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

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

  1. 1. 62 EMERGING MARKETS FINANCE AND TRADE 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, MEI-CHEN LIN, AND MIN-TEH YU Margins and Price Limits in Taiwan’s Stock Index Futures Market Abstract: This study extends the framework of Brennan (1986) to find the cost-minimizing combination of spot limits, futures limits, and margins for stock and index futures in the Taiwan market. Our empirical results show that the cost-minimization combination of margins, spot price limits, and futures price limits is 7 percent, 6 percent, and 6 percent, respectively, when the index level is less than 7,000. When the index level ranges from 7,000 to 9,000, the efficient futures contract calls for a combination of 6.5 percent, 5 percent, 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 contract performance, with a default risk lower than the 0.3 percent rate that is accepted by the Taiwan Futures Exchange. On the other hand, though im- posing equal price limits of 7 percent on both the spot and futures markets does not coincide with the efficient contract design, it does have a lower contract 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. Pin-Huang Chou (choup@cc.ncu.edu.tw) is a professor in the Department of Finance, National Central University, Jung-Li, Taiwan; Mei-Chen Lin (meclin@nuu .edu.tw) is a professor in the Department of Finance, National United University, Miao Li, Taiwan; Min-Teh Yu (mtyu@pu.edu.tw) is a professor at Providence Uni- versity, Taichung, Taiwan. 62
  2. 2. JANUARY–FEBRUARY 2006 63 A margin on a futures contract helps protect the integrity and reputation of 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 costs of margins that are too high and too low. Lower margins increase a trader’s default risk when a daily adverse price movement exceeds the balance on his margin account; higher margins raise the cost of futures trading.1 Previous studies on the method of determining margins focus mainly on how to set margin levels to reduce default risk to a specified level (Booth et al. 1997; Cotter 2001; Dewachter and Gielens 1999; Edwards and Neftci 1988; Figlewski 1984; Gay et al. 1986; and Longin 1999) or to 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 price limits can decrease price and default probability. The margin require- ments, which are usually set simultaneously with daily price limits, may then 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 effective when precise information about the equilibrium futures price is avail- able. The price of index futures can be estimated from the prices of constituent shares, and limits have little effect on the reneging decision. Because information about the equilibrium futures price can be derived from the spot market for the underlying commodity, imposing price limits on the spot market creates additional noise for traders to forecast the equilibrium 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 contract for market participants, there is a liquidity cost associated with using them. Clearly, the tighter the spot limits are, the more often spot market trading is interrupted, and the greater the losses in liquidity to traders in spot markets. Thus, a policy maker faces a trade-off between reducing default risk in futures contracts and increasing liquidity costs in spot markets. From the broader view of the policy maker, the objective would be 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
  3. 3. 64 EMERGING MARKETS FINANCE AND TRADE In Taiwan, price limits are imposed on both the spot and the futures markets, 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 Taiwan market. Our analytical framework follows from Brennan (1986), in which price 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 information available 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 index futures contracts—spot month, the next calendar month, and the next three-quarter months—each with a different maturity, can be listed at the same time. Each contract has, at most, one year of life. The second- oldest contract becomes the new nearby contract when the current nearby contract expires at its maturity date. Margin levels are adjusted and announced by the margin committee in accordance with the Standards and Collecting Methods for Clearing Margins of the Taiwan Futures Exchange (TAIFEX). The clearing mar- gin for a Taiwan Stock Exchange (TAIEX) futures contract is the TAIEX futures index, multiplied by the value of each index point, multiplied by the risk coefficient. The risk coefficient is the possibility of losses aris- ing from the TAIEX futures contract due to market price changes, which is calculated based on the movement of a TAIEX futures index, expressed in points, within a certain period, usually the past three months. The coefficient is a range covering at least 99.7 percent of the recorded range of daily fluctuations in the prices of TAIEX futures.5 If the difference between the current clearing margin level and the level calculated daily according to the risk coefficient reaches 15 percent or more, the TAIFEX may adjust the collection level of the margin. An adjustment to the level of a clearing margin by the TAIFEX takes effect at the close of trading on the next business day after announcement. The exchange sets the minimum level of margin that a member FCM must demand from its customers; the FCMs can require a larger margin. Table 1 presents some of the main features of TAIEX futures contracts.
  4. 4. JANUARY–FEBRUARY 2006 65 Table 1 Main Features of TAIEX Futures Contracts Item Description Underlying assets TAIEX capitalization weighted stock index Contract size New Taiwan $200 multiplied by TAIEX index Deliver months Spot month, next calendar month, and next three quarter-months Last trading day Third Wednesday of delivery month of each contract Daily settlement price Last trading price of closing session, or otherwise determined by TAIFEX according to trading rules Settlement procedure Cash settlement Minimum 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 and its nearby futures contract. We adopt daily closing prices from July 21, 1998, to March 5, 2001, for the TAIEX and its futures contracts in our study. The means and medians are negative for both contracts. Futures volatility (measured by standard deviation) is greater than spot market volatility over the entire sample period. The skewness of the closing returns 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 and its futures returns, respectively.6 Table 3 displays the initial margins, maintenance margins, and ratios of initial margins over the contract value during the sample period. It shows that, most of the time, the TAIFEX set margins to be more than 8 percent of the contract value. Brennan’s Model: Price Limits on Futures Contracts Suppose that a representative risk-neutral trader enters into a futures contract at time 0 and deposits an initial margin, m, with his broker. The futures price at time 0, P0, 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 if he does, that it will be unsuccessful. Let γ be the sum of the expected
  5. 5. 66 EMERGING MARKETS FINANCE AND TRADE Table 2 Summary Statistics for Closing Returns on TAIEX and TAIEX Futures Statistics TAIEX TAIEX futures Sample size 697 697 Mean –0.00022 –0.00023 Median –0.00061 –0.00030 Standard deviation 0.0187 0.0213 Skewness 0.1555 0.0875 Excess kurtosis 1.0556 1.7302 Notes: This table reports the descriptive statistics for the stock index return and its nearby futures contract. We adopt daily closing prices from July 21, 1998 through March 5, 2001 for the TAIEX and its futures contracts in our study. reputation and legal costs the trader must bear for reneging. Then the trader in a short position will have an incentive to renege if Π[P1 – P0 – 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 absolute price change exceeds the effective margin, M = m + Π–1γ, that is, P1 − P0 > M . (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 limit is hit at time 1, and the trader can observe a signal, Y1, 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. Knowing that his losses exceed the futures price limit, but not by how much, he will shift his attention to the expected position at time 2. Ignoring dis- counting, his decision whether to renege will be based on the expected losses at time 2, conditional on the limit move at time 1 and the avail- able information. Reneging occurs for a positive price change if and only if f1 ≥ L f and
  6. 6. Table 3 Initial Margins and Maintenance Margins for TAIEX Futures, 1998–2001 Maintenance Average Adjusted Initial Ratio2 ratio day Index1 margin Margin (percent) (percent) July 21, 1998 8,242 140,000 110,000 8.5–9.5 9.00 August 19, 1998 7,405 120,000 90,000 8.0–10.8 9.40 February 24, 1999 6,289 140,000 110,000 8.0–11.0 9.50 August 18, 1999 8,190 160,000 130,000 9.4–10.8 10.10 October 14, 1999 7,828 140,000 110,000 6.9–9.5 8.20 February 15, 2000 9,980 160,000 130,000 7.8–10.2 9.00 August 7, 2000 7,879 140,000 110,000 8.4–10.5 9.45 September 28, 2000 6,798 120,000 90,000 9.0–13.1 11.05 February 13, 2001 5,755 110,000 90,000 9.0–10.0 14.05 Notes: 1 TAIEX futures index of nearby contract before adjusted day. 2 Ratio of initial margin over the contract value. JANUARY–FEBRUARY 2006 67
  7. 7. 68 EMERGING MARKETS FINANCE AND TRADE ( E f1 + f2 f1 ≥ L f , Y1 > M , ) (2) where ft = Pt – Pt–1. The expected loss is unknown, because it is conditional on the level of the signal, Y1. There exists a critical value of the signal, Y1*(M, Lf), 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 to renege whenever f1 ≥ Lf and Y1 ≥ Y1*(M, Lf). Assuming that the joint distribution of (ft, Yt) is symmetric about the origin, the probability of reneging is given by ( 2 Pr f1 ≥ L f , Y1 ≥ Y1* ( M , L f ) . ) 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 the unit cost of margin per unit of time. The cost of reneging is assumed to be proportional to the probability of reneging, that is, 2βPr(f1 ≥ Lf, Y1 ≥ Y1*(M, Lf)). Assuming that the cost of trading interruptions is propor- tional to the probability that a limit is triggered, the cost of price limits at time 1 can then be written as Pr ( f1 ≥ L f ) α . Pr ( f1 ≤ L f ) The total cost for the representative trader at time 1, C(M, Lf), now be- comes Pr ( f1 ≥ L f ) C ( M , L f ) = kM + α Pr ( f1 ≤ L f ) (3) + 2 βPr ( f1 ≥ L f , Y1 ≥ Y1* ( M , L f )) . Futures price limits may be conceptually useful to alleviate default problems and reduce the effective margin, because they obscure the ex- act amount of losses that the trader incurs. There are situations in which reneging would have occurred without price limits, but are avoided with
  8. 8. 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 by the correlation coefficient between the signal Yt and the equilibrium price change ft. Without assuming a specific distribution for a futures price change, solving for the above optimization problem, Equation (3), would be impossible. Even if a specific distribution—say, the normal distribu- tion—is assumed, finding analytical solutions for optimization is still quite 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 price limits can be a partial substitute for margin requirements in ensuring contract performance. The study finds that it may be optimal to run some risk 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 price is observed. Up to this point, we have assumed that no price limits exist in the spot 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 the critical value of the spot price change may not be observable if it falls outside 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 improve the role of price limits in ensuring futures contract performance. The case 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 markets have price limits, investigating their corresponding contract costs. When price limits are triggered in the futures market, the trader will turn his attention to additional information, Yt, to help him decide whether to renege. Because such a signal can be derived from the spot price change of the underlying commodity, st, imposing spot limits may create an “ambiguity effect.” If the information derived from the spot price is not constrained by the price-limit rule, then reneging would occur, as be-
  9. 9. 70 EMERGING MARKETS FINANCE AND TRADE fore, whenever the expected loss conditional on the spot price change exceeds the margin M. The situation is more complicated, however, when a price limit is imposed 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 is required to mark to the market. Nevertheless, the fundamental principle remains the same. That is, under an effective price-limit rule, reneging occurs for a positive price change if and only if f1 ≥ L f and ( ) E f1 + f2 f1 ≥ L f , s1 < M . 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 is always observable, the critical level of the spot price change is unob- served when it falls outside the limits. Because the decision to renege is affected by whether or not the price limit in the spot market is hit, the following two cases need to be considered. In case 1, we consider when limits are hit in both markets; in case 2, we consider when limits are hit only in the futures market. Case 1: Limits in Both Markets If the spot price limit is also hit when the price limit in the futures market is triggered, then the information available to the loser about the extent of his losses is constrained again. A trader is then forced to speculate about the size of a loss based on the fact that price limits are triggered in both the spot and futures markets. A trader with a short position will renege if the following condition holds for the up-limit case: f1 ≥ L f s1 ≥ Ls and
  10. 10. JANUARY–FEBRUARY 2006 71 ( ) E f1 + f2 f1 ≥ L f , s1 ≥ Ls ≥ M , (4) where Ls is the spot price limit. Given that price limits are hit in both markets, reneging occurs if the expected conditional loss, E(f1 + f2|f1 ≥ Lf, s1 ≥ Ls), exceeds the effec- tive margin, M. Assuming that the joint distribution of (ft, st) is symmet- ric about the origin, the probability of reneging is given by 2 Pr ( f1 ≥ L f , s1 ≥ Ls )⋅θ, where θ is an indicator function that takes the value of 1 if E(f1 + f2|f1 ≥ Lf, s1 ≥ Ls) ≥ M and 0 otherwise. If condition (4) is violated, then reneging will not occur even if spot and futures limits are triggered, f1 ≥ Lf and s1 ≥ Ls. Hence, based on the current spot limit, the futures exchange can always choose a futures limit that is small enough such that the expected loss, conditional on the knowledge 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 futures price limit, M(Ls, Lf ).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, Lf, decreases with the spot limit Ls. That is, imposing spot price limits can release the futures price limits to ensure that a contract is self-enforcing. However, the spot market may actually not hit the limit even if the limits 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 price limit rule is not triggered in the spot market. In this case, a trader in a short position will have an incentive to renege whenever f1 ≥ Lf and s1 ≥ Ls ≥ s*(M, Lf). However, if the critical level of the spot price change, s*(M, Lf), is larger than Ls, then the default probability of reneging will
  11. 11. 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, st) is symmetric about the origin, the probability of reneging is given by ( 2 Pr f1 ≥ L f , Ls ≥ s1 ≥ s* ( M , L f ) . ) Total Cost in Futures and Spot Markets Although spot price limits reduce the default risk of futures contracts to market participants, there is a liquidity cost associated with using them. A policy maker faces a trade-off between reducing the default risks in the futures market and increasing liquidity costs in the spot market. The policy maker is assumed to design an efficient contract to minimize the total cost of participation in both markets C: the cost in the futures mar- ket, Cf, and the cost in the spot markets, Cs. As in Brennan’s model, the cost for the futures contract, Cf (M, Lf, Ls), contains three components: the cost of margin, the liquidity cost due to trading interruptions, and the cost of reneging. As for the cost for the spot contract, Cs(Ls), we assume that it contains only the cost caused by the spot price limits. The efficient contract design may thus be written as MinC ( M , L f , Ls ) = C f ( M , L f , Ls ) + Cs ( Ls ) . (5) Following Brennan (1986), the total cost of futures contracting for the representative trader at time 1, Cf(M, Lf, Ls), now becomes Pr ( f1 ≥ L f ) C f ( M , L f , Ls ) = kM + α + β⋅ DP, Pr ( f1 ≤ L f ) (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 ( DP = 2 Pr f1 ≥ L f , Ls ≥ s1 ≥ s* ( M , L f ) ) + 2 Pr ( f1 ≥ L f , s1 ≥ Ls )⋅θ.
  12. 12. JANUARY–FEBRUARY 2006 73 Here, θ is an indicator function, which takes the value of 1 if E(f1 + f2|f1 ≥ Lf, s1 ≥ Ls) ≥ M and 0 otherwise. Suppose now that only the cost of spot price limits constitutes the spot contract cost, and the cost of spot price limits is also proportional to the probability that a spot limit is hit. The spot contract cost for the representative trader at time 1, Cs(Ls), is then Pr ( s1 ≥ Ls ) Cs ( L s ) = α . Pr ( s1 ≤ Ls ) (7) The contracting costs for the futures and spot markets are next fully displayed, but it is still impossible to solve the optimization problem unless a distribution is specified. In fact, as mentioned above, even if the distribution is known, it is still unlikely that one can obtain an analytical solution to the problem. Hence, in the next section, we fol- low Brennan’s numerical example and assume that the futures price changes and spot price changes are bivariate and normally distributed, so as to estimate the optimal combination of margins, spot limits, and futures limits.9 Numerical Analysis In this section, numerical examples for normally distributed futures and spot price processes are presented to estimate the cost-minimizing com- bination of margins, spot limits, and futures limits. The case where the spot limit is set to be 7 percent is also considered. The same values as in Brennan (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 index and its associated nearby futures contract from July 21, 1998, to March 5, 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 the spot price change and the futures price change, is also empirically esti- mated, and is 0.933. The high ρ is associated with an extremely accurate signal from the spot counterpart. Starting from the 6,000 futures index point, contract costs for each of the three scenarios are computed for margins at intervals of 0.25 percent of the value of the contract, and for limits at intervals of 1 percent of the value of the contract for each margin.11 The optimization problem is
  13. 13. 74 Table 4 Least Costs of TAIEX Futures and TAIEX Spot Contracts for Combinations of Margin Requirement, Spot Price Limits, and Futures Price Limits With futures and spot limits Without limits Index Margin Ls 1 Lf DP LPf LPs Cost s Total Margin DP (percent) (percent) (percent) (percent) (percent) Costf (percent) (percent) cost 2 (percent) (percent) Costf Panel A: Cost-minimization combination of margin, spot limits, futures limits 6,000 7.00 6 6 0.0003 0.4849 17.8069 0.1334 0.2672 18.0740 8.25 0.0107 20.87 6,500 7.00 6 6 0.0003 0.4849 19.2069 0.1334 0.2672 19.4740 8.25 0.0107 22.52 7,000 6.50 5 6 0.0018 0.4849 19.3563 0.7500 1.5113 20.8676 8.00 0.0173 24.13 7,500 6.50 5 6 0.0018 0.4849 20.6563 0.7500 1.5113 22.1676 8.00 0.0173 25.73 8,000 6.50 5 6 0.0018 0.4849 21.9563 0.7500 1.5113 23.4676 8.00 0.0173 27.33 EMERGING MARKETS FINANCE AND TRADE 8,500 6.50 5 6 0.0018 0.4849 23.2563 0.7500 1.5113 24.7676 8.00 0.0173 28.93 9,000 6.50 5 6 0.0018 0.4849 24.5563 0.7500 1.5113 26.0676 8.00 0.0173 30.53 Panel B: When spot price limits are set as 7 percent 6,000 7.50 7 6 4.40E-05 0.4849 18.9789 0.0182 0.0363 19.0152 8.25 0.0107 20.87 6,500 7.50 7 6 4.40E-05 0.4849 20.4789 0.0182 0.0363 20.5152 8.25 0.0107 22.52 7,000 7.50 7 6 4.40E-05 0.4849 21.9789 0.0182 0.0363 22.0152 8.00 0.0173 24.13 7,500 7.50 7 6 4.40E-05 0.4849 23.4789 0.0182 0.0363 23.5152 8.00 0.0173 27.53 8,000 7.50 7 6 4.40E-05 0.4849 24.9789 0.0182 0.0363 25.0152 8.00 0.0173 27.33 8,500 7.50 7 6 4.40E-05 0.4849 26.4789 0.0182 0.0363 26.5152 8.00 0.0173 28.93 9,000 7.50 7 6 4.40E-05 0.4849 27.9789 0.0182 0.0363 28.0152 8.00 0.0173 30.53
  14. 14. Panel C: When both futures price limits and spot price limits are set as 7 percent 6,000 7.75 7 7 0.0024 0.1015 19.0423 1.0182 1.82E-04 19.0425 8.25 1.0107 20.87 6,500 7.75 7 7 0.0026 0.1015 20.6113 1.0182 1.82E-04 20.6115 8.25 0.0107 22.52 7,000 7.75 7 7 0.0027 0.1015 22.1766 1.0182 1.82E-04 22.1767 8.00 0.0173 24.13 7,500 7.75 7 7 0.0028 0.1015 23.7381 1.0182 1.82E-04 23.7383 8.00 0.0173 25.73 8,000 7.75 7 7 0.0029 0.1015 25.2963 1.0182 1.82E-04 25.2965 8.00 0.0173 27.33 8,500 7.75 7 7 0.0030 0.1015 26.8518 1.0182 1.82E-04 26.8520 8.00 0.0173 28.93 9,000 7.75 7 7 0.0030 0.1015 28.4052 1.0182 1.82E-04 28.4054 8.00 0.0173 30.53 Notes: This table presents various combinations of margin requirement and futures price limit when spot price limits are imposed. Under a price-limit rule, the optimal margin M and limit Lf are set to minimize the futures contract cost, that is, −L f   −L  2 αΦ  2 αΦ s  σ     f1   σ s1  MinC ( M , L f , Ls ) = kM + + 2 β γ Pr f1 ≥ L f , Ls ≥ s1 ≥ s* + 2 βΦ −α f Φ (−α s )⋅θ + ( ) ( ) ,  Lf  L  2Φ − 1 2Φ s − 1 σ  σ   s1   f1  where θ is an indicator function that takes the value of 1 if E(f1 + f2|f1 ≥ Lf, s1 ≥ Ls) ≥ M and 0 otherwise. The parameters of the cost function are k = 0.02 percent, α =1, β = 50, σf1 = 1,000, and σs1 = 800. Only the results of the cost-minimizing limit for each margin are presented. The probabilities of futures limit moves and spot limit moves are labeled by LPf and LPs, respectively. DP labels the default probability. 1 The combination of Ls and Lf is set to minimize the total spot and futures contract cost for a given margin. Results for only the cost-minimizing limit for each margin are shown in the table. 2 The spot and futures contract cost of the optimal limit for a given margin. JANUARY–FEBRUARY 2006 75
  15. 15. 76 EMERGING MARKETS FINANCE AND TRADE solved numerically, and only the results of the cost-minimizing limit for each index are reported. The results are given in Table 4. Panel A of Table 4 shows that, without imposing price limits on either the spot or the futures market, the optimal margin requirement is 8.25 percent, for which the corresponding reneging probability and futures contract costs are 0.0107 percent and 20.87, respectively. However, if there are price limits on both the futures and spot markets, the optimal margin for the 6,000 index level decreases to 7 percent of the value of the contract, and the corresponding contract cost reduces to 18.0740. The efficient futures contract calls for spot limits of 6 percent and futures limits of 6 percent, and the reneging probability also reduces to 0.0003 percent. The results for 6,000 and 6,500 index levels are similar. When the 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, 6 percent, and 6 percent, respectively, with a corresponding total contract cost of 19.4740 and a futures contract cost of 19.2069. When the index ranges from 7,000 to 9,000, the results are somewhat different. As shown in the third to seventh rows, when the index ranges from 7,000 to 9,000, the efficient futures contract calls for a margin of 6.5 percent 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 third row, when the futures index is 7,000 points, compared with the mini- mum cost attainable without price limits of 24.13, imposing price limits also reduces the total contract costs to 20.8676 (the corresponding fu- tures contract cost is 19.3563). Furthermore, imposing limits reduces the efficient margin from 8.25 percent without price limits to 6.5 per- cent, and reduces the default probability from 0.0107 percent to 0.0018 percent. In comparison, the efficient futures contract cost is reduced in the reneging probability; the cost of imposing price limits is likewise reduced in the optimal margin requirement. That the cost-minimizing margin requirement and contract cost under price limits are less than those without imposing price limits indicates that imposing price limits reduces 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 optimal combination of limits and margins is always less than the 0.3 percent accepted by the TAIFEX. The optimal margin requirements (6.5 percent
  16. 16. 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 to control the 99.7 percent risk of daily price volatility, then the optimal margin should be about three times the margin–volatility ratio. Recall that the estimated futures return volatility is 0.0213. When the futures index is less than 7,000, the margin–volatility ratio is 3.03 (6.5 percent divided 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 prudently required. Because margin requirements in excess of such a level increase the cost of trading with no substantial benefit in return, it seems proper for 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 less than (no less than) 7,000, which is extremely low. In the real world, no limits have been triggered since TAIEX futures were introduced. By comparison, the optimal margin levels for TAIEX futures (less than 7 percent) are significantly smaller than those actually required by the exchange (from Table 3, usually more than 8 percent). This provides further evidence, with empirical data, that it may be optimal to run some risk 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 price limits imposed on the spot market are not constrained at 7 percent. In the real world, the Taiwan Securities Exchange imposes 7 percent price lim- its on the spot market. Panel B of Table 4 displays the results when such price limits are imposed. The results show that, given Ls = 7 percent, when the futures index is 6,000, the optimal combination of margin and futures price limits is 7.5 percent and 6 percent. This combination holds for the second to seventh rows of Panel B, where the futures index ranges from 6,500 to 9,000. When the futures index is 6,000, the corresponding total 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,
  17. 17. 78 EMERGING MARKETS FINANCE AND TRADE without the constraint. The efficient contract cost and required margin with a given 7 percent price limit on the spot market are larger overall than 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, default probability, and effective margin requirement are still smaller than they are without imposing the price limits. Moreover, even given Ls = 7 per- cent, the default risk of the cost-minimizing contract, regardless of the index level, is still lower than the 0.3 percent accepted by the exchange. Compared with the optimal combination of margins, spot limits, and futures price-limit levels of 7 percent, 6 percent, and 6 percent when the futures index is less than 7,000, and 6.5 percent, 5 percent, and 6 percent when the futures index ranges from 7,000 to 9,000, it can be found that, when Ls = 7 percent, the optimal margin increases to 7.5 percent. The reason is that the looser the spot price limit is, the less the information from the spot market is restricted, and thus, the larger a margin is re- quired. In addition to futures price limits, spot price limits appear to partially substitute for margins in controlling contract performance. Price Limits of 7 Percent on Both the Spot and Futures Markets Because in the real world, 7 percent price limits are imposed on Taiwan’s spot and futures markets, we also consider the case of identical limits of 7 percent. The results are presented in Panel C of Table 4. They show that, when Ls = Lf = 7 percent, the optimal margin requirement is 7.75 percent of a contract’s value, regardless of the index level. Compared with 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 equals 6,000 points, the cost-minimizing total contract cost and margin require- ments with Ls = Lf = 7 percent are 19.0425 and 7.75 percent, which are larger than those attainable without this equality constraint of 18.0740 and 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, spot limits, and futures price-limits levels is 7 percent, 6 percent, and 6
  18. 18. JANUARY–FEBRUARY 2006 79 percent when the futures index is less than 7,000, and 6.5 percent, 5 percent, and 6 percent when the futures index ranges from 7,000 to 9,000. The wider the spot and futures limits are, the more margin is required. This is because the tighter the spot and the futures price lim- its are, the more information is restricted, and thus, the less the futures margin 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 liquidity cost from imposing spot price limits is the same as that from futures price limits. Because we cannot ascertain whether they are the same, we con- duct a sensitivity analysis to see if and how much the results are affected by the choices of parameter values about liquidity cost. The value αs = 1 shows that the liquidity cost of spot limits is the same as that of futures limits; αs = 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. They show that, when index points are within 6,000 and 6,500, the optimal combination of margin, spot limits, and futures limits is not sensitive to the choices of the liquidity cost for spot limits (αs). Likewise, when index points increase to 7,000 or more, the optimal combination about the margin and limits for αs = 0.8 and αs = 1 also comes to the same thing. Nevertheless, as the liquidity cost of spot limits is larger than that of futures limits (αs = 1.2) and index points are no less than 7,000, the margin required increases from 6.5 percent to 6.75 percent; the spot price limits are relaxed from 5 percent to 6 percent; and the default probability increases from 0.0018 percent to 0.0073 percent. Even faced with higher default probability and contract cost due to higher liquidity cost from imposing spot limits, the default probability for the optimal combination of limits and margins is also always less than 0.3 percent, which the exchange accepts to control the 99.7 percent risk of daily price volatility. In addition, the margin–volatility ratio is 3.169 (6.75 percent divided by 2.13 percent), which is larger than three. Thus, either from the cost-minimization model or from the acceptable default risk model, the optimal margin is less than that actually required by the TAIFEX, even when other parameter values of liquidity cost for spot limits are used.
  19. 19. 80 Table 5 Optimal Combinations of Margin Requirement, Spot Price Limits, and Futures Price Limits When Liquidity Cost of Spot Limits Differs from Futures Limits Margin Ls 1 Lf DP LPf LPs Costs Total Index αs (percent) (percent) (percent) (percent) (percent) Costf (percent) (percent) cost2 6,000 0.8 7 6 6 0.0003 0.4849 17.8069 0.0013 0.2137 18.0206 1 7 6 6 0.0003 0.4849 17.8069 0.0013 0.2672 18.0740 1.2 7 6 6 0.0003 0.4849 17.8069 0.0013 0.3206 18.1275 6,500 0.8 7 6 6 0.0003 0.4849 19.2069 0.0013 0.2137 19.4206 1 7 6 6 0.0003 0.4849 19.2069 0.0013 0.2672 19.4740 EMERGING MARKETS FINANCE AND TRADE 1.2 7 6 6 0.0003 0.4849 19.2069 0.0013 0.3206 19.5275 7,000 0.8 6.5 5 6 0.0018 0.4849 19.3563 0.0075 1.2090 20.5654 1 6.5 5 6 0.0018 0.4849 19.3563 0.0075 1.5513 20.8676 1.2 6.75 6 6 0.0073 0.4849 20.6014 0.0013 0.3206 20.9220 7,500 0.8 6.5 5 6 0.0018 0.4849 20.6563 0.0075 1.2090 21.8654 1 6.5 5 6 0.0018 0.4849 20.6563 0.0075 1.5113 22.1676 1.2 6.75 6 6 0.0073 0.4849 21.9515 0.0013 0.3206 22.2721 8,000 0.8 6.5 5 6 0.0018 0.4849 21.9563 0.0075 1.2090 23.1654 1 6.5 5 6 0.0018 0.4849 21.9563 0.0075 1.5513 23.4676 1.2 6.75 6 6 0.0073 0.4849 23.3015 0.0013 0.3206 23.6221
  20. 20. 8,500 0.8 6.5 5 6 0.0018 0.4849 23.2563 0.0075 1.2090 24.4654 1 6.5 5 6 0.0018 0.4849 23.2563 0.0075 1.5513 24.7676 1.2 6.75 6 6 0.0073 0.4849 24.6516 0.0013 0.3206 24.9722 9,000 0.8 6.5 5 6 0.0018 0.4849 24.5563 0.0075 1.2090 25.7654 1 6.5 5 6 0.0018 0.4849 24.5563 0.0075 1.5513 26.0676 1.2 6.75 6 6 0.0073 0.4849 26.0016 0.0013 0.3206 26.3222 Notes: This table presents the combinations of margin requirements, futures price-limits, and spot price limits for various index points. Under a price-limit rule, the optimal margin M and limit Lf are set to minimize the futures contract cost, that is, −L f  −L   2 α f Φ     2 αs Φ s   σ f1  *  σ s1  MinC ( M , L f , Ls ) = kM + + 2 βPγ f1 ≥ L f , Ls ≥ s1 ≥ s + 2 βΦ −α f Φ (−αs ) ⋅θ + ( ) ( ) ,  Lf  L  2Φ − 1  2Φ s − 1 σ   σ s1   f1  where α is an indicator function that takes the value of 1 if E(f1 + f2|f1 ≥ Lf, s1 ≥ Ls) ≥ M and 0 otherwise. The parameters of the cost function are k = 0.02 percent, αf = 1, β = 50, σf1 = 1,000, and σs1 = 800. Only the results of the cost-minimizing limit for each margin are presented. The probabilities of futures limit move and spot limit move are labeled by LPf and LPs, respectively. DP labels the default probability. 1 The combination of Ls and Lf is set to minimize the total spot and futures contract cost for a given margin. 2 The spot and futures contract cost of the optimal limit for a given margin. JANUARY–FEBRUARY 2006 81
  21. 21. 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 off the market and aid in resolving prices. Then, if price limits have an effect on the underlying price-generating process when a limit is hit, the story may be different. To help understand the cool-off effect of price limits on price behavior, assume that the true price change fol- lows an independent normal distribution with mean zero and variance σft2, that is, ft ~ N(0,σft2). Suppose further that time t – 1 is not a limit day, and an upper limit is hit at time t, then the potential price change following an up-limit move has the following conditional mean under the normality assumption: φ ( α1 ) ( ) E ft+1 + ft − L f ft ≥ L f = σ ft 1−Φ ( α1 ) −Lf , where α1 = Lf/σft · ϕ(·) and Φ(·) are the standard normal density and distribution functions, respectively. This indicates that the expected price change following an upper limit will increase. Likewise, the expected price change following a lower limit will decrease. If part of the price change is not fundamental, but transitory, and can be eliminated by introducing price limits, then as alleged by some price- limit proponents, price limits might reduce extreme price movement in the same direction by pulling the price back. Under normal distribution, the conditional mean after an upper limit is hit takes the following form:  φ ( α1 )  ( ) E ft+1 + ft ft ≥ L f = γ σ ft  1−Φ ( α ) .   1  It implies that the price-limit rule has a cool-off effect when γ has a value smaller than one. Table 6 presents the results for the case when price limits reduce the potential price change by 20 percent following a limit hit. That is, γ = 0.8. Panel A of Table 6 summarizes the results in Panel A of Table 4, in which price limits delay the price (γ = 1), and Panel 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-
  22. 22. Table 6 Optimal Combinations of Margin Requirements, Spot Price Limits, and Futures Price Limits for a Given Index When Price Limits Have a Cool-Off Effect Margin Ls 1 Lf DP LPf LPs Costs Total Index (percent) (percent) (percent) (percent) (percent) Costf (percent) (percent) cost2 Panel A: γ = 1 6,000 7.0 6.0 6.0 0.0003 0.4849 17.8069 0.1334 0.2672 18.0740 6,500 7.0 6.0 6.0 0.0003 0.4849 19.2069 0.1334 0.2672 19.4740 7,000 6.5 5.0 6.0 0.0018 0.4849 19.3563 0.7500 1.5113 20.8676 7,500 6.5 5.0 6.0 0.0018 0.4849 20.6563 0.7500 1.5113 22.1676 8,000 6.5 5.0 6.0 0.0018 0.4849 21.9563 0.7500 1.5113 23.4676 8,500 6.5 5.0 6.0 0.0018 0.4849 23.2563 0.7500 1.5113 24.7676 9,000 6.5 5.0 6.0 0.0018 0.4849 24.5563 0.7500 1.5113 26.0676 (continues) JANUARY–FEBRUARY 2006 83
  23. 23. 84 Table 6 (Continued) Margin Ls 1 Lf DP LPf LPs Costs Total Index (percent) (percent) (percent) (percent) (percent) Costf (percent) (percent) cost2 Panel B: γ = 0.8 6,000 6.0 6.0 7.0 6.77E-05 0.1015 14.6099 0.1334 0.2672 14.8771 6,500 6.0 6.0 7.0 6.77E-05 0.1015 15.8099 0.1334 0.2672 16.0771 7,000 6.0 6.0 7.0 6.77E-05 0.1015 17.0099 0.1334 0.2672 17.2771 7,500 6.0 6.0 7.0 6.77E-05 0.1015 18.2099 0.1334 0.2672 18.4771 8,000 5.0 6.0 6.0 3.23E-04 0.4849 18.6099 0.1334 0.2672 18.8740 8,500 5.0 6.0 6.0 3.23E-04 0.4849 19.7069 0.1334 0.2672 19.9740 9,000 5.0 6.0 6.0 3.23E-04 0.4849 20.8069 0.1334 0.2672 21.0740 EMERGING MARKETS FINANCE AND TRADE Notes: This table presents the combinations of margin requirements, futures price limits, and spot price limits for various index points when price limits have a cool-off effect to reduce extreme price movement in the same direction by pulling the price back. A value for γ smaller than 1 implies that the price-limit rule has a cool-off effect. The conditional mean after an up limit might take the following form:  φ ( α1 )  E ft+1 + ft γt ≥ L f = γ σ ft ( )  1−Φ ( α ) .   1  The parameters of the cost function are k = 0.02 percent, α = 1, β = 50, σf1 = 1,000, and σs1 = 800. Only the results of the cost- minimizing combinations of margin and limits for each index are presented. The probabilities of futures limit move and spot limit move are labeled by LPf and LPs, respectively. DP labels the default probability. 1 The combination of Ls and Lf is set to minimize the total spot and futures contract cost for a given margin. 2 The spot and futures contract cost of the optimal limit for a given margin.
  24. 24. JANUARY–FEBRUARY 2006 85 pected price, it can reduce margin, default probability, and contract cost to a greater degree. For example, when γ = 1, the optimal combinations of margin requirements, spot price limits, and futures price limits are 7 percent, 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 a cool-off effect (γ = 0.8), the optimal margin decreases to 6 percent of the value of the contract, and the corresponding contract cost reduces to 14.6099. The efficient futures contract calls for spot limits of 6 percent and futures limits of 7 percent, and the reneging probability reduces to 0.0000677 percent. Conclusion This study investigates the cost-minimizing combination of spot limits, futures limits, and margins for stock and index futures in the Taiwan market. Because price limits can lower price volatility and default prob- ability, margin requirements after price limits are imposed may be lower than those without price limits. Our empirical results support this view. The cost-minimization combination of margin, spot price limits, and futures price limits is 7 percent, 6 percent, and 6 percent when the index level 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 spot and futures limit pair of 5 percent and 6 percent, where the optimal margin, reneging probability, and corresponding contract cost are less than those without price limits. The default risk is also less than the 0.3 percent probability of the price move exceeding the margin, which the exchange accepts. This supports the finding that price limits, by preventing investors from real- izing the magnitude of their loss in the futures markets, may partially substitute for margin requirements in ensuring contract performance. On the other hand, when equal price limits of 7 percent are imposed on the futures and spot markets, the efficient contract cost is larger than that without this constraint. Though this may not coincide with efficient contract design, the common practice of imposing equal price limits of 7 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 are significantly smaller than those actually required by the exchange. Be-
  25. 25. 86 EMERGING MARKETS FINANCE AND TRADE cause excess margin requirements increase trading costs without clear benefits in return, it would be more efficient for the exchange to reduce the margin requirements. Notes 1. Telser (1981) argues that margins use up part of the trader’s precautionary balances, 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 knows that the adverse price movement exceeds the limit, but not exactly how much he will lose, he must conjecture about the size of his losses. Price limits thus create noise when the trader is forming an expectation about the unobserved equilibrium futures price. 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 in many stock markets, such as Austria, Belgium, China, France, Greece, Japan, Mexico, Spain, South Korea, Taiwan, and Thailand. For example, both the stocks and index futures of the Tokyo Stock Exchange and the TAIEX are traded under price limits. 4. Some recent reports by government regulatory agencies call for coordinating regulatory activities across financial markets. These studies include those conducted by the Chicago Board of Trade (1987), the Securities and Exchange Commission (SEC 1988), the Commodity Futures Trading Commission (CFTC 1988), and the General Accounting Office (GAO 1988). Specifically, the CFTC (1988) recommends that any price limits placed in force must consider their effects on other related markets. The GAO (1988) suggests that circuit breakers, such as price limits, must be coordinated across markets. 5. Assuming that daily volatility is 0.02427 and the index point is 5,000, the volatility coverage is 5,000 * 0.02427 * 3 = 364.5 index points. (Under normal distribution, 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 does not take into account the added risk inherent in leptokurtotic data (see, e.g., Warshawsky 1989). However, as documented by Hull (1993), among the factors that may affect the committee’s margin-setting decision (including underlying asset price levels, underly- ing asset price volatility, volume, and so on), volatility in the underlying asset price is the primary factor affecting the margin-level decision. As a result, we focus on the price 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 all parties to fulfill it without the threat of legal action (Brennan 1986).
  26. 26. 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 and fat-tailedness by scaling up the values of the parameters α and β to accommodate the potential underestimates of probabilities due to the normality assumption. While not reported in the paper, we find that our results are not sensitive to the choices of the parameter values, suggesting that our results are not sensitive to the presence of nonnormality. 10. Closing returns are used in this paper because the margin is determined with them, according to the criteria for covering at least 99.7 percent of the daily fluctuations. 11. The specific functional forms for each of the costs and the associated cost components 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 about 9.935 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 Science 43, 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 Stock Index 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 the Presidential Task Force on Market Mechanisms.” Cotter, J. 2001. “Margin Exceedences for European Stock Index Futures Using Extreme 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 Index Futures 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 Futures Contract 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.
  27. 27. 88 EMERGING MARKETS FINANCE AND TRADE Longin, François M. 1999. “Optimal Margin Level in Futures Markets: Extreme Price Movements.” Journal of Futures Markets 19, no. 2: 127–152. SEC (Securities and Exchange Commission). 1988. “The October 1987 Market Break.” 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 of Futures Markets 8, no. 3: 420–437. To order reprints, call 1-800-352-2210; outside the United States, call 717-632-3535.

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