Margins and Price Limits in Taiwan's Stock Index Futures Market

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


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.


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


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


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.


( )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


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-


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


( )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


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


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


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


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.


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


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,


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


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.


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


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.


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-


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

)


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

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. The

con

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afte

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up

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llow

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form

:

()

()

()

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tf

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ff

L1

11

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+

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The

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are

k =

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, β =

50,

σf1 =

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00, a

nd σ

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800

. Onl

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the

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led

by L

Pf a

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t pro

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lity.

1 The

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s and

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tal s

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act c

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he s

pot a

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f th

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timal

lim

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r a

give

n m

argi

n.


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-


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).


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.


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.

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Margins and Price Limits in Taiwan's Stock Index Futures Market