 
 2. The Economy 
 
 Consider an economy with one consumption good that can be consumed only at 
 time t = 1 , with two securities that are actually state contingent claims on 
 the conumption good at t = 1 . 
 
 A probability space (ft,F,P) is given, and weft represents a state of the 
 world. 
 
 At time t = 1 , the securities pay 
 
 P^l.w) = e r , P 2 (1,w) = e a+b W(1 ' w) > 
 
 where r, a, and b are constants, and W = (W(t); 0<_ t<_1} is a standard Brownian 
 motion on (Q,F,P). Denote by F fc the a algebra generated by (W(u) ; 0£u<tl and 
 assume that F^ = p. Each agent is informed at time t which sets BeF contain 
 the true state of the world. 
 
 There are N identical agents who consume at time t = 1. At time t = 
 
 each agent holds 1/N of both securities. 
 
 Assume prices of the securities at time t, if the state i3 w, are given by 
 
 p /. >. rt p /,. \ at + b W(t,w) rr>\ 
 
 *l(t,w) = e , P 2^» w ^ = © Kd) 
 
 i 
 Assume agent i can hold at time t, q (t,w) of security j, as long as he 
 
 is using admissible self-financing trading strategies, as defined in Harrison 
 
 and Kreps (1979). 
 
 Agent i will choose an admissible self-financing trading strategy (q 1 ,q 2 ) 
 
 to maximize 
 
 U K(1,w)e r + q2(1,w)e a+b W( 1 ' w) ) dP(w), 
 
-5- 
 
 over all the admissible self financing trading strategies, where U, the agent's 
 utility function, is a strictly concave function defined on the real line. 
 We assume further that there exists a function g(h) such that g(h) -*■ as 
 h •*■ and 
 
 h U(-h" 1 g(h)) -► — , as h + 0. 
 
 Assume also that prices (2) are equilibrium prices in the sense that 
 
 N 
 
 I Qj(t,w) = 1 J * 1,2 0<t<1 
 
 is1 
 
 Harrison and Kreps (1979) and Kreps (1979) show that if the equilibrium prices 
 of the basic securities are given by (2) , then the equilibrium prices of any 
 options that are traded (in zero net supply) will be their arbitrage prices. 
 
 We will show that if a large number of options are traded, then the 
 assumption that agents are price takers, believing that the prices of the basic 
 securities are determined by (2) , is unreasonable. 
 
-6- 
 
 3. An Example 
 
 Let C(x,P 2 ,t) be the Black-Scholes prices at time t of the European call 
 option with striking price x (that pays (Pp(l) - x) at t=1). Assume that 
 options with striking prices x, x+h and x-h are traded, where x and h are 
 further described below. 
 
 Suppose agents 2, 3, • •'•, N agree at time t = to trade options with 
 agent 1 in exchange for the appropriate "hedge portfolios" of the stock and 
 the bond. If agent 1 pays a bit more bond for the transaction, the other 
 agents will be happy to make this transaction because they think they will 
 make arbitrage profits. 
 
 Agent 1 buys from all other agents g(h)h~ 2 options with striking prices 
 x+h and x-h, and agent 1 sells 2g(h)h~ 2 options with striking price x. Thus 
 agent 1 purchases the contingent claim 
 
 g(h)h- 2 ((P 2 (i) _ x+ h) + - 2(P 2 (1) - x) + + (P 2 (l) - x - h) + ). 
 
 The payoff of this claim as a function of P ? (1) is depicted in figure 1 
 The "price" of this package of options (at the Black-Scholes prices) is 
 g(h) h " 2 (C(x+h,P 2 ,o) - 2C(x,P 2 ,0) + C(x-h,P 2 ,0) ) , 
 
 or, approximately, g(h) — C(x,P_ o) . This is small if h is small. 
 
 3X 2 
 
-7- 
 
 payof f 
 
 i" gOO f 
 
 ^ 
 
 Figure 1: payoff of the contingent claim at date t=l 
 
 P 2 (D 
 
-8- 
 The payment to agents 2, 3, . . . , N at t = includes 
 
 g(h)h" 2 (^|-(C(x-h,P 2 ,0) -2C(x,P 2 ,0) + C(x+h,P 2 ,0)) 
 of security 2. This is approximately 
 
 g(h) —2 =-C(x,P 2 ,0) , 
 
 3P 2 3 x 
 
 and is also small if h is small. Thus agent 1, in exchange for the "tent"' 
 depicted in figure 1, has given up a small amount of both the stock and the 
 bond. 
 
 As h -*• 0, therefore, even if agent 1 does not trade after t = 0, the 
 expected utility from this portfolio will be, at worst, a bit less than 
 before the options' transaction. 
 
 Agents 2 through N expect to make money by continuously trading with 
 agent 1, in order to carry out the Black-Scholes hedge. But suppose agent 1, 
 after time 0, announces that he will no longer engage in such trades. If 
 agent 1 stays out of the market after date t = 0, the other agents must then 
 trade among themselves. As time passes and securities' prices change, the 
 other agents try to buy and sell securities in order to hedge their portfolio. 
 But since they are identical, nobody will sell. If agent 1 never trades with 
 them again, they wind up (at time 1) with a little extra of both the stock and 
 the bond, and they are short the "tent" depicted in figure 1. The extra stock 
 and bond raises their utility slightly, but the "tent" lowers their expected 
 utility by a term on the order of h U(-g(h)h ). That is, with probability 
 on the order of h, they must make good on the "tent," lowering their consump- 
 tion by this amount. For small enough h, our assumption on U ensures that 
 they view this as a precarious position indeed. What will happen at this 
 
-9- 
 
 point we cannot say precisely, because agent 1 finds himself in the position 
 of a monopolist. There are clearly gains to trade between agent 1 and all 
 the rest, and as agent 1 holds the key to those gains, we expect that he will 
 extract much of the surplus. That is, agent 1 will at some time before t = 1 
 trade again, but on very favorable terms. 
 
 This example can be altered by assuming that only agent 2 agrees to sell 
 (or buy) every option to (from). agent 1 at its arbitrage price. Agent 1 will sell 
 the claim (with a very small h) to agent 2 and then will get out of the market. 
 If securities prices change agent 2 would like to buy and sell securities in 
 order to hedge his portfolio. Since agent 1 is out of the market the demand 
 for these securities will be different than the demand in the case in which 
 agent 1 is trading. So prices will change. Agent 2 will observe he can 
 not hedge this portfolio using the arbitrage method. He will then be willing 
 to buy back the claim at a loss. The other agents can gain a little by 
 selling agent 2 some of the claims at a high price. However, they cannot 
 sell too much because the moment they sell this claim they would be in the 
 same trouble as agent 2. Agent 1 is the only one in the economy who can sell 
 large quantities of this claim, and thus can gain much. This example can 
 also be generalized to the case in which the agents are not identical and 
 also instead of one agent of type 1 and one agent of type 2 there are several 
 agents of both kinds. 
 
 The conclusion is that an agent who buys the above claim, with h small 
 enough, can gain. Moreover, it is unimportant from whom he buys. Therefore, 
 if everything else remains the same, all agents would like to buy this claim 
 at its arbitrage price while nobody would be willing to sell at that price. 
 We do not know what the market equilibrium will be but we do know what it 
 will not be. 
 
-10- 
 
 4 . Conclusion 
 
 The prices (2) of the basic securities together with the arbitrage prices 
 of the options are an equilibrium in an economy of the kind described by Kreps 
 in which all agents are price takers. The price taking assumption is usually 
 made in the case in which a small agent cannot gain very much by not being a 
 price taker. This is certainly not the case here. 
 
 Although the example in Section 3 is pathological (when h is very small) , 
 it demonstrates, in an extreme case, what may happen in less pathological 
 cases. Prices (2) of the basic securities together with the arbitrage prices 
 of the options, may be reasonable equilibrium prices if only a small number 
 of options are allowed to be traded. However if many options are traded it 
 seems that the equilibrium of price takers is no longer reasonable and an 
 equilibrium which takes into account the possibility of manipulation is needed. 
 
 In Section 3 we could have used out of the money call options instead of 
 the option described there. There is evidence that in real options' markets 
 the prices of options that are very well out of the money are much higher than 
 predicted by the Black Scholes model. The sort of "market manipulation" 
 described in this paper may provide an explanation for this phenomena. 
 
 The example of this paper is not unique to prices given by (1) but can 
 be generalized to prices given by much more general stochastic processes. 
 Furthermore a similar example can be constructed for a discrete time model 
 like the one given in Cox, Ross and Rubinstein (1979). 
 
-11- 
 
 REFERENCES 
 
 F. Black and M. Scholes, 1973, The Pricing of Options and Corporate 
 Liabilities, Journal of Political Economy 3, 637-654. 
 
 J. Cox, S. Ross and M. Rubinstein, 1979, Option Pricing: A Simplified 
 Approach, Journal of Financial Economy 7, 229-263. 
 
 M. Harrison and D. Kreps, 1979, Martingales and Arbitrages in Multiperiod 
 Securities Markets, Journal of Economics Theory 20, 381-408. 
 
 0. Hart, 1979, On Shareholders Unanimity in Large Stock Market Economies, 
 Econometrica 47, 1057-1083. 
 
 D. Kreps, 1979, Three Essays on Capital Market, Technical Report No. 298, 
 The Economic Series, Institute for Mathemetical Studies in the Social Sciences, 
 Stanford University. 
 
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