margins and price limits in taiwan's stock index futures market
TRANSCRIPT
62 EMERGING MARKETS FINANCE AND TRADE
PIN-HUANG CHOU, MEI-CHEN LIN, AND
MIN-TEH YU
Margins and Price Limits in Taiwan’sStock Index Futures Market
Abstract: This study extends the framework of Brennan (1986) to find thecost-minimizing combination of spot limits, futures limits, and margins forstock and index futures in the Taiwan market. Our empirical results showthat the cost-minimization combination of margins, spot price limits, andfutures price limits is 7 percent, 6 percent, and 6 percent, respectively, whenthe index level is less than 7,000. When the index level ranges from 7,000 to9,000, the efficient futures contract calls for a combination of 6.5 percent, 5percent, and 6 percent. The optimal margin, reneging probability, and cor-responding contract cost are less than those without price limits. Price lim-its may partially substitute for margin requirements in ensuring contractperformance, with a default risk lower than the 0.3 percent rate that isaccepted by the Taiwan Futures Exchange. On the other hand, though im-posing equal price limits of 7 percent on both the spot and futures marketsdoes not coincide with the efficient contract design, it does have a lowercontract cost and margin requirement (7.75 percent) than that without im-posing price limits (8.25 percent).
Key words: default risk, futures, margin requirement, price limits.
62
Emerging Markets Finance and Trade, vol. 42, no. 1,January–February 2006, pp. 62–88.© 2006 M.E. Sharpe, Inc. All rights reserved.ISSN 1540–496X/2006 $9.50 + 0.00.
Pin-Huang Chou ([email protected]) is a professor in the Department ofFinance, National Central University, Jung-Li, Taiwan; Mei-Chen Lin ([email protected]) is a professor in the Department of Finance, National United University,Miao Li, Taiwan; Min-Teh Yu ([email protected]) is a professor at Providence Uni-versity, Taichung, Taiwan.
JANUARY–FEBRUARY 2006 63
A margin on a futures contract helps protect the integrity and reputationof the futures exchange. It also protects the futures commission mer-chant (FCM) from losses resulting from customer default. When the fu-tures exchange sets margins, a trade-off must be made between the costsof margins that are too high and too low. Lower margins increase a trader’sdefault risk when a daily adverse price movement exceeds the balanceon his margin account; higher margins raise the cost of futures trading.1
Previous studies on the method of determining margins focus mainlyon how to set margin levels to reduce default risk to a specified level(Booth et al. 1997; Cotter 2001; Dewachter and Gielens 1999; Edwardsand Neftci 1988; Figlewski 1984; Gay et al. 1986; and Longin 1999) orto minimize the contract cost (Fenn and Kupiec 1993). In the latter model,default risk is a major determinant of margins, and the volatility of fu-tures prices matters when the exchange sets margins to a specified prob-ability of default. However, previous studies never consider that pricelimits can decrease price and default probability. The margin require-ments, which are usually set simultaneously with daily price limits, maythen be lower than those without price limits. Specifically, Brennan (1986)argues that price limits, by preventing investors from realizing the mag-nitude of their loss in the futures markets, may reduce both investors’incentive to default and the margins required by the exchange.2
Brennan (1986) also documents that price limits are less effectivewhen precise information about the equilibrium futures price is avail-able. The price of index futures can be estimated from the prices ofconstituent shares, and limits have little effect on the reneging decision.Because information about the equilibrium futures price can be derivedfrom the spot market for the underlying commodity, imposing price limitson the spot market creates additional noise for traders to forecast theequilibrium futures price.3 Thus, with spot price limits, the role of fu-tures price limits in ensuring futures contract performance improves.
While spot price limits reduce the default risk of a futures contractfor market participants, there is a liquidity cost associated with usingthem. Clearly, the tighter the spot limits are, the more often spot markettrading is interrupted, and the greater the losses in liquidity to traders inspot markets. Thus, a policy maker faces a trade-off between reducingdefault risk in futures contracts and increasing liquidity costs in spotmarkets. From the broader view of the policy maker, the objective wouldbe to find a combination of rules that minimizes the total cost of partici-pation in the markets, including both the futures and the spot markets.4
64 EMERGING MARKETS FINANCE AND TRADE
In Taiwan, price limits are imposed on both the spot and the futuresmarkets, so the margin requirements set by the exchange should be co-ordinated with the daily price limits. Taking the view of the policy maker,this study intends to find the cost-minimizing combination of spot lim-its, futures limits, and margins for stock and index futures in the Taiwanmarket. Our analytical framework follows from Brennan (1986), in whichprice limits, in conjunction with margins, are shown to be useful to con-trol default risk and reduce the cost of futures contracting. Moreover,their effectiveness is a decreasing function of the amount of informationavailable to traders about the equilibrium futures prices.
TAIEX Futures Contract Specifications
Stock index futures were introduced on July 21, 1998, as the first finan-cial derivative product of an organized exchange in Taiwan. Five indexfutures contracts—spot month, the next calendar month, and the nextthree-quarter months—each with a different maturity, can be listed atthe same time. Each contract has, at most, one year of life. The second-oldest contract becomes the new nearby contract when the current nearbycontract expires at its maturity date.
Margin levels are adjusted and announced by the margin committeein accordance with the Standards and Collecting Methods for ClearingMargins of the Taiwan Futures Exchange (TAIFEX). The clearing mar-gin for a Taiwan Stock Exchange (TAIEX) futures contract is the TAIEXfutures index, multiplied by the value of each index point, multiplied bythe risk coefficient. The risk coefficient is the possibility of losses aris-ing from the TAIEX futures contract due to market price changes, whichis calculated based on the movement of a TAIEX futures index, expressedin points, within a certain period, usually the past three months. Thecoefficient is a range covering at least 99.7 percent of the recorded rangeof daily fluctuations in the prices of TAIEX futures.5 If the differencebetween the current clearing margin level and the level calculated dailyaccording to the risk coefficient reaches 15 percent or more, the TAIFEXmay adjust the collection level of the margin. An adjustment to the levelof a clearing margin by the TAIFEX takes effect at the close of tradingon the next business day after announcement. The exchange sets theminimum level of margin that a member FCM must demand from itscustomers; the FCMs can require a larger margin. Table 1 presents someof the main features of TAIEX futures contracts.
JANUARY–FEBRUARY 2006 65
Table 1
Main Features of TAIEX Futures Contracts
Item Description
Underlying assets TAIEX capitalization weighted stock indexContract size New Taiwan $200 multiplied by TAIEX indexDeliver months Spot month, next calendar month, and next three
quarter-monthsLast trading day Third Wednesday of delivery month of each
contractDaily settlement price Last trading price of closing session, or otherwise
determined by TAIFEX according to trading rules Settlement procedure Cash settlementMinimum price fluctuation One index point (New Taiwan $200)Daily price limit 7 percent of previous day’s settlement price
Table 2 reports the descriptive statistics for the stock index return andits nearby futures contract. We adopt daily closing prices from July 21,1998, to March 5, 2001, for the TAIEX and its futures contracts in ourstudy. The means and medians are negative for both contracts. Futuresvolatility (measured by standard deviation) is greater than spot marketvolatility over the entire sample period. The skewness of the closingreturns is 0.1555 and 0.0875 for the TAIEX and its futures returns, re-spectively. The excess kurtosis is 1.0556 and 1.7302 for the TAIEX andits futures returns, respectively.6
Table 3 displays the initial margins, maintenance margins, and ratiosof initial margins over the contract value during the sample period. Itshows that, most of the time, the TAIFEX set margins to be more than 8percent of the contract value.
Brennan’s Model: Price Limits on Futures Contracts
Suppose that a representative risk-neutral trader enters into a futurescontract at time 0 and deposits an initial margin, m, with his broker. Thefutures price at time 0, P
0, is given and is not subject to price limits. At
time 1, the position must be settled. The trader has an incentive to re-nege if the expected default benefits exceed the expected default costs.Let Π be the probability that the broker will not take legal action, or ifhe does, that it will be unsuccessful. Let γ be the sum of the expected
66 EMERGING MARKETS FINANCE AND TRADE
Table 2
Summary Statistics for Closing Returns on TAIEX and TAIEX Futures
Statistics TAIEX TAIEX futures
Sample size 697 697Mean –0.00022 –0.00023Median –0.00061 –0.00030Standard deviation 0.0187 0.0213Skewness 0.1555 0.0875Excess kurtosis 1.0556 1.7302
Notes: This table reports the descriptive statistics for the stock index return and itsnearby futures contract. We adopt daily closing prices from July 21, 1998 throughMarch 5, 2001 for the TAIEX and its futures contracts in our study.
reputation and legal costs the trader must bear for reneging. Then thetrader in a short position will have an incentive to renege if Π[P
1– P
0–
m] > γ. Because each contract includes both a long and a short position,one of the parties will have an incentive to renege whenever the absoluteprice change exceeds the effective margin, M = m + Π–1γ, that is,
P P M1 0 .− > (1)
Now, suppose that a futures price limit, Lf, is imposed on the absolute
price change. Consider the losing party’s decision when the price limitis hit at time 1, and the trader can observe a signal, Y
1, a random variable
correlated with the change in the equilibrium futures price, ft. Such a
signal can be derived from the spot market for the underlying commod-ity, the markets for other futures contracts, or other sources. Knowingthat his losses exceed the futures price limit, but not by how much, hewill shift his attention to the expected position at time 2. Ignoring dis-counting, his decision whether to renege will be based on the expectedlosses at time 2, conditional on the limit move at time 1 and the avail-able information. Reneging occurs for a positive price change if andonly if
ff L1 ≥
and
JANUARY–FEBRUARY 2006 67
Tabl
e 3
Init
ial M
arg
ins
and
Mai
nte
nan
ce M
arg
ins
for
TAIE
X F
utu
res,
199
8–20
01
Mai
nten
ance
Ave
rage
Adj
uste
dIn
itial
Rat
io2
ratio
day
Inde
x1m
argi
nM
argi
n(p
erce
nt)
(per
cent
)
July
21,
199
88,
242
140,
000
110,
000
8.5–
9.5
9.00
Aug
ust 1
9, 1
998
7,40
512
0,00
090
,000
8.0–
10.8
9.40
Feb
ruar
y 24
, 199
96,
289
140,
000
110,
000
8.0–
11.0
9.50
Aug
ust 1
8, 1
999
8,19
016
0,00
013
0,00
09.
4–10
.810
.10
Oct
ober
14,
199
97,
828
140,
000
110,
000
6.9–
9.5
8.20
Feb
ruar
y 15
, 200
09,
980
160,
000
130,
000
7.8–
10.2
9.00
Aug
ust 7
, 200
07,
879
140,
000
110,
000
8.4–
10.5
9.45
Sep
tem
ber
28, 2
000
6,79
812
0,00
090
,000
9.0–
13.1
11.0
5F
ebru
ary
13, 2
001
5,75
511
0,00
090
,000
9.0–
10.0
14.0
5
Not
es:
1 TA
IEX
fut
ures
inde
x of
nea
rby
cont
ract
bef
ore
adju
sted
day
. 2 Rat
io o
f in
itial
mar
gin
over
the
cont
ract
val
ue.
68 EMERGING MARKETS FINANCE AND TRADE
( )fE f f f L Y M1 2 1 1, ,+ ≥ > (2)
where ft = P
t – P
t–1.
The expected loss is unknown, because it is conditional on the levelof the signal, Y
1. There exists a critical value of the signal, Y
1*(M, L
f),
defined by the equality in (2), beyond which the conditional loss ex-ceeds the margin. A trader in a short position then has an incentive torenege whenever f
1 ≥ L
f and Y
1 ≥ Y
1*(M, L
f). Assuming that the joint
distribution of (ft, Y
t) is symmetric about the origin, the probability of
reneging is given by
( )( )r f fP f L Y Y M L*1 1 12 , , .≥ ≥
The cost for the futures contract contains three major components:the cost of margin, liquidity cost due to possible trading interruptions,and the cost of reneging.7 Let the cost of capital be kM, where k is theunit cost of margin per unit of time. The cost of reneging is assumed tobe proportional to the probability of reneging, that is, 2βP
r(f
1 ≥ L
f, Y
1 ≥
Y1*(M, L
f)). Assuming that the cost of trading interruptions is propor-
tional to the probability that a limit is triggered, the cost of price limitsat time 1 can then be written as
( )( )
r f
r f
P f L
P f L
1
1
.≥
α≤
The total cost for the representative trader at time 1, C(M, Lf), now be-
comes
( )( )( )
( )( )
r f
f
r f
f f
P f LC M L kM
P f L
Pr f L Y Y M L
1
1
*1 1 1
,
2 , , .
≥= +α
≤
+ β ≥ ≥(3)
Futures price limits may be conceptually useful to alleviate defaultproblems and reduce the effective margin, because they obscure the ex-act amount of losses that the trader incurs. There are situations in whichreneging would have occurred without price limits, but are avoided with
JANUARY–FEBRUARY 2006 69
price limits. However, if precise information about the true price be-comes available, then the “ambiguity effect” might disappear.
The precision of the additional information can be characterized bythe correlation coefficient between the signal Y
t and the equilibrium price
change ft. Without assuming a specific distribution for a futures price
change, solving for the above optimization problem, Equation (3), wouldbe impossible. Even if a specific distribution—say, the normal distribu-tion—is assumed, finding analytical solutions for optimization is stillquite difficult. As a result, Brennan (1986) uses some numerical ex-amples to examine whether futures price limits are useful.
Based on the above setting, Brennan (1986) shows that futures pricelimits can be a partial substitute for margin requirements in ensuringcontract performance. The study finds that it may be optimal to run somerisk of trading interruption by imposing price limits to reduce the mar-gin requirement. The effectiveness of futures price limits, however, de-teriorates as precise information about the unobserved equilibrium priceis observed.
Up to this point, we have assumed that no price limits exist in thespot market. Hence, in the event of a limit move, the trader can ob-serve the critical spot price change above which reneging will occur.Of course, price limits do exist in many stock (spot) markets, and thecritical value of the spot price change may not be observable if it fallsoutside the limits. The spot price limits may thus further restrict infor-mation to the losing party about the extent of the loss when he is re-quired to mark to the market, and consequently, may further improvethe role of price limits in ensuring futures contract performance. Thecase of having price limits in both the spot and futures markets is ana-lyzed in the following section.
Price Limits in Both Spot and Futures Contracts
We now consider a model in which both the spot and futures marketshave price limits, investigating their corresponding contract costs. Whenprice limits are triggered in the futures market, the trader will turn hisattention to additional information, Y
t, to help him decide whether to
renege. Because such a signal can be derived from the spot price changeof the underlying commodity, s
t, imposing spot limits may create an
“ambiguity effect.” If the information derived from the spot price is notconstrained by the price-limit rule, then reneging would occur, as be-
70 EMERGING MARKETS FINANCE AND TRADE
fore, whenever the expected loss conditional on the spot price changeexceeds the margin M.
The situation is more complicated, however, when a price limit isimposed on the spot market. This is because it further restricts informa-tion to the losing party about the extent of the loss when the trader isrequired to mark to the market. Nevertheless, the fundamental principleremains the same. That is, under an effective price-limit rule, renegingoccurs for a positive price change if and only if
ff L1 ≥
and
( )fE f f f L s M1 2 1 1, .+ ≥ <
There is a critical level of the spot price change, s*(M, Lf), above
which the conditional loss will exceed the margin and reneging occurs.Unlike the previous case, in which the equilibrium spot price change isalways observable, the critical level of the spot price change is unob-served when it falls outside the limits. Because the decision to renege isaffected by whether or not the price limit in the spot market is hit, thefollowing two cases need to be considered. In case 1, we consider whenlimits are hit in both markets; in case 2, we consider when limits are hitonly in the futures market.
Case 1: Limits in Both Markets
If the spot price limit is also hit when the price limit in the futuresmarket is triggered, then the information available to the loser aboutthe extent of his losses is constrained again. A trader is then forced tospeculate about the size of a loss based on the fact that price limits aretriggered in both the spot and futures markets. A trader with a shortposition will renege if the following condition holds for the up-limitcase:
ff L1 ≥
ss L1 ≥
and
JANUARY–FEBRUARY 2006 71
( )f sE f f f L s L M1 2 1 1, ,+ ≥ ≥ ≥ (4)
where Ls is the spot price limit.
Given that price limits are hit in both markets, reneging occurs if theexpected conditional loss, E(f
1 + f
2|f
1 ≥ L
f, s
1 ≥ L
s), exceeds the effec-
tive margin, M. Assuming that the joint distribution of (ft, s
t) is symmet-
ric about the origin, the probability of reneging is given by
( )r f sP f L s L1 12 , ,≥ ≥ ⋅θ
where θ is an indicator function that takes the value of 1 if E(f1 + f
2|f
1≥
Lf, s
1≥ L
s) ≥ M and 0 otherwise.
If condition (4) is violated, then reneging will not occur even if spotand futures limits are triggered, f
1 ≥ L
f and s
1 ≥ L
s. Hence, based on the
current spot limit, the futures exchange can always choose a futureslimit that is small enough such that the expected loss, conditional on theknowledge of limits in both markets, is smaller than the specified mar-gin such that the condition as formulated in Equation (4) is violated.Therefore, the equality in Equation (4) defines the optimal self-enforc-ing contract margin level as a function of the spot price limit and futuresprice limit, M(L
s, L
f ).8
Note that ∂M/∂Ls ≥ 0, meaning that a spot price limit rule can reduce
the margin required for a futures contract to be completely self-enforc-ing. In addition, for a given margin, condition (4) indicates that the fu-tures limit, L
f, decreases with the spot limit L
s. That is, imposing spot
price limits can release the futures price limits to ensure that a contractis self-enforcing.
However, the spot market may actually not hit the limit even if thelimits in the futures markets are triggered. The default risk for this situ-ation is analyzed in the following case.
Case 2: Default Risk for Limits in the Futures Market Only
Contrary to the previous case, we consider the case in which the pricelimit rule is not triggered in the spot market. In this case, a trader in ashort position will have an incentive to renege whenever f
1 ≥ L
f and s
1≥
Ls≥ s*(M, L
f). However, if the critical level of the spot price change,
s*(M, Lf), is larger than L
s, then the default probability of reneging will
72 EMERGING MARKETS FINANCE AND TRADE
be zero, because the realized price change is less than Ls, and the critical
level is unattainable. Assuming that the joint distribution of (ft, s
t) is
symmetric about the origin, the probability of reneging is given by
( )( )r f s fP f L L s s M L*1 12 , , .≥ ≥ ≥
Total Cost in Futures and Spot Markets
Although spot price limits reduce the default risk of futures contracts tomarket participants, there is a liquidity cost associated with using them.A policy maker faces a trade-off between reducing the default risks inthe futures market and increasing liquidity costs in the spot market. Thepolicy maker is assumed to design an efficient contract to minimize thetotal cost of participation in both markets C: the cost in the futures mar-ket, C
f, and the cost in the spot markets, C
s.
As in Brennan’s model, the cost for the futures contract, Cf (M, L
f, L
s),
contains three components: the cost of margin, the liquidity cost due totrading interruptions, and the cost of reneging. As for the cost for thespot contract, C
s(L
s), we assume that it contains only the cost caused by
the spot price limits. The efficient contract design may thus be writtenas
( ) ( ) ( )f s f f s s sC M L L C M L L C LMin , , , , .= + (5)
Following Brennan (1986), the total cost of futures contracting for therepresentative trader at time 1, C
f(M, L
f, L
s), now becomes
( )( )( )
r f
f f s
r f
P f LC M L L kM DP
P f L
1
1
, , ,≥
= + α + β⋅≤ (6)
where DP is the default probability. As shown in case 1 and case 2,where spot price limits exist as discussed above, the default probability,DP, becomes
( )( )( )
r f s f
r f s
DP P f L L s s M L
P f L s L
*1 1
1 1
2 , ,
2 , .
= ≥ ≥ ≥
+ ≥ ≥ ⋅θ
JANUARY–FEBRUARY 2006 73
Here, θ is an indicator function, which takes the value of 1 if E(f1
+f2|f
1≥ L
f, s
1 ≥ L
s) ≥ M and 0 otherwise.
Suppose now that only the cost of spot price limits constitutes thespot contract cost, and the cost of spot price limits is also proportional tothe probability that a spot limit is hit. The spot contract cost for therepresentative trader at time 1, C
s(L
s), is then
( )( )( )
r ss s
r s
P s LC L
P s L
1
1
.≥
= α≤ (7)
The contracting costs for the futures and spot markets are next fullydisplayed, but it is still impossible to solve the optimization problemunless a distribution is specified. In fact, as mentioned above, even ifthe distribution is known, it is still unlikely that one can obtain ananalytical solution to the problem. Hence, in the next section, we fol-low Brennan’s numerical example and assume that the futures pricechanges and spot price changes are bivariate and normally distributed,so as to estimate the optimal combination of margins, spot limits, andfutures limits.9
Numerical Analysis
In this section, numerical examples for normally distributed futures andspot price processes are presented to estimate the cost-minimizing com-bination of margins, spot limits, and futures limits. The case where thespot limit is set to be 7 percent is also considered. The same values as inBrennan (1986) are used for the parameters of the cost function: k =0.02, α = 1, and β = 50. Daily closing prices on the stock market indexand its associated nearby futures contract from July 21, 1998, to March5, 2001, are used.10 The futures return volatility of 0.0213 and spot re-turn volatility of 0.0187 are estimated from actual data. The extra-mar-ket signal, measured by ρ, which represents the correlation between thespot price change and the futures price change, is also empirically esti-mated, and is 0.933. The high ρ is associated with an extremely accuratesignal from the spot counterpart.
Starting from the 6,000 futures index point, contract costs for each ofthe three scenarios are computed for margins at intervals of 0.25 percentof the value of the contract, and for limits at intervals of 1 percent of thevalue of the contract for each margin.11 The optimization problem is
74 EMERGING MARKETS FINANCE AND TRADE
Tabl
e 4
Lea
st C
ost
s o
f TA
IEX
Fu
ture
s an
d T
AIE
X S
po
t C
on
trac
ts f
or
Co
mb
inat
ion
s o
f M
arg
in R
equ
irem
ent,
Sp
ot
Pri
ce L
imit
s, a
nd
Fu
ture
s P
rice
Lim
its
With
fut
ures
and
spo
t lim
itsW
ithou
t lim
its
Inde
xM
argi
nL s
1L f
DP
LPf
LPs
Cos
t sTo
tal
Mar
gin
DP
(per
cent
)(p
erce
nt)
(per
cent
)(p
erce
nt)
(per
cent
)C
ost f
(per
cent
)(p
erce
nt)
cost
2(p
erce
nt)
(per
cent
)C
ost f
Pan
el A
: Cos
t-m
inim
izat
ion
com
bina
tion
of m
argi
n, s
pot
limits
, fu
ture
s lim
its
6,00
07.
006
60.
0003
0.48
4917
.806
90.
1334
0.26
7218
.074
08.
250.
0107
20.8
76,
500
7.00
66
0.00
030.
4849
19.2
069
0.13
340.
2672
19.4
740
8.25
0.01
0722
.52
7,00
06.
505
60.
0018
0.48
4919
.356
30.
7500
1.51
1320
.867
68.
000.
0173
24.1
37,
500
6.50
56
0.00
180.
4849
20.6
563
0.75
001.
5113
22.1
676
8.00
0.01
7325
.73
8,00
06.
505
60.
0018
0.48
4921
.956
30.
7500
1.51
1323
.467
68.
000.
0173
27.3
38,
500
6.50
56
0.00
180.
4849
23.2
563
0.75
001.
5113
24.7
676
8.00
0.01
7328
.93
9,00
06.
505
60.
0018
0.48
4924
.556
30.
7500
1.51
1326
.067
68.
000.
0173
30.5
3
Pan
el B
: Whe
n sp
ot p
rice
limits
are
set
as
7 pe
rcen
t
6,00
07.
507
64.
40E
-05
0.48
4918
.978
90.
0182
0.03
6319
.015
28.
250.
0107
20.8
76,
500
7.50
76
4.40
E-0
50.
4849
20.4
789
0.01
820.
0363
20.5
152
8.25
0.01
0722
.52
7,00
07.
507
64.
40E
-05
0.48
4921
.978
90.
0182
0.03
6322
.015
28.
000.
0173
24.1
37,
500
7.50
76
4.40
E-0
50.
4849
23.4
789
0.01
820.
0363
23.5
152
8.00
0.01
7327
.53
8,00
07.
507
64.
40E
-05
0.48
4924
.978
90.
0182
0.03
6325
.015
28.
000.
0173
27.3
38,
500
7.50
76
4.40
E-0
50.
4849
26.4
789
0.01
820.
0363
26.5
152
8.00
0.01
7328
.93
9,00
07.
507
64.
40E
-05
0.48
4927
.978
90.
0182
0.03
6328
.015
28.
000.
0173
30.5
3
JANUARY–FEBRUARY 2006 75
Pan
el C
: Whe
n bo
th f
utur
es p
rice
limits
and
spo
t pr
ice
limits
are
set
as
7 pe
rcen
t
6,00
07.
757
70.
0024
0.10
1519
.042
31.
0182
1.82
E-0
419
.042
58.
251.
0107
20.8
76,
500
7.75
77
0.00
260.
1015
20.6
113
1.01
821.
82E
-04
20.6
115
8.25
0.01
0722
.52
7,00
07.
757
70.
0027
0.10
1522
.176
61.
0182
1.82
E-0
422
.176
78.
000.
0173
24.1
37,
500
7.75
77
0.00
280.
1015
23.7
381
1.01
821.
82E
-04
23.7
383
8.00
0.01
7325
.73
8,00
07.
757
70.
0029
0.10
1525
.296
31.
0182
1.82
E-0
425
.296
58.
000.
0173
27.3
38,
500
7.75
77
0.00
300.
1015
26.8
518
1.01
821.
82E
-04
26.8
520
8.00
0.01
7328
.93
9,00
07.
757
70.
0030
0.10
1528
.405
21.
0182
1.82
E-0
428
.405
48.
000.
0173
30.5
3
Not
es:
Thi
s ta
ble
pres
ents
var
ious
com
bina
tions
of
mar
gin
requ
irem
ent
and
futu
res
pric
e lim
it w
hen
spot
pri
ce l
imits
are
im
pose
d. U
nder
a p
rice
-lim
it ru
le, t
heop
timal
mar
gin
M a
nd l
imit
Lf ar
e se
t to
min
imiz
e th
e fu
ture
s co
ntra
ct c
ost,
that
is,
()
()
()
()
fs
fs
fs
rf
sf
sf
s sf
LL
CM
LL
kMP
fL
Ls
sL
L
11
*1
1
11
22
Min
,,
2,
2,
21
21
γ
−
−
αΦ
αΦ
σσ
=+
+β
≥≥
≥+
βΦ−
αΦ
−α
⋅θ
+
Φ
−Φ
−
σ
σ
whe
re θ
is
an i
ndic
ator
fun
ctio
n th
at t
akes
the
val
ue o
f 1
if E
(f1
+ f 2|f
1 ≥
Lf,
s 1 ≥
Ls)
≥ M
and
0 o
ther
wis
e. T
he p
aram
eter
s of
the
cos
t fu
nctio
n ar
e k
= 0
.02
perc
ent,
α=
1, β
= 5
0, σ
f1 =
1,0
00, a
nd σ
s1 =
800
. Onl
y th
e re
sults
of
the
cost
-min
imiz
ing
limit
for
each
mar
gin
are
pres
ente
d. T
he p
roba
bilit
ies
of f
utur
es l
imit
mov
es a
nd s
pot
limit
mov
es a
re l
abel
ed b
y L
Pf a
nd L
Ps,
resp
ectiv
ely.
DP
lab
els
the
defa
ult
prob
abili
ty. 1
The
com
bina
tion
of L
s and
Lf i
s se
t to
min
imiz
e th
eto
tal
spot
and
fut
ures
con
trac
t co
st f
or a
giv
en m
argi
n. R
esul
ts f
or o
nly
the
cost
-min
imiz
ing
limit
for
each
mar
gin
are
show
n in
the
tab
le. 2
The
spo
t an
d fu
ture
sco
ntra
ct c
ost
of t
he o
ptim
al l
imit
for
a gi
ven
mar
gin.
76 EMERGING MARKETS FINANCE AND TRADE
solved numerically, and only the results of the cost-minimizing limit foreach index are reported. The results are given in Table 4.
Panel A of Table 4 shows that, without imposing price limits on eitherthe spot or the futures market, the optimal margin requirement is 8.25percent, for which the corresponding reneging probability and futurescontract costs are 0.0107 percent and 20.87, respectively. However, ifthere are price limits on both the futures and spot markets, the optimalmargin for the 6,000 index level decreases to 7 percent of the value of thecontract, and the corresponding contract cost reduces to 18.0740. Theefficient futures contract calls for spot limits of 6 percent and futureslimits of 6 percent, and the reneging probability also reduces to 0.0003percent. The results for 6,000 and 6,500 index levels are similar. Whenthe futures index is at the 6,500 point, the cost-minimization combina-tion of margin, spot price limits, and futures price limits is 7 percent, 6percent, and 6 percent, respectively, with a corresponding total contractcost of 19.4740 and a futures contract cost of 19.2069. When the indexranges from 7,000 to 9,000, the results are somewhat different.
As shown in the third to seventh rows, when the index ranges from7,000 to 9,000, the efficient futures contract calls for a margin of 6.5percent and a spot and futures limits pair of 5 percent and 6 percent,where the optimal margin, reneging probability, and corresponding con-tract costs are less than those without price limits. As shown in the thirdrow, when the futures index is 7,000 points, compared with the mini-mum cost attainable without price limits of 24.13, imposing price limitsalso reduces the total contract costs to 20.8676 (the corresponding fu-tures contract cost is 19.3563). Furthermore, imposing limits reducesthe efficient margin from 8.25 percent without price limits to 6.5 per-cent, and reduces the default probability from 0.0107 percent to 0.0018percent. In comparison, the efficient futures contract cost is reduced inthe reneging probability; the cost of imposing price limits is likewisereduced in the optimal margin requirement. That the cost-minimizingmargin requirement and contract cost under price limits are less thanthose without imposing price limits indicates that imposing price limitsreduces the margin requirements and futures contract costs. This sup-ports Brennan’s (1986) argument that price limits can be a partial sub-stitute for margin requirements in ensuring contract performance.
Regardless of the index levels, the default probability for the optimalcombination of limits and margins is always less than the 0.3 percentaccepted by the TAIFEX. The optimal margin requirements (6.5 percent
JANUARY–FEBRUARY 2006 77
or 7 percent) are also smaller than those that the exchange actually re-quires (more than 8 percent).12 On the other hand, from the margin-volatility ratio in Fenn and Kupiec (1993), if the exchange wishes tocontrol the 99.7 percent risk of daily price volatility, then the optimalmargin should be about three times the margin–volatility ratio. Recallthat the estimated futures return volatility is 0.0213. When the futuresindex is less than 7,000, the margin–volatility ratio is 3.03 (6.5 percentdivided by 2.13 percent). When the futures index is no less than 7,000,the margin–volatility ratio is 3.29 (7 percent divided by 2.13 percent).Both of them are larger than three. Thus, either from the cost-minimiza-tion model, or from the acceptable default risk model, the margin actu-ally required by the TAIFEX is always less than what is prudentlyrequired. Because margin requirements in excess of such a level increasethe cost of trading with no substantial benefit in return, it seems properfor the exchange’s margin committee to decrease the margin.
Note that the probability of limit moves for a cost-minimizing con-tract is 0.1015 percent (0.2276 percent) when the futures index is lessthan (no less than) 7,000, which is extremely low. In the real world, nolimits have been triggered since TAIEX futures were introduced. Bycomparison, the optimal margin levels for TAIEX futures (less than 7percent) are significantly smaller than those actually required by theexchange (from Table 3, usually more than 8 percent). This providesfurther evidence, with empirical data, that it may be optimal to run somerisk of a trading interruption due to price limits, because they can de-crease default probability and margin requirements.
Price Limits of 7 Percent on the Spot Market
So far, our analysis has been based on the situation in which the pricelimits imposed on the spot market are not constrained at 7 percent. In thereal world, the Taiwan Securities Exchange imposes 7 percent price lim-its on the spot market. Panel B of Table 4 displays the results when suchprice limits are imposed. The results show that, given L
s = 7 percent,
when the futures index is 6,000, the optimal combination of margin andfutures price limits is 7.5 percent and 6 percent. This combination holdsfor the second to seventh rows of Panel B, where the futures index rangesfrom 6,500 to 9,000. When the futures index is 6,000, the correspondingtotal contract cost and margins are 19.0152 and 7.5 percent of the con-tract value, which are larger than those attainable, 18.2676 and 7 percent,
78 EMERGING MARKETS FINANCE AND TRADE
without the constraint. The efficient contract cost and required marginwith a given 7 percent price limit on the spot market are larger overallthan they are without this constraint. This example indicates that 7 per-cent price limits on the spot market do not correspond to the cost-mini-mizing contract design. Nevertheless, the total contract cost, defaultprobability, and effective margin requirement are still smaller than theyare without imposing the price limits. Moreover, even given L
s= 7 per-
cent, the default risk of the cost-minimizing contract, regardless of theindex level, is still lower than the 0.3 percent accepted by the exchange.
Compared with the optimal combination of margins, spot limits, andfutures price-limit levels of 7 percent, 6 percent, and 6 percent when thefutures index is less than 7,000, and 6.5 percent, 5 percent, and 6 percentwhen the futures index ranges from 7,000 to 9,000, it can be found that,when L
s = 7 percent, the optimal margin increases to 7.5 percent. The
reason is that the looser the spot price limit is, the less the informationfrom the spot market is restricted, and thus, the larger a margin is re-quired. In addition to futures price limits, spot price limits appear topartially substitute for margins in controlling contract performance.
Price Limits of 7 Percent on Both the Spot andFutures Markets
Because in the real world, 7 percent price limits are imposed on Taiwan’sspot and futures markets, we also consider the case of identical limits of7 percent. The results are presented in Panel C of Table 4. They showthat, when L
s = L
f = 7 percent, the optimal margin requirement is 7.75
percent of a contract’s value, regardless of the index level. Comparedwith the results in Panel B, where only 7 percent spot limits are given,the additional 7 percent futures limit increases the margin from 7.5 per-cent to 7.75 percent, and the corresponding contract cost also rises, re-gardless of the index level. Specifically, when the index level equals6,000 points, the cost-minimizing total contract cost and margin require-ments with L
s = L
f = 7 percent are 19.0425 and 7.75 percent, which are
larger than those attainable without this equality constraint of 18.0740and 7 percent. However, imposing 7 percent price limits on both mar-kets, though inefficient, will lower the contract cost and margin require-ment versus that without price limits.
Compared with Panel A, the optimal combination of margins, spotlimits, and futures price-limits levels is 7 percent, 6 percent, and 6
JANUARY–FEBRUARY 2006 79
percent when the futures index is less than 7,000, and 6.5 percent, 5percent, and 6 percent when the futures index ranges from 7,000 to9,000. The wider the spot and futures limits are, the more margin isrequired. This is because the tighter the spot and the futures price lim-its are, the more information is restricted, and thus, the less the futuresmargin is required. The higher price limits correspond to higher mar-gin requirements.
Sensitivity Analysis
The above results are obtained based on the assumption that the liquiditycost from imposing spot price limits is the same as that from futures pricelimits. Because we cannot ascertain whether they are the same, we con-duct a sensitivity analysis to see if and how much the results are affectedby the choices of parameter values about liquidity cost. The value α
s = 1
shows that the liquidity cost of spot limits is the same as that of futureslimits; α
s = 0.8 indicates that the liquidity cost of spot limits is 80 percent
of futures limits; αs = 1.2 means that the liquidity cost of spot limits is
1.2 times of futures limits. The results are reported in Table 5. Theyshow that, when index points are within 6,000 and 6,500, the optimalcombination of margin, spot limits, and futures limits is not sensitive tothe choices of the liquidity cost for spot limits (α
s). Likewise, when
index points increase to 7,000 or more, the optimal combination aboutthe margin and limits for α
s = 0.8 and α
s = 1 also comes to the same
thing. Nevertheless, as the liquidity cost of spot limits is larger than thatof futures limits (α
s = 1.2) and index points are no less than 7,000, the
margin required increases from 6.5 percent to 6.75 percent; the spotprice limits are relaxed from 5 percent to 6 percent; and the defaultprobability increases from 0.0018 percent to 0.0073 percent. Even facedwith higher default probability and contract cost due to higher liquiditycost from imposing spot limits, the default probability for the optimalcombination of limits and margins is also always less than 0.3 percent,which the exchange accepts to control the 99.7 percent risk of dailyprice volatility. In addition, the margin–volatility ratio is 3.169 (6.75percent divided by 2.13 percent), which is larger than three. Thus, eitherfrom the cost-minimization model or from the acceptable default riskmodel, the optimal margin is less than that actually required by theTAIFEX, even when other parameter values of liquidity cost for spotlimits are used.
80 EMERGING MARKETS FINANCE AND TRADE
Tabl
e 5
Op
tim
al C
om
bin
atio
ns
of
Mar
gin
Req
uir
emen
t, S
po
t P
rice
Lim
its,
an
d F
utu
res
Pri
ce L
imit
s W
hen
Liq
uid
ity
Co
st o
f S
po
tL
imit
s D
iffe
rs f
rom
Fu
ture
s L
imit
s
Mar
gin
L s 1
L fD
PLP
fLP
sC
ost s
Tota
lIn
dex
αs
(per
cent
)(p
erce
nt)
(per
cent
)(p
erce
nt)
(per
cent
)C
ost f
(per
cent
)(p
erce
nt)
cost
2
6,00
00.
87
66
0.00
030.
4849
17.8
069
0.00
130.
2137
18.0
206
17
66
0.00
030.
4849
17.8
069
0.00
130.
2672
18.0
740
1.2
76
60.
0003
0.48
4917
.806
90.
0013
0.32
0618
.127
56,
500
0.8
76
60.
0003
0.48
4919
.206
90.
0013
0.21
3719
.420
61
76
60.
0003
0.48
4919
.206
90.
0013
0.26
7219
.474
01.
27
66
0.00
030.
4849
19.2
069
0.00
130.
3206
19.5
275
7,00
00.
86.
55
60.
0018
0.48
4919
.356
30.
0075
1.20
9020
.565
41
6.5
56
0.00
180.
4849
19.3
563
0.00
751.
5513
20.8
676
1.2
6.75
66
0.00
730.
4849
20.6
014
0.00
130.
3206
20.9
220
7,50
00.
86.
55
60.
0018
0.48
4920
.656
30.
0075
1.20
9021
.865
41
6.5
56
0.00
180.
4849
20.6
563
0.00
751.
5113
22.1
676
1.2
6.75
66
0.00
730.
4849
21.9
515
0.00
130.
3206
22.2
721
8,00
00.
86.
55
60.
0018
0.48
4921
.956
30.
0075
1.20
9023
.165
41
6.5
56
0.00
180.
4849
21.9
563
0.00
751.
5513
23.4
676
1.2
6.75
66
0.00
730.
4849
23.3
015
0.00
130.
3206
23.6
221
JANUARY–FEBRUARY 2006 81
8,50
00.
86.
55
60.
0018
0.48
4923
.256
30.
0075
1.20
9024
.465
41
6.5
56
0.00
180.
4849
23.2
563
0.00
751.
5513
24.7
676
1.2
6.75
66
0.00
730.
4849
24.6
516
0.00
130.
3206
24.9
722
9,00
00.
86.
55
60.
0018
0.48
4924
.556
30.
0075
1.20
9025
.765
41
6.5
56
0.00
180.
4849
24.5
563
0.00
751.
5513
26.0
676
1.2
6.75
66
0.00
730.
4849
26.0
016
0.00
130.
3206
26.3
222
Not
es:
Thi
s ta
ble
pres
ents
the
com
bina
tions
of
mar
gin
requ
irem
ents
, fut
ures
pri
ce-l
imits
, and
spo
t pri
ce li
mits
for
var
ious
inde
x po
ints
.U
nder
a p
rice
-lim
it ru
le, t
he o
ptim
al m
argi
n M
and
lim
it L
f are
set
to m
inim
ize
the
futu
res
cont
ract
cos
t, th
at is
,
()
()
()
()
fs
fs
fs
fs
fs
fs
fs s
f
LL
CM
LL
kMP
fL
Ls
sL
L
11
*1
1
11
22
Min
,,
2,
2,
21
21
γ
−
−
αΦ
αΦ
σσ
=+
+β
≥≥
≥+
βΦ−
αΦ
−α
⋅θ
+
Φ
−Φ
−
σ
σ
whe
re α
is a
n in
dica
tor
func
tion
that
take
s th
e va
lue
of 1
if E
(f1
+ f 2|f
1 ≥ L
f, s 1
≥ L
s) ≥
M a
nd 0
oth
erw
ise.
The
par
amet
ers
of th
e co
stfu
nctio
n ar
e k
= 0
.02
perc
ent,
αf =
1, β
= 5
0, σ
f1 =
1,0
00, a
nd σ
s1 =
800
. Onl
y th
e re
sults
of
the
cost
-min
imiz
ing
limit
for
each
mar
gin
are
pres
ente
d. T
he p
roba
bilit
ies
of f
utur
es li
mit
mov
e an
d sp
ot li
mit
mov
e ar
e la
bele
d by
LP
f and
LP
s, re
spec
tivel
y. D
P la
bels
the
defa
ult
prob
abili
ty. 1 T
he c
ombi
natio
n of
Ls a
nd L
f is
set t
o m
inim
ize
the
tota
l spo
t and
fut
ures
con
trac
t cos
t for
a g
iven
mar
gin.
2 The
spo
t and
futu
res
cont
ract
cos
t of
the
optim
al li
mit
for
a gi
ven
mar
gin.
82 EMERGING MARKETS FINANCE AND TRADE
The Cool-Off Effect in Price Limits
Because investors are given additional time to process relevant infor-mation under price limits, it is possible that price limits may cool offthe market and aid in resolving prices. Then, if price limits have aneffect on the underlying price-generating process when a limit is hit,the story may be different. To help understand the cool-off effect ofprice limits on price behavior, assume that the true price change fol-lows an independent normal distribution with mean zero and varianceσ
ft2, that is, f
t~ N(0,σ
ft2). Suppose further that time t – 1 is not a limit
day, and an upper limit is hit at time t, then the potential price changefollowing an up-limit move has the following conditional mean underthe normality assumption:
( ) ( )( )t t f t f ft fE f f L f L L1
11
,1+
φ α+ − ≥ = σ −
−Φ α
where α1 = L
f/σ
ft · ϕ(·) and Φ(·) are the standard normal density and
distribution functions, respectively. This indicates that the expected pricechange following an upper limit will increase. Likewise, the expectedprice change following a lower limit will decrease.
If part of the price change is not fundamental, but transitory, and canbe eliminated by introducing price limits, then as alleged by some price-limit proponents, price limits might reduce extreme price movement inthe same direction by pulling the price back. Under normal distribution,the conditional mean after an upper limit is hit takes the following form:
( ) ( )( )t t t f ftE f f f L 1
11
.1+
φ α + ≥ = γ σ −Φ α
It implies that the price-limit rule has a cool-off effect when γ has avalue smaller than one. Table 6 presents the results for the case whenprice limits reduce the potential price change by 20 percent following alimit hit. That is, γ = 0.8. Panel A of Table 6 summarizes the results inPanel A of Table 4, in which price limits delay the price (γ = 1), andPanel B presents the results that price limits have a cool-off effect (γ =0.8). When the price-limit rule has the real effect of changing the ex-
JANUARY–FEBRUARY 2006 83
Tabl
e 6
Op
tim
al C
om
bin
atio
ns
of
Mar
gin
Req
uir
emen
ts, S
po
t P
rice
Lim
its,
an
d F
utu
res
Pri
ce L
imit
s fo
r a
Giv
en In
dex
Wh
en P
rice
Lim
its
Hav
e a
Co
ol-
Off
Eff
ect
Mar
gin
L s 1
L fD
PLP
fLP
sC
ost s
Tota
lIn
dex
(per
cent
)(p
erce
nt)
(per
cent
)(p
erce
nt)
(per
cent
)C
ost f
(per
cent
)(p
erce
nt)
cost
2
Pan
el A
: γ =
1
6,00
07.
06.
06.
00.
0003
0.48
4917
.806
90.
1334
0.26
7218
.074
06,
500
7.0
6.0
6.0
0.00
030.
4849
19.2
069
0.13
340.
2672
19.4
740
7,00
06.
55.
06.
00.
0018
0.48
4919
.356
30.
7500
1.51
1320
.867
67,
500
6.5
5.0
6.0
0.00
180.
4849
20.6
563
0.75
001.
5113
22.1
676
8,00
06.
55.
06.
00.
0018
0.48
4921
.956
30.
7500
1.51
1323
.467
68,
500
6.5
5.0
6.0
0.00
180.
4849
23.2
563
0.75
001.
5113
24.7
676
9,00
06.
55.
06.
00.
0018
0.48
4924
.556
30.
7500
1.51
1326
.067
6(c
onti
nues
)
84 EMERGING MARKETS FINANCE AND TRADE
Tabl
e 6
(Con
tinue
d)
Mar
gin
L s 1
L fD
PLP
fLP
sC
ost s
Tota
lIn
dex
(per
cent
)(p
erce
nt)
(per
cent
)(p
erce
nt)
(per
cent
)C
ost f
(per
cent
)(p
erce
nt)
cost
2
Pan
el B
: γ =
0.8
6,00
06.
06.
07.
06.
77E
-05
0.10
1514
.609
90.
1334
0.26
7214
.877
16,
500
6.0
6.0
7.0
6.77
E-0
50.
1015
15.8
099
0.13
340.
2672
16.0
771
7,00
06.
06.
07.
06.
77E
-05
0.10
1517
.009
90.
1334
0.26
7217
.277
17,
500
6.0
6.0
7.0
6.77
E-0
50.
1015
18.2
099
0.13
340.
2672
18.4
771
8,00
05.
06.
06.
03.
23E
-04
0.48
4918
.609
90.
1334
0.26
7218
.874
08,
500
5.0
6.0
6.0
3.23
E-0
40.
4849
19.7
069
0.13
340.
2672
19.9
740
9,00
05.
0 6
.06.
03.
23E
-04
0.48
4920
.806
90.
1334
0.26
7221
.074
0
Not
es:
Thi
s ta
ble
pres
ents
the
com
bina
tions
of
mar
gin
requ
irem
ents
, fut
ures
pri
ce li
mits
, and
spo
t pri
ce li
mits
for
var
ious
inde
xpo
ints
whe
n pr
ice
limits
hav
e a
cool
-off
eff
ect t
o re
duce
ext
rem
e pr
ice
mov
emen
t in
the
sam
e di
rect
ion
by p
ullin
g th
e pr
ice
back
.A
val
ue f
or γ
sm
alle
r th
an 1
impl
ies
that
the
pric
e-lim
it ru
le h
as a
coo
l-of
f ef
fect
. The
con
ditio
nal m
ean
afte
r an
up
limit
mig
htta
ke th
e fo
llow
ing
form
:
()
()
()
tt
tf
ftE
ff
L1
11
.1
+
φα
+γ
≥=
γσ
−Φ
α
The
par
amet
ers
of th
e co
st f
unct
ion
are
k =
0.0
2 pe
rcen
t, α
= 1
, β =
50,
σf1 =
1,0
00, a
nd σ
s1 =
800
. Onl
y th
e re
sults
of
the
cost
-m
inim
izin
g co
mbi
natio
ns o
f m
argi
n an
d lim
its f
or e
ach
inde
x ar
e pr
esen
ted.
The
pro
babi
litie
s of
fut
ures
lim
it m
ove
and
spot
lim
it m
ove
are
labe
led
by L
Pf a
nd L
Ps,
resp
ectiv
ely.
DP
labe
ls th
e de
faul
t pro
babi
lity.
1 The
com
bina
tion
of L
s and
Lf i
s se
t to
min
imiz
e th
e to
tal s
pot a
nd f
utur
es c
ontr
act c
ost f
or a
giv
en m
argi
n. 2 T
he s
pot a
nd f
utur
es c
ontr
act c
ost o
f th
e op
timal
lim
it fo
r a
give
n m
argi
n.
JANUARY–FEBRUARY 2006 85
pected price, it can reduce margin, default probability, and contract costto a greater degree. For example, when γ = 1, the optimal combinationsof margin requirements, spot price limits, and futures price limits are 7percent, 6 percent, and 6 percent for index point = 6,000. The corre-sponding reneging probability and futures contract costs are 0.0003 per-cent and 17.8069, respectively. However, when price limits have acool-off effect (γ = 0.8), the optimal margin decreases to 6 percent ofthe value of the contract, and the corresponding contract cost reduces to14.6099. The efficient futures contract calls for spot limits of 6 percentand futures limits of 7 percent, and the reneging probability reduces to0.0000677 percent.
Conclusion
This study investigates the cost-minimizing combination of spot limits,futures limits, and margins for stock and index futures in the Taiwanmarket. Because price limits can lower price volatility and default prob-ability, margin requirements after price limits are imposed may be lowerthan those without price limits. Our empirical results support this view.The cost-minimization combination of margin, spot price limits, andfutures price limits is 7 percent, 6 percent, and 6 percent when the indexlevel is less than 7,000. When the index level ranges from 7,000 to 9,000,the efficient futures contract calls for a margin of 6.5 percent and a spotand futures limit pair of 5 percent and 6 percent, where the optimalmargin, reneging probability, and corresponding contract cost are lessthan those without price limits.
The default risk is also less than the 0.3 percent probability of theprice move exceeding the margin, which the exchange accepts. Thissupports the finding that price limits, by preventing investors from real-izing the magnitude of their loss in the futures markets, may partiallysubstitute for margin requirements in ensuring contract performance.On the other hand, when equal price limits of 7 percent are imposed onthe futures and spot markets, the efficient contract cost is larger thanthat without this constraint. Though this may not coincide with efficientcontract design, the common practice of imposing equal price limits of7 percent on both markets has lower contract cost and margin require-ments (7.75 percent) than without imposing spot price limits (8.25 per-cent). The optimal margin levels for Taiwan’s stock index futures aresignificantly smaller than those actually required by the exchange. Be-
86 EMERGING MARKETS FINANCE AND TRADE
cause excess margin requirements increase trading costs without clearbenefits in return, it would be more efficient for the exchange to reducethe margin requirements.
Notes
1. Telser (1981) argues that margins use up part of the trader’s precautionarybalances, making them unavailable to deal with unexpected events.
2. Price limits can alleviate the default problem, because they can hide the in-formation from the losing party about the extent of his losses. When a trader knowsthat the adverse price movement exceeds the limit, but not exactly how much he willlose, he must conjecture about the size of his losses. Price limits thus create noisewhen the trader is forming an expectation about the unobserved equilibrium futuresprice. As a result, there are situations in which reneging would have occurred with-out price limits, but is avoided with them.
3. Price limits are a standard in futures markets and can easily be found inmany stock markets, such as Austria, Belgium, China, France, Greece, Japan,Mexico, Spain, South Korea, Taiwan, and Thailand. For example, both the stocksand index futures of the Tokyo Stock Exchange and the TAIEX are traded underprice limits.
4. Some recent reports by government regulatory agencies call for coordinatingregulatory activities across financial markets. These studies include those conductedby the Chicago Board of Trade (1987), the Securities and Exchange Commission(SEC 1988), the Commodity Futures Trading Commission (CFTC 1988), and theGeneral Accounting Office (GAO 1988). Specifically, the CFTC (1988) recommendsthat any price limits placed in force must consider their effects on other relatedmarkets. The GAO (1988) suggests that circuit breakers, such as price limits, mustbe coordinated across markets.
5. Assuming that daily volatility is 0.02427 and the index point is 5,000, thevolatility coverage is 5,000 * 0.02427 * 3 = 364.5 index points. (Under normaldistribution, the probability that one observation falls within three standard devia-tions of its mean is 99.7 percent.)
6. The data seem to be leptokurtotic, so the assumption of normality may under-estimate the probability of margin violation because the Gaussian assumption doesnot take into account the added risk inherent in leptokurtotic data (see, e.g., Warshawsky1989). However, as documented by Hull (1993), among the factors that may affect thecommittee’s margin-setting decision (including underlying asset price levels, underly-ing asset price volatility, volume, and so on), volatility in the underlying asset priceis the primary factor affecting the margin-level decision. As a result, we focus on theprice level and volatility of the underlying asset to determine the margins.
7. In the Fenn and Kupiec (1993) model, both margin and settlement frequen-cies are used to reduce settlement risk, and contract costs include settlement costs.This paper ignores settlement costs, concerning itself solely with the costs of mar-gins, limits, and contract enforcement. This is because the TAIFEX fixes the num-ber of daily settlements at 3, and hence, settlement costs are fixed.
8. A contract may be regarded as self-enforcing if it is in the interest of allparties to fulfill it without the threat of legal action (Brennan 1986).
JANUARY–FEBRUARY 2006 87
9. Although Brennan (1986) assumes normality for asset returns in his numeri-cal analysis, his model can be conceptually extended to incorporate skewness andfat-tailedness by scaling up the values of the parameters α and β to accommodatethe potential underestimates of probabilities due to the normality assumption. Whilenot reported in the paper, we find that our results are not sensitive to the choices ofthe parameter values, suggesting that our results are not sensitive to the presence ofnonnormality.
10. Closing returns are used in this paper because the margin is determinedwith them, according to the criteria for covering at least 99.7 percent of the dailyfluctuations.
11. The specific functional forms for each of the costs and the associated costcomponents can be obtained from the authors upon request.
12. On March 1, 2001, the settlement price of a TAIEX contract was 5,536 points.The corresponding initial margins were set at New Taiwan $110,000, which is about9.935 percent of the contract value. The maintenance margins were New Taiwan$90,000, about 81.82 percent of the initial margin.
References
Booth, G.G.; J.P. Broussard; T. Martikainen; and V. Puttonen. 1997. “Prudent Mar-gin Levels in the Finnish Stock Index Futures Market.” Management Science43, no. 8: 1177–1188.
Brennan, M.J. 1986. “Theory of Price Limits in Futures Markets.” Journal of Finan-cial Economics 16, no. 2: 213–233.
CFTC (Commodity Futures Trading Commission). 1988. “Final Report on StockIndex Futures and Cash Market Activity During October 1987 to the U.S.” Com-modity Futures Trading Commission, Division of Economic Analysis and Divi-sion of the Trading and Markets.
Chicago Board of Trade. 1987. “The Report of the Chicago Board of Trade to thePresidential Task Force on Market Mechanisms.”
Cotter, J. 2001. “Margin Exceedences for European Stock Index Futures UsingExtreme Value Theory.” Journal of Banking and Finance 25, no. 8: 1475–1502.
Dewachter, H., and G. Gielens. 1999. “Setting Futures Margins: The Extremes Ap-proach.” Applied Financial Economics 9, no. 2: 173–181.
Edwards, F.R., and S.N. Neftci. 1988. “Extreme Price Movements and Margin Lev-els in Futures Markets.” Journal of Futures Markets 4, no. 6: 369–392.
Fenn, G.W., and P. Kupiec. 1993. “Prudential Margin Policy in a Future-Style Settle-ment System.” Journal of Futures Markets 13, no. 4: 389–408.
Figlewski, S. 1984. “Margins and Market Integrity: Margin Setting for Stock IndexFutures and Options.” Journal of Futures Markets 4, no. 3: 385–416.
GAO (General Accounting Office). 1988. “Financial Markets: Preliminary Obser-vations on the October 1987 Crash.” Report to the Congressional Requesters,Washington, DC.
Gay, G.D.; W.C. Hunter; and R.W. Kolb. 1986. “A Comparative Analysis of FuturesContract Margins.” Journal of Futures Markets 6, no. 2: 307–324.
Hull, J.C. 1993. Options, Futures and Other Derivative Securities. Englewood Cliffs,NJ: Prentice Hall.
88 EMERGING MARKETS FINANCE AND TRADE
To order reprints, call 1-800-352-2210; outside the United States, call 717-632-3535.
Longin, François M. 1999. “Optimal Margin Level in Futures Markets: ExtremePrice Movements.” Journal of Futures Markets 19, no. 2: 127–152.
SEC (Securities and Exchange Commission). 1988. “The October 1987 MarketBreak.” Division of Market Regulation Report, Washington, DC.
Telser, Lester G. 1981. “Margins and Futures Contracts.” Journal of Futures Mar-kets 1, no. 2: 225–253.
Warshawsky, M.J. 1989. “The Adequacy and Consistency of Margin Requirements:The Cash, Futures and Options Segments of the Equity Markets.” Review ofFutures Markets 8, no. 3: 420–437.