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Execution lags and imperfect arbitrage
Economics Letters 0165-1765/93/$Oh.O0
43 (1993) 103- 109 0 1993 Elsevier Science
Execution
103 Publishers
B.V. All rights
reserved
lags and imperfect
arbitrage
The case of stock index arbitrage Anne
Fremault
.School
of Manqemenf.
Rcccivcd Accepted
Vila” Boston Universit_y, 704 Commonwdth
A venue, Bosion. MA 0221.5, USA
1 June 1993 29 June 1993
Abstract This paper presents the resulting execution to persist.
a simple equilibrium model of stock index arbitrage with endogenous execution lags. We show that or immediacy risk reduces arbitrage activity and causes price differences between the two markets
Introduction
The law of One Price is one of the centerpieces of modern financial economics: in the absence of market imperfections, financial assets that offer the same future pay-offs must be trading at the same price. If this were not the case, then arbitrageurs could make riskless profits by buying the cheaper asset and simultaneously selling the more expensive one. This would in turn restore the Law of One Price and with it optimal risk-sharing in the market. In the presence of market frictions, however, this mechanism no longer works smoothly, as recognized in the emerging literature on equilibria with market imperfections [see, for example, Aiyagari and Gertler (1991) and Tuckman and Vila (1992)]. Arbitrage becomes a risky and/or costly activity and price differences may continue to persist in equilibrium. Execution lags are one example of such imperfections. Execution lags arise in financial markets because of institutional arrangements, a lack of technology, insufficient liquidity, or other reasons. As a result, the price at which an order is executed may differ from the price at which the order was (optimally) decided upon. This execution risk - also referred to as immediacy risk - affects arbitrageurs since the actual spread they realize on their program may significantly deviate from the initial spread at which the program was initiated. ’ Therefore, in a volatile market environment, execution lags add considerable risk to arbitrage. This risk, however, differs across markets. In the futures market, orders are typically executed without much delay, whereas in the stock market substantially more time is required. This is in
. This paper is a revised version of Chapter 3 of my Ph.D. dissertation at the University of Pennsylvania and was previously circulated under the title: ‘Index Arbitrage with Immediacy Risk’. I thank Beth Allen, Richard Kihlstrom, Jean-Luc Vila and seminar participants at Boston University for their helpful comments. I also thank Gregory Kuserk from the Commodity Futures Trading Commission for providing very useful information. ’ Limit orders do not resolve this problem because of uncertainty about their execution.
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Letters
43 (1993) 103-109
part due to the fact that stock portfolio orders have to be decomposed into single stock orders for the actual execution of the trade. Once all the stocks have been traded, the respective price quotes are recomposed into a single index price, which is the relevant information for the stock index arbitrageur. ’ The asymmetry in execution time may also stem from the fact that the New York Stock Exchange’s (NYSE) specialists are slower in adjusting quotes than futures market floor traders, primarily because of their obligation to maintain a fair and orderly market. Delayed openings and stale stock prices have similar effects. Trading restrictions on the stock market will further accentuate this asymmetry in execution time. One example is the NYSE’s ‘side car procedure’ which delays Designated Order Turnaround (DOT) orders, identified as program trading orders, for 5 minutes after a loo-point fall in the Dow Jones Industrial Average. 3 The NYSE rule 80A is another cause for asymmetric execution. This rule constrains index arbitrageurs to execute their stock market orders on an uptick (downtick) if the Dow has fallen (risen) by more than 50 points since the previous day’s closing price. Finally, price limits and trading halts may be enacted at different times, since prices of stock and futures markets do not necessarily fall (or rise) at the same speed. Empirical tests by Chung (1991) show that arbitrage opportunities are substantially smaller when execution lags are taken into account. The theoretical literature on stock index arbitrage, however, largely ignores the structural asymmetry between stock and futures markets and the resulting immediacy risk. Transaction costs that give rise to arbitrage opportunities are typically assumed to be exogenous [see, for example, Holden (1990) and Fremault (1991)]. In this paper we endogenize execution lags in a simple equilibrium framework and analyze their impact of stock index arbitrage. We introduce an execution lag in the following way. In period 0, arbitrageurs observe a profitable spread and trade in the futures market. They also decide about their stock positions, but because of the described delays have to wait until period 1 before these trades are executed. In period 2, they liquidate their positions. Immediacy risk in this simple model is defined as the risk of a price change between 0 and 1. In this context, we show that immediacy risk reduces index arbitrage activity. Consequently, price discrepancies between the two markets continue to persist. Index arbitrageurs cease to provide the necessary liquidity to the two markets. Our results further imply that index arbitrageurs will be absent in illiquid markets, precisely when their presence is most needed to restore market liquidity. This indicates that policy interventions should be aimed at reducing those factors that are responsible for immediacy risk, and for the resulting lack of index arbitrage, rather than trying to curtail index arbitrage itself. For example, circuit-breakers, although useful to disseminate information, would intensify the earlier described structural asymmetry, rather than reduce it. Hence, in our context, they clearly are not the optimal policy instrument. Miller’s (1990) suggestion of contingent limit orders, which would allow stock market specialists to change price quotes based on price changes in the stock index futures market, may be a (costly) way to remove some of the structural inefficiencies seen in the stock market, while at the same time eliminating or reducing arbitrage opportunities.
’ In 1989, the NYSE introduced basket trading in Exchange Stock Portfolios (ESP), assigned to a single specialist, and which would allow investors to trade the S&P500 as a whole. Owing to insufficient interest, trading of the ESP was recently discontinued. ’ The DOT (also known as Superdot) is the electronic order routing system of the NYSE. It allows member firms to send orders directly to the specialist’s booth.
A.F.
The section
Vila I Economics
Letters
43 (1993)
103-109
105
remainder of the paper is organized as follows. In section 2 we present the model. In 3 we define the equilibrium and derive our main results. Section 4 concludes the paper.
2. The model In the model there are three periods and three types of assets. The assets are stocks, futures and bonds. The futures contract is written on an (equally weighted) index which comprises all the stocks in the economy. The timing of the model is as follows: the futures contract is traded at time 0, and liquidated at time 2. The stocks are traded at time 0 as well, but not all traders will have their order executed at that time. These traders - to be identified below - submit their orders at time 0, but have to wait until time 1 for execution. Hence, at the time they are placing their order, they ignore which price they will eventually obtain. The resulting price difference constitutes precisely the earlier described immediacy risk. All assets pay off in terms of the consumption good of the economy. We suppose that the interest rate is constant and equal to zero, and normalize the bond’s period 0 price to 1. The other asset prices are defined as follows: p,, is the period 0 price of the stock market index, p1 is the period 1 price of the stock market index and pf is the (period 0) futures price. The liquidation value of the stock index, and of the futures contract, is a random variable, 6. By hypothesis, v” is equal to v”= V + u”l + &, where the u,‘s are normally distributed with E( u”;) = 0; var( ii) = of and cov( u”,, &) = 0. The random variables, z?,, represent public information about v”released in period There are two types of traders in the model: one-market traders and index arbitrageurs. By definition, index arbitrageurs trade on both the futures market and the stock market. One-market traders are, for various exogenous reasons, confined to one market, either the stock market or the futures market. ’ Execution delays affect these traders in different ways. First, we suppose that there are no such delays on the futures market. As explained in the introduction, immediacy risk is less severe on the futures market, which justifies the above simplification. Second, delays do affect index arbitrageurs in their execution of the stock market part of the arbitrage program. As a result, index arbitrageurs will be trading futures in period 0, but will have to wait until period 1 before their stock orders are executed. ’ Finally, we assume that one-market traders on the stock market do not face execution delays. This assumption is made for analytical convenience only and does not affect the results. One could also interpret them as traders who have paid costs of
’ This term is a little imprecise, in the sense that all traders in the model trade on the bond market as well. However, for most of the analysis, we focus on the stock and the futures markets, hence the choice of the term. The distinction between one-market traders and index arbitrageurs allows us to concentrate on the behavior of the latter. As such, we do not model the trader’s choice between the two markets, which may be based on transaction costs, liquidity, etc. For an explicit model of this choice problem, see Fremault (1989). . Since p”, is unknown as of f = 0 when arbitrageurs decide about x, they do not know whether this choice will be feasible given the actual realization of ~7,. So, a non-negativity constraint should be added to the arbitrageur’s objective function. However, the use of the CARA utility function makes this constraint irrelevant, since demand is independent of wealth. This is a well-known criticism of the CARA model.
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Table I Timing of execution Period 0
Period q
Stock market
One-market
Futures
One-market traders Index arbitrageurs
market
traders
One-market traders Index arbitrageurs
immediacy to guarantee the immediate execution of their orders. The trading situation is given in Table 1. All traders are assumed to be price-takers. One-market traders have initial endowments of stocks, denoted by e’. ’ All traders have initial endowments of the bond B’. Traders are supposed to maximize a Constant Absolute Risk Aversion utility function defined over final wealth: U( I+:) = -exp(I+;), where I!$ is the period 2 wealth of agent i and his coefficient of risk aversion is set equal to 1. Demand functions are denoted as follows: 9, is the demand function of one-market traders on the futures market; yi, and y’, are the demand functions of one-market traders in the stock market at 0 and 1, respectively; X’ and fi are the arbitrageur’s demand functions in the stock market (executed at 1) and the futures market, respectively. Furthermore, we define X” and X’ as the aggregate stock endowments held by one-market traders on the stock market and the futures market, respectively; N’ and N‘ as the number of one-market traders in the futures market and the stock market, respectively; and N” as the number of index arbitrageurs. Finally, we assume that the aggregate endowment in the stock market at both time 0 and 1 are identical. This assumption reduces analytical complexity, without affecting the results. Given the above assumptions, the arbitrageur’s budget constraint is
(1) One-market
traders
on the futures
market
face the following
@,=(v”-pf)q’+v”e’+B’. One-market
traders
on the stock market
I$ = (p”, -p,)yl,
+ (v”-p”,)yi
budget
constraint:
(2) face the constraint: +p,)e’ +
B’
(3)
Recall that futures market traders cannot sell their stock endowments, e’, directly, but trade futures instead. As said, one-market traders on the stock market can trade in both period 0 and period 1. Using the properties of the exponential utility function, we obtain the arbitrageur’s demand functions: ‘One-market traders
can either be hedgers or speculators. In the former case, they are trading to hedge their initial endowments of stocks, denoted by e’, whereas in the latter case they are supposed to have no risky endowments (e’ = 0) and are willing to take up endowment risk. The assumption that index arbitrageurs have no initial endowments simplifies the analysis without affecting the results.
A.F.
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43 (1993) 103-109
(4) and f’
=
_
WI -P,> 2
.
(Tl
of the expected spread, Expression (5) indicates that the arbitrage position, f’, is a function pI - E(p”,). Expression (4) shows the extent to which the arbitrage position is covered: the first term is the index arbitrageur’s speculative position (x’ +fi). The index arbitrage will be a one-to-one arbitrage only if the risk premium (V - Ep”,) is zero. Since the arbitrage is initiated on the futures market at t = 0, the decision to offset this earlier taken position depends on the conditions in the stock market prevailing at t = 1, as expected at date 0.
3. Equilibrium Proposition
1. There exists an equilibrium
in the model
with index arbitrage
where the spreads
are
given by the following equations:
ai(Nf +N”) Pr -Po
Pr -
=
A
x’
X”
*L N‘ - Nt+
W,) = (pf -PO) -N’
N”
(6)
1’
UTX” 7
where A = 6 + N”IN’+ N” * (1 - (T)I(N’ + N”) and immediacy risk relative to total price risk.
we define
6 = ~:/(a:
+ cr:) as the ratio
of
From Proposition 1 and Eqs. (4) and (5) we learn that the sign of the actual spread, pf -p,,, which signals the existence of arbitrage possibilities, depends on the net hedging demand in the economy. Indeed, if the spread is positive, i.e. if the ‘per capita risk’ in the period 0 stock market (X’lN”) is higher than the ‘per capita risk’ in the futures market (X’/Nf + N”), then the index arbitrageur will undertake a long arbitrage program: buy stocks and sell futures. Similarly, when the spread is negative, the arbitrage will be of the short type: sell stocks short and buy futures. ’ Notice that when the immediacy risk, a:, becomes very small, then the equilibrium spreads converge to zero. Consequently, there will be no index arbitrage activity in equilibrium. The risk in the economy is optimally shared by index arbitrageurs (acting as pure speculators though), and one-market traders on the futures and the stock market. Moreover, the stock markets at t = 0 and t = I, and the futures market, collapse into one single market. Hence, when VT--+ 0, the equilibrium converges to the first-best optimum. By contrast, when the immediacy risk becomes very large, then the (risk-averse) arbitrageurs will be unwilling to undertake the (risky) arbitrage activity and the two markets will diverge completely. Apart from these extreme cases, it is straightforward to show that an increase in ai ’ The interested reader is referred and of the comparison between
to Fremault (1991) for a more detailed index arbitrageurs and market-makers.
explanation
of the mechanism
of index arbitrage
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Letters 43 (1993) 103-109
always lowers the size of the index arbitrage position, irrespective of the sign of the spread. This reduction in index arbitrage activity stems from the fact that a rise in g, 2 increases the difference between the initial spread and the expectation of the actual spread, and thus makes the arbitrage activity more risky. Furthermore, the index arbitrageur will keep a larger (smaller) portion uncovered in the case of a long (short) arbitrage program. This follows from the fact that the risk premium (V - Ep”,) will be higher (lower) in the case of a long (short) arbitrage, all other things equal.
4. Conclusion Our results indicate that an increase in execution risk reduces index arbitrage. Index arbitrageurs will perceive their activity as too risky if the stock market cannot absorb their orders in a timely fashion. As a result, index arbitrageurs will be absent in illiquid markets, precisely when their presence is most needed to restore liquidity.
Appendix:
Proof of Proposition
1
Step 1. Using the properties of the exponential one-market traders on the futures market: b-P,>
q; =
(4 + 4)
function,
we compute
the demand
that arbitrageurs as a group price for their market:
Xl-A’
p,=v-(u;+a:)N’.
of
(Al)
-ee’.
Given their demands and their knowledge one-market traders conjecture the following
The maximization backwards. Notice
utility
will trade
Af = xi f’,
(A21
problem of the one-market traders on the stock market is being solved that at t = 1, the only uncertainty stems from the risk premium ( v”-pl): (A31
Just like the futures market traders, these stock market traders can conjecture the equilibrium in their market, conditional on their knowledge that arbitrageurs as a group have a stock exposure A’, following their futures trade at t = 0: _
p,
2
v + u, -
=
m2
X"+Af N”
+
(A41
N”
Finally, at t = 0, one-market traders on the stock market solve their maximization problem, conditional on their optimal choice of yi, which is independent of &,. Hence, their period 0 objective function reduces to
(P,
-
PJYf, + cc - P,)Y: - ;b:(\;;J2
+ 4Y:)‘)
.
109
A.F. Vila I Economics Letters 33 (109.?) 103- 109
Using
(A4),
their
optimal
ul
Since
all trading
is
af(X’ + A’)
+p,,
Yi, + 2
demand
&(N’
at t = 0 is done
(A5)
+ N”) by one-market
7_XS ,X'+A' p,,= V-a,N‘-u’Ns+Na.
traders
selling
X”, p,, is equal
to
(‘46)
Step 2. Using the price conjectures (A2), (A4) and (A6), we now compute the arbitrageur’s futures demand, h = A’IN”. Then, after some tedious, but straightforward calculations, we obtain the spreads (6) and (7).
References Aiyagari. R.S. and M. Gertler, 1991, Asset returns with transaction costs and uninsured individual risk: A stage III exercise. Journal of Monetary Economics 27, 3099331. Chung, P.. 1991, A transactions data test of stock index futures market efficiency and index arbitrage profitability. Journal of Finance 46, 1791-1809. Fremault, A.. 1989. The role of stock index futures in a general equilibrium model with entry costs. Boston University. Working Paper 9 1-31. Frcmault, A.. 1991, Stock index futures and index arbitrage in a rational expectations model, Journal of Business 64. 523-547. Holden, CW.. 1990, A theory of arbitrage trading in financial market equilibrium, Discussion Paper 478, Indiana University, November. Miller, M., 1990, Index arbitrage and volatility, Report for Market Volatility and Investor Confidence Panel. New York Stock Exchange, June. Tuckman. B. and J.L. Vila, 1992, Holding costs and equilibrium arbitrage, MIT working paper, June.
43 (1993) 103- 109 0 1993 Elsevier Science
Execution
103 Publishers
B.V. All rights
reserved
lags and imperfect
arbitrage
The case of stock index arbitrage Anne
Fremault
.School
of Manqemenf.
Rcccivcd Accepted
Vila” Boston Universit_y, 704 Commonwdth
A venue, Bosion. MA 0221.5, USA
1 June 1993 29 June 1993
Abstract This paper presents the resulting execution to persist.
a simple equilibrium model of stock index arbitrage with endogenous execution lags. We show that or immediacy risk reduces arbitrage activity and causes price differences between the two markets
Introduction
The law of One Price is one of the centerpieces of modern financial economics: in the absence of market imperfections, financial assets that offer the same future pay-offs must be trading at the same price. If this were not the case, then arbitrageurs could make riskless profits by buying the cheaper asset and simultaneously selling the more expensive one. This would in turn restore the Law of One Price and with it optimal risk-sharing in the market. In the presence of market frictions, however, this mechanism no longer works smoothly, as recognized in the emerging literature on equilibria with market imperfections [see, for example, Aiyagari and Gertler (1991) and Tuckman and Vila (1992)]. Arbitrage becomes a risky and/or costly activity and price differences may continue to persist in equilibrium. Execution lags are one example of such imperfections. Execution lags arise in financial markets because of institutional arrangements, a lack of technology, insufficient liquidity, or other reasons. As a result, the price at which an order is executed may differ from the price at which the order was (optimally) decided upon. This execution risk - also referred to as immediacy risk - affects arbitrageurs since the actual spread they realize on their program may significantly deviate from the initial spread at which the program was initiated. ’ Therefore, in a volatile market environment, execution lags add considerable risk to arbitrage. This risk, however, differs across markets. In the futures market, orders are typically executed without much delay, whereas in the stock market substantially more time is required. This is in
. This paper is a revised version of Chapter 3 of my Ph.D. dissertation at the University of Pennsylvania and was previously circulated under the title: ‘Index Arbitrage with Immediacy Risk’. I thank Beth Allen, Richard Kihlstrom, Jean-Luc Vila and seminar participants at Boston University for their helpful comments. I also thank Gregory Kuserk from the Commodity Futures Trading Commission for providing very useful information. ’ Limit orders do not resolve this problem because of uncertainty about their execution.
104
A.F.
Vila I Economics
Letters
43 (1993) 103-109
part due to the fact that stock portfolio orders have to be decomposed into single stock orders for the actual execution of the trade. Once all the stocks have been traded, the respective price quotes are recomposed into a single index price, which is the relevant information for the stock index arbitrageur. ’ The asymmetry in execution time may also stem from the fact that the New York Stock Exchange’s (NYSE) specialists are slower in adjusting quotes than futures market floor traders, primarily because of their obligation to maintain a fair and orderly market. Delayed openings and stale stock prices have similar effects. Trading restrictions on the stock market will further accentuate this asymmetry in execution time. One example is the NYSE’s ‘side car procedure’ which delays Designated Order Turnaround (DOT) orders, identified as program trading orders, for 5 minutes after a loo-point fall in the Dow Jones Industrial Average. 3 The NYSE rule 80A is another cause for asymmetric execution. This rule constrains index arbitrageurs to execute their stock market orders on an uptick (downtick) if the Dow has fallen (risen) by more than 50 points since the previous day’s closing price. Finally, price limits and trading halts may be enacted at different times, since prices of stock and futures markets do not necessarily fall (or rise) at the same speed. Empirical tests by Chung (1991) show that arbitrage opportunities are substantially smaller when execution lags are taken into account. The theoretical literature on stock index arbitrage, however, largely ignores the structural asymmetry between stock and futures markets and the resulting immediacy risk. Transaction costs that give rise to arbitrage opportunities are typically assumed to be exogenous [see, for example, Holden (1990) and Fremault (1991)]. In this paper we endogenize execution lags in a simple equilibrium framework and analyze their impact of stock index arbitrage. We introduce an execution lag in the following way. In period 0, arbitrageurs observe a profitable spread and trade in the futures market. They also decide about their stock positions, but because of the described delays have to wait until period 1 before these trades are executed. In period 2, they liquidate their positions. Immediacy risk in this simple model is defined as the risk of a price change between 0 and 1. In this context, we show that immediacy risk reduces index arbitrage activity. Consequently, price discrepancies between the two markets continue to persist. Index arbitrageurs cease to provide the necessary liquidity to the two markets. Our results further imply that index arbitrageurs will be absent in illiquid markets, precisely when their presence is most needed to restore market liquidity. This indicates that policy interventions should be aimed at reducing those factors that are responsible for immediacy risk, and for the resulting lack of index arbitrage, rather than trying to curtail index arbitrage itself. For example, circuit-breakers, although useful to disseminate information, would intensify the earlier described structural asymmetry, rather than reduce it. Hence, in our context, they clearly are not the optimal policy instrument. Miller’s (1990) suggestion of contingent limit orders, which would allow stock market specialists to change price quotes based on price changes in the stock index futures market, may be a (costly) way to remove some of the structural inefficiencies seen in the stock market, while at the same time eliminating or reducing arbitrage opportunities.
’ In 1989, the NYSE introduced basket trading in Exchange Stock Portfolios (ESP), assigned to a single specialist, and which would allow investors to trade the S&P500 as a whole. Owing to insufficient interest, trading of the ESP was recently discontinued. ’ The DOT (also known as Superdot) is the electronic order routing system of the NYSE. It allows member firms to send orders directly to the specialist’s booth.
A.F.
The section
Vila I Economics
Letters
43 (1993)
103-109
105
remainder of the paper is organized as follows. In section 2 we present the model. In 3 we define the equilibrium and derive our main results. Section 4 concludes the paper.
2. The model In the model there are three periods and three types of assets. The assets are stocks, futures and bonds. The futures contract is written on an (equally weighted) index which comprises all the stocks in the economy. The timing of the model is as follows: the futures contract is traded at time 0, and liquidated at time 2. The stocks are traded at time 0 as well, but not all traders will have their order executed at that time. These traders - to be identified below - submit their orders at time 0, but have to wait until time 1 for execution. Hence, at the time they are placing their order, they ignore which price they will eventually obtain. The resulting price difference constitutes precisely the earlier described immediacy risk. All assets pay off in terms of the consumption good of the economy. We suppose that the interest rate is constant and equal to zero, and normalize the bond’s period 0 price to 1. The other asset prices are defined as follows: p,, is the period 0 price of the stock market index, p1 is the period 1 price of the stock market index and pf is the (period 0) futures price. The liquidation value of the stock index, and of the futures contract, is a random variable, 6. By hypothesis, v” is equal to v”= V + u”l + &, where the u,‘s are normally distributed with E( u”;) = 0; var( ii) = of and cov( u”,, &) = 0. The random variables, z?,, represent public information about v”released in period There are two types of traders in the model: one-market traders and index arbitrageurs. By definition, index arbitrageurs trade on both the futures market and the stock market. One-market traders are, for various exogenous reasons, confined to one market, either the stock market or the futures market. ’ Execution delays affect these traders in different ways. First, we suppose that there are no such delays on the futures market. As explained in the introduction, immediacy risk is less severe on the futures market, which justifies the above simplification. Second, delays do affect index arbitrageurs in their execution of the stock market part of the arbitrage program. As a result, index arbitrageurs will be trading futures in period 0, but will have to wait until period 1 before their stock orders are executed. ’ Finally, we assume that one-market traders on the stock market do not face execution delays. This assumption is made for analytical convenience only and does not affect the results. One could also interpret them as traders who have paid costs of
’ This term is a little imprecise, in the sense that all traders in the model trade on the bond market as well. However, for most of the analysis, we focus on the stock and the futures markets, hence the choice of the term. The distinction between one-market traders and index arbitrageurs allows us to concentrate on the behavior of the latter. As such, we do not model the trader’s choice between the two markets, which may be based on transaction costs, liquidity, etc. For an explicit model of this choice problem, see Fremault (1989). . Since p”, is unknown as of f = 0 when arbitrageurs decide about x, they do not know whether this choice will be feasible given the actual realization of ~7,. So, a non-negativity constraint should be added to the arbitrageur’s objective function. However, the use of the CARA utility function makes this constraint irrelevant, since demand is independent of wealth. This is a well-known criticism of the CARA model.
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A. F. Vila I Economics
Letters 43 (1993)
103-109
Table I Timing of execution Period 0
Period q
Stock market
One-market
Futures
One-market traders Index arbitrageurs
market
traders
One-market traders Index arbitrageurs
immediacy to guarantee the immediate execution of their orders. The trading situation is given in Table 1. All traders are assumed to be price-takers. One-market traders have initial endowments of stocks, denoted by e’. ’ All traders have initial endowments of the bond B’. Traders are supposed to maximize a Constant Absolute Risk Aversion utility function defined over final wealth: U( I+:) = -exp(I+;), where I!$ is the period 2 wealth of agent i and his coefficient of risk aversion is set equal to 1. Demand functions are denoted as follows: 9, is the demand function of one-market traders on the futures market; yi, and y’, are the demand functions of one-market traders in the stock market at 0 and 1, respectively; X’ and fi are the arbitrageur’s demand functions in the stock market (executed at 1) and the futures market, respectively. Furthermore, we define X” and X’ as the aggregate stock endowments held by one-market traders on the stock market and the futures market, respectively; N’ and N‘ as the number of one-market traders in the futures market and the stock market, respectively; and N” as the number of index arbitrageurs. Finally, we assume that the aggregate endowment in the stock market at both time 0 and 1 are identical. This assumption reduces analytical complexity, without affecting the results. Given the above assumptions, the arbitrageur’s budget constraint is
(1) One-market
traders
on the futures
market
face the following
@,=(v”-pf)q’+v”e’+B’. One-market
traders
on the stock market
I$ = (p”, -p,)yl,
+ (v”-p”,)yi
budget
constraint:
(2) face the constraint: +p,)e’ +
B’
(3)
Recall that futures market traders cannot sell their stock endowments, e’, directly, but trade futures instead. As said, one-market traders on the stock market can trade in both period 0 and period 1. Using the properties of the exponential utility function, we obtain the arbitrageur’s demand functions: ‘One-market traders
can either be hedgers or speculators. In the former case, they are trading to hedge their initial endowments of stocks, denoted by e’, whereas in the latter case they are supposed to have no risky endowments (e’ = 0) and are willing to take up endowment risk. The assumption that index arbitrageurs have no initial endowments simplifies the analysis without affecting the results.
A.F.
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Letters
107
43 (1993) 103-109
(4) and f’
=
_
WI -P,> 2
.
(Tl
of the expected spread, Expression (5) indicates that the arbitrage position, f’, is a function pI - E(p”,). Expression (4) shows the extent to which the arbitrage position is covered: the first term is the index arbitrageur’s speculative position (x’ +fi). The index arbitrage will be a one-to-one arbitrage only if the risk premium (V - Ep”,) is zero. Since the arbitrage is initiated on the futures market at t = 0, the decision to offset this earlier taken position depends on the conditions in the stock market prevailing at t = 1, as expected at date 0.
3. Equilibrium Proposition
1. There exists an equilibrium
in the model
with index arbitrage
where the spreads
are
given by the following equations:
ai(Nf +N”) Pr -Po
Pr -
=
A
x’
X”
*L N‘ - Nt+
W,) = (pf -PO) -N’
N”
(6)
1’
UTX” 7
where A = 6 + N”IN’+ N” * (1 - (T)I(N’ + N”) and immediacy risk relative to total price risk.
we define
6 = ~:/(a:
+ cr:) as the ratio
of
From Proposition 1 and Eqs. (4) and (5) we learn that the sign of the actual spread, pf -p,,, which signals the existence of arbitrage possibilities, depends on the net hedging demand in the economy. Indeed, if the spread is positive, i.e. if the ‘per capita risk’ in the period 0 stock market (X’lN”) is higher than the ‘per capita risk’ in the futures market (X’/Nf + N”), then the index arbitrageur will undertake a long arbitrage program: buy stocks and sell futures. Similarly, when the spread is negative, the arbitrage will be of the short type: sell stocks short and buy futures. ’ Notice that when the immediacy risk, a:, becomes very small, then the equilibrium spreads converge to zero. Consequently, there will be no index arbitrage activity in equilibrium. The risk in the economy is optimally shared by index arbitrageurs (acting as pure speculators though), and one-market traders on the futures and the stock market. Moreover, the stock markets at t = 0 and t = I, and the futures market, collapse into one single market. Hence, when VT--+ 0, the equilibrium converges to the first-best optimum. By contrast, when the immediacy risk becomes very large, then the (risk-averse) arbitrageurs will be unwilling to undertake the (risky) arbitrage activity and the two markets will diverge completely. Apart from these extreme cases, it is straightforward to show that an increase in ai ’ The interested reader is referred and of the comparison between
to Fremault (1991) for a more detailed index arbitrageurs and market-makers.
explanation
of the mechanism
of index arbitrage
108
A.F.
Vila I Economics
Letters 43 (1993) 103-109
always lowers the size of the index arbitrage position, irrespective of the sign of the spread. This reduction in index arbitrage activity stems from the fact that a rise in g, 2 increases the difference between the initial spread and the expectation of the actual spread, and thus makes the arbitrage activity more risky. Furthermore, the index arbitrageur will keep a larger (smaller) portion uncovered in the case of a long (short) arbitrage program. This follows from the fact that the risk premium (V - Ep”,) will be higher (lower) in the case of a long (short) arbitrage, all other things equal.
4. Conclusion Our results indicate that an increase in execution risk reduces index arbitrage. Index arbitrageurs will perceive their activity as too risky if the stock market cannot absorb their orders in a timely fashion. As a result, index arbitrageurs will be absent in illiquid markets, precisely when their presence is most needed to restore liquidity.
Appendix:
Proof of Proposition
1
Step 1. Using the properties of the exponential one-market traders on the futures market: b-P,>
q; =
(4 + 4)
function,
we compute
the demand
that arbitrageurs as a group price for their market:
Xl-A’
p,=v-(u;+a:)N’.
of
(Al)
-ee’.
Given their demands and their knowledge one-market traders conjecture the following
The maximization backwards. Notice
utility
will trade
Af = xi f’,
(A21
problem of the one-market traders on the stock market is being solved that at t = 1, the only uncertainty stems from the risk premium ( v”-pl): (A31
Just like the futures market traders, these stock market traders can conjecture the equilibrium in their market, conditional on their knowledge that arbitrageurs as a group have a stock exposure A’, following their futures trade at t = 0: _
p,
2
v + u, -
=
m2
X"+Af N”
+
(A41
N”
Finally, at t = 0, one-market traders on the stock market solve their maximization problem, conditional on their optimal choice of yi, which is independent of &,. Hence, their period 0 objective function reduces to
(P,
-
PJYf, + cc - P,)Y: - ;b:(\;;J2
+ 4Y:)‘)
.
109
A.F. Vila I Economics Letters 33 (109.?) 103- 109
Using
(A4),
their
optimal
ul
Since
all trading
is
af(X’ + A’)
+p,,
Yi, + 2
demand
&(N’
at t = 0 is done
(A5)
+ N”) by one-market
7_XS ,X'+A' p,,= V-a,N‘-u’Ns+Na.
traders
selling
X”, p,, is equal
to
(‘46)
Step 2. Using the price conjectures (A2), (A4) and (A6), we now compute the arbitrageur’s futures demand, h = A’IN”. Then, after some tedious, but straightforward calculations, we obtain the spreads (6) and (7).
References Aiyagari. R.S. and M. Gertler, 1991, Asset returns with transaction costs and uninsured individual risk: A stage III exercise. Journal of Monetary Economics 27, 3099331. Chung, P.. 1991, A transactions data test of stock index futures market efficiency and index arbitrage profitability. Journal of Finance 46, 1791-1809. Fremault, A.. 1989. The role of stock index futures in a general equilibrium model with entry costs. Boston University. Working Paper 9 1-31. Frcmault, A.. 1991, Stock index futures and index arbitrage in a rational expectations model, Journal of Business 64. 523-547. Holden, CW.. 1990, A theory of arbitrage trading in financial market equilibrium, Discussion Paper 478, Indiana University, November. Miller, M., 1990, Index arbitrage and volatility, Report for Market Volatility and Investor Confidence Panel. New York Stock Exchange, June. Tuckman. B. and J.L. Vila, 1992, Holding costs and equilibrium arbitrage, MIT working paper, June.