Chapter 16
Margin Setting and Extreme Value Theory
John Cotter1 and Kevin Dowd2
1UCD School of Business, University College Dublin, Dublin, Ireland
2Economics and Finance, Durham University Business School, Durham, United Kingdom
AMS 2000 subject classification. Primary 62G32, 62E20; Secondary 65C05.
16.1 Introduction
This chapter outlines some key issues in the modeling of futures margins using extreme value theory (EVT). We examine its application in the context of setting initial futures margins. There is a developing volume of literature that examines using EVT in setting margins, and this study provides an overview of the key issues that have been examined.1 The chapter illustrates how EVT can be a useful approach in the setting of margins with examples for a set of stock index futures.
The successful operation of futures exchanges for the relevant stakeholders such as traders and the exchange necessitates that there be a tradeoff between optimizing liquidity and prudence.2 The imposition of margins is the mechanism by which these objectives are met.3 Margin requirements act as collateral that investors are required to pay to reduce default risk. Default risk is incurred if the effect of the futures price change is at such a level that the investor's margin does not cover it, leading to nonpayment by one of the parties to the contract.4 Margin committees face a dilemma, however, in determining the magnitude of the margin requirement imposed on futures traders. On one hand, setting a high margin level reduces default risk. On the other hand, if the margin level is set too high, then the futures contracts will be less attractive for investors due to higher costs and decreased liquidity, and finally less profitable for the exchange itself. This quandary has forced margin committees to impose investor deposits, which represent a practical compromise between meeting the objectives of adequate prudence and liquidity of the futures contracts.
The clearinghouse sets initial margins so that this deposit protects against a vast range of possible price movements with a relatively low probability that actual price changes exceed the margin. This is equivalent to modeling the initial margin as a quantile or value at risk (VaR) estimate. Using this, the clearinghouse imposes margins based on a statistical analysis of price changes, and adjusts this to take account of other factors such open interest, volume of trade, concentration in futures positions, and the margins of competing exchanges. This chapter deals with the statistical modeling element of margin setting and examines different approaches in the context of using EVT and the application of nonparametric measures for the optimal margin level. Given the true distribution of futures price changes being nonnormal and, in fact, unknown (e.g., Cotter and McKillop, 2000; Hall et al., 1989), it is appropriate to examine margins from using a number of possible distributions. We do so in by comparing margins that are set using EVT, the normal distribution, and the Student-t distribution.
The chapter proceeds as follows. In the next section we provide details of margin setting. This is followed by a discussion of the methods, primarily EVT, that should be used for modeling margins. We then illustrate some empirical results for a selection of stock index futures. Finally, we provide some concluding comments.
16.2 Margin Setting
We begin with a discussion of margin setting in futures markets. We outline the different elements of a margin account and provide examples of how margins are set in practice and in the literature.
Futures clearinghouses use margin requirement accounts to minimize default risk and act as counterparty to all trades that take place within its exchanges. This provides stability and encourages trading. Setting margins ensures that individual traders do not have to concern themselves with credit risk exposures to other traders, because the clearinghouse assumes all such risks itself. Margin requirements consist of an initial margin (deposit) and a variation or daily margin, assuming a minimum or maintenance margin is breached. The variation margin will result in a margin call, where the broker asks the trader to top up their margin account. The focus of this chapter, the initial margin, represents the deposit a futures trader must give to a clearinghouse to initiate a trade.
The modeling approaches followed for initial and variation margins are usually quite distinct. Setting initial margins utilizes the unconditional distribution of returns. Typically, there is a focus on the tails of the distribution and modeling the asset price movements for extreme confidence levels for extraordinary market events so as to minimize the probability that the associated quantile is exceeded. Most of the previous literature has focused on this approach as we do in this chapter. The variation margin comes into play once the futures contract is trading, and can thought of as supporting the initial margin after it has been breached to help avoid trader default. Here, the focus is on the conditional distribution of returns where we would be interested in the levels of volatility during the lifetime of the futures contract.
We now provide an example of the way the initial margins are typically set by clearinghouses and the exchange on which the futures are traded.
Let us describe as an example the way margins are set on the London International Financial Futures and Options Exchange (LIFFE). For products traded on this exchange, margin requirements are set by the LCH.Clearnet Group (London Clearing House, LCH). The LCH risk committee is responsible for all decisions relating to margin requirements for LIFFE contracts. Margin committees generally involve experienced market participants who have widespread knowledge in dealing with margin setting and implementation, through their exposure to various market conditions and their ability to respond to changing environments. The LCH risk committee is independent of the commercial function of the Exchange. In order to measure and manage risk, the LCH uses the London systematic portfolio analysis of risk (SPAN) system, a specifically developed variation of the SPAN system originally introduced by the Chicago Mercantile Exchange (CME). The London SPAN system is a nonparametric, risk-based model that provides output of actual margin requirements that are sufficient to cover potential default losses in all but the most extreme circumstances.5 The inputs to the system are a set of estimated margin requirements relying on price movements that are not expected to be exceeded over a day or a couple of days. These estimated values are based on diverse criteria incorporating a focus on a contract's price history, its close-to-close price movements, its liquidity, its seasonality, and forthcoming price sensitive events. Market volatility is especially a key factor to set margin levels. Most important, however, is the extent of the contract's price movements with a policy for a minimum margin requirement that covers three standard deviations of historic price volatility based on the higher of 1-day or 2-day price movements over the previous 60-day trading period. This is akin to using the normal distribution, where multiples of standard deviation covers certain price movements at various probability levels.6
In the literature, margins have been typically modeled as a quantile or VaR, and we interchange between these terms in this chapter. The clearinghouse then selects a particular confidence level, and sets the margin as the VaR at this confidence level.7
16.3 Theory and Methods
We concentrate our discussion of setting margins using EVT. Here we will discuss two commonly used approaches: estimation assuming maximum domain of attraction (MDA), and fitting excesses (extremes) over a threshold. There is also a third approach, where one would estimate the parameters for an extreme value distribution. This is less applied in margin setting but an application is outlined in Longin (1999). We also provide a brief discussion of non-EVT approaches such as assuming alternative distributions, the most common being normality and nonparametric approaches, for example, historical simulation.
16.3.1 Maximum Domain of Attraction
The statistical properties of financial returns have interested many, and none more so than the modeling of tail returns and the fitting of candidate distributions. For initial margin setting and associated measures of violation probabilities, it is common to use the theoretical framework of EVT. Three alternative extreme value distributions are detailed, the Weibull, the Gumbell, and the fat-tailed Fréchet distribution.
The distributional assumptions of EVT are applicable through the MDA, allowing for approximation to certain distributional characteristics rather than being required to belong to a specific distribution (Leadbetter et al., 1983; Fraga Alves and Neves, 2017). Thus, financial returns do not have to exactly fit a particular set of distributional assumptions. Rather, our analysis assumes that the return series have extreme values that are approximated by a Fréchet-type distribution, and this implies that the series belong to the MDA of the Fréchet distribution. Advantageously, it avoids our having to ascertain the exact form of a, for example, fat-tailed distribution that matches the data we are analyzing. While there is a general agreement on the existence of fat tails for financial data, its exact form for all financial returns is unknown. For this reason, it is appropriate to deal with approximation of the Fréchet distribution in the sense of being in the MDA. This property will lead to the use of nonparametric statistics in modeling tail returns.
The three extreme value distributions can be divided into three separate types depending on the value of their shape parameter α. The classification of a Weibull distribution (α < 0) includes the uniform example where the tail is bounded by having a finite right end point and is a short-tailed distribution. The more commonly assumed class of distributions used for asset price changes includes the set of thin-tailed densities. This second classification of densities includes the normal and gamma distributions, and these belong to the Gumbell distribution, having a characteristic of tails decaying exponentially. Of primary concern to the analysis of fat-tailed distributions is the Fréchet classification, and examples of this type generated here are the Cauchy, Student-t, ordinary Fréchet, and the Pareto distributions. This important classification of distributions for extreme asset price movements has tail values that decay by a power function. A vast literature on asset returns has recognized the existence of fat-tailed characteristics, and thus we will focus on the Fréchet type of extreme value distribution.
Turning to the task of margin setting using futures, we examine a sequence of futures returns {R} arranged in ascending order and expressed in terms of the maxima (Mn) of n random variables belonging to the true unknown cumulative probability density function F, where
The corresponding density function of Mn is obtained from the cumulative probability relationship, and this represents the probability of exceeding a margin level on a short position for n returns:
where rshort represents the margin level on a short position.
With margin setting required for both long and short positions, we are interested in both the upper and lower tails of the distribution Fn(r). We also can apply EVT to examine the associated lower order statistics where lower tail price movements are relevant for margin requirements of a long position in a futures contract. The theoretical framework for examining sample minima tail statistics can easily be converted by applying the identity Min{R1, R2, … , Rn} = −Max{−R1, −R2, … , −Rn}. The corresponding probability expression for exceeding a margin level on a long position for n returns is
where rlong represents the margin level on a long position.
16.3.2 Tail and Probability Estimators
Because of the semiparametric specification of being in the MDA of the fat-tailed Fréchet distribution, it is appropriate to apply nonparametric measures of our tail estimates. We apply the commonly used nonparametric Hill index (1975) that determines the tail estimates of the stock index futures (see Beirlant et al., 2017). It is given as
This tail estimator is asymptotically normal, that is, 
(Hall, 1982).
As this study is examining the probability of a sequence of returns exceeding a particular margin level relying on expressions (16.2) and (16.3), an empirical issue arises in determining the number of returns entailed in the tail of a distribution. From a large number of methods of identifying the optimal tail threshold, we adopt the approach proposed by Phillips et al. (1996). The optimal threshold value Mn, which minimizes the mean square error of the tail estimate γ, is 
, where λ is estimated adaptively by 
.
In setting margins, futures exchanges and clearinghouses would be interested in variations in the upper and lower tail values. If these are invariant across tails, the clearinghouse can set similar margins for long and short trading positions. To investigate this, the tail index estimator is used to determine each tail individually, but it is also used to measure a common margin requirement encompassing the extreme price movements of both tails. The relative stability of the tail measures determines the optimal margin policy. Stability across the tails supports the hypothesis of having a common margin requirement regardless of trading position, and instability suggests the need for separate margin levels. Tail stability is tested using a statistic suggested by Loretan and Phillips (1994):
where γ+ (γ−) is the estimate of the right (left) tail.
The application of EVT allows us produce different margin measures. We outline a measure that allows us to determine the probability of exceeding a certain price movement. From this, the setting of optimal margin requirements can be made based on an examination of the violation probability for a range of price movements in association with the tradeoff between optimizing liquidity and prudence for an exchange's contract. The nonparametric measure detailing the probability p of exceeding a certain large price change rp for any tail measure is
Using (16.6), a related nonparametric measure examines the margin level or quantile that would not be violated for particular extreme price movements rp at different probabilities p:
We can then compare these measures with alternative approaches such as using a Student-t distribution at similar confidence levels.
16.3.3 Peaks Over Threshold
An alternative extreme value approach is to use the peaks over threshold (POT) (generalized Pareto) distribution (see Cotter and Dowd, 2006). Here, the risks of extremely high losses can be modeled with the POT approach based on the generalized Pareto distribution (GPD).8 This approach focuses on the realizations of a random variable R (in our case futures returns) over a high threshold u. More particularly, if R has the distribution function F(r), we are interested in the distribution function 
of exceedances of R over a high threshold u. As u gets large (as would be the case for the thresholds relevant to clearinghouses), the distribution of exceedances tends to a GPD. The shape ξ and scale β > 0 parameters of the GPD are estimated conditional on the threshold u (Embrechts et al., 1997, pp. 162–164). We have described a variant of the shape parameter before as the inverse of the tail index parameter, and which we are using the well-known Hill estimator. The GPD parameters can be estimated by maximum likelihood methods. Maximum likelihood estimates are then found by maximizing the log-likelihood function using suitable (e.g., numerical optimization) methods.
16.3.4 Further Models
We turn our attention briefly to some non-EVT approaches to setting margins. We provide a brief overview of different statistical models that have been used to compute the margin level for a given probability. Many of these, for example, the use of the normal distribution and historical simulation methods, are heavily relied upon in practice and also discussed extensively in the literature.
First, we discuss the normal distribution. It is commonly applied in risk management and in margin setting. For example, the SPAN margin system estimates normal margin requirements. It requires the estimation of two parameters only, the mean μ and the variance σ2. For a given probability p, the margin level corresponds to the quantile or VaR where one is examining what margin level is sufficient to exceed futures price changes over a time-period of length T for the probability level p. Because of the fat-tailed phenomena of financial returns, it tends to underestimate tail behavior and the associated margins associated with futures price movements. An alternative distribution often suggested as overcoming this weakness but keeps its strong links with normality is the the Student-t distribution. Here, the number of moments of the distribution is detailed by the t parameter. A further approach not relying on distributions is the use of historical simulation. This is heavily used in practice, where margins are calculated as quantiles of the historical distribution of returns. As there is no model per se, it avoids model risk. Returns of the historical distribution are ordered in an ascending manner and the margin is read of as a quantile of the distribution.
All the above approaches assume that we are modeling an unconditional distribution of futures returns. However, during the lifespan of a contract the inherent volatility that drives the level of deposit in the margin account may vary substantially. This would potentially involve an initial margin being breached and a margin call taking place requiring variation margin being posted. To model this, it is important to look at the conditional distribution of returns and look at time-varying approaches. The most commonly applied of these is the use of generalized autoregressive conditional heteroscedasticity (GARCH) models. For example, one study exclusively uses GARCH models in the setting of variation margins (see Cotter and Dowd, 2012). As the variation margins require modeling of returns as the contract trades, the modeling of the associated volatility is of paramount importance. As we have seen clearly before, during and after financial crises, market conditions may vary substantially over time, and a conditional model like a GARCH process would attempt to capture this variability.
Now we turn to the use of EVT in setting margins with an illustration of modeling margins in practice.
16.4 Empirical Results
We are going to illustrate the use of EVT in setting initial margins. We will also show results for other approaches. First though, we will describe the types of preliminary analysis that would be followed in setting margin requirements. The analysis requires some futures returns data. This will be analyzed both as a full distribution of returns but also for break-outs of upper and lower distributions. The main focus of analysis is the tails of the distribution.
To set the scene, Figure 16.1 shows the distribution of asset returns where we have marked out the upper tail of the distribution. Going from the lower distribution that would deal with the returns of a long position, the upper distribution focuses on a short position. Here, a quantile of the distribution is identified, and this represents an initial margin or margin requirement. Returns in excess of this would represent violations of the associated initial margins. These violations are large price movements and would be those that might result in investor default. It is these large returns that we will model using EVT, and we will determine whether a large movement is covered by an initial margin or whether it represents a violation of the size of the margin deposit.
Figure 16.1 Margin requirements for a short position and a distribution of returns. This figure illustrates a distribution of futures returns with special emphasis on the short position. At the upper tail of the distribution, a certain margin requirement is identified. Any price movement in excess of this margin requirement, given by the shaded area, represents a violation of this by the investor.
Figure 16.1 is the well-known and heavily applied normal distribution. Financial returns, for example, those of futures, have relatively fat tails, and this characteristic is shown in Figure 16.2. Here, the probability mass for the fat-tailed distribution is greater around the tails vis-à-vis the normal distribution. This implies that volatility levels calculated assuming normality would be less (and incorrectly so) than that which reality dictates. This leads us to using EVT and the modeling of tail returns with the Fréchet distribution.
Figure 16.2 Fat-tailed and normal distributions. This figure illustrates the tail distribution of futures returns with fat tails (dashed curve) and a normal distribution (solid curve). The fat tails ensure more probability mass in the tails than the normal distribution.
We now present some EVT statistics where we model the tail behavior for a selection of European stock index futures. We convert the price data into returns using the first difference of the natural logarithm of closing-day quotes. Some summary statistics for the series is given in Table 16.1. We report findings in common with those reported for financial returns. Returns do not belong to a normal distribution, and there is excess kurtosis and, in general, excess skewness. Note the excess kurtosis describes the fat-tailed characteristic of the data. The lack of normality also suggests that using this distribution to model margins is inappropriate. The lack of normality and how it relates to the tail returns can be clearly seen from a Q–Q plot. An example is given for the DAX index in Figure 16.3. Here it is clear that there is extensive deviation for this contract's returns from the normal distribution as given by the straight line.
Table 16.1 Summary statistics for stock index futures
| Contract | Meana | Minimuma | Maximuma | Skewness | Kurtosis | Normality |
| BEL20 | 0.06 | −5.26 | 5.48 | −0.11b | 4.40 | 0.07 |
| KFX | 0.04 | −7.80 | 6.65 | −0.33 | 5.19 | 0.08 |
| CAC40 | 0.04 | −7.74 | 8.63 | −0.08b | 3.36 | 0.05 |
| DAX | 0.06 | −12.85 | 8.38 | −0.56 | 8.31 | 0.08 |
| AEX | 0.06 | −7.70 | 7.28 | −0.33 | 5.94 | 0.08 |
| MIF30 | 0.08 | −7.84 | 7.07 | −0.06b | 2.27 | 0.06 |
| OBX | 0.04 | −19.55 | 21.00 | 0.32 | 97.69 | 0.18 |
| PSI20 | 0.13 | −11.55 | 6.96 | −0.87 | 7.95 | 0.11 |
| IBEX35 | 0.07 | −10.84 | 7.25 | −0.49 | 4.72 | 0.07 |
| OMX | 0.06 | −11.92 | 10.81 | −0.27 | 8.93 | 0.07 |
| SWISS | 0.08 | −9.09 | 7.08 | −0.50 | 9.30 | 0.06 |
| FTSE100 | 0.05 | −16.72 | 8.09 | −1.18 | 18.78 | 0.05 |
The summary statistics are presented for each future's index returns. The mean, minimum, and maximum values represent the average, lowest, and highest returns, respectively. The skewness statistic is a measure of distribution asymmetry, with symmetric returns having a value of zero. The kurtosis statistic measures the shape of a distribution vis-à-vis a normal distribution, with a normal density function having a value of zero. Normality is formally examined with the Kolmogorov–Smirnov test, which indicates a normal distribution with a value of zero.
a Statistics are expressed in percentages.
b Represents insignificant at the 5% level, whereas all other skewness, kurtosis, and normality coefficients are significant different from zero.
Figure 16.3 Q–Q plot of DAX log returns series. This figure plots the quantile of the empirical distribution of the DAX futures index returns against the normal distribution. The plot shows whether the distribution of the DAX returns matches a normal distribution. The straight line represents a normal quantile plot, whereas the curved line represents the quantile plot of the empirical distribution of the DAX contract. If the full set of DAX returns followed a normal distribution, then its quantile plot should match the normal plot and also be a straight line. The extent to which these DAX returns diverge from the straight line indicates the relative lack of normality.
We now turn to our explicit modeling of the futures returns using EVT. We start by showing the Hill index tail estimates in Table 16.2. The table shows the optimal number of tail returns and nonparametric Hill index estimates for the lower tail, the upper tail, and both tails, corresponding to the shape parameter used in the calculation of long, short, and common margin requirements. The optimal number of returns in each tail appears to be reasonably constant, hovering around the 5% mark for each contract.
Table 16.2 Optimal tail estimates for stock index futures
| Contract | m− | γ− | m+ | γ+ | m* | γ* | γ+ − γ− |
| BEL20 | 63 | 2.81 | 68 | 2.85 | 92 | 3.20 | 0.07 |
| (0.35) | (0.35) | (0.33) | |||||
| KFX | 72 | 2.65 | 78 | 3.01 | 111 | 2.98 | 0.78 |
| (0.31) | (0.34) | (0.28) | |||||
| CAC40 | 100 | 2.97 | 103 | 3.33 | 137 | 3.89 | 0.83 |
| (0.30) | (0.33) | (0.33) | |||||
| AEX | 95 | 2.72 | 95 | 2.93 | 141 | 2.95 | 0.51 |
| (0.28) | (0.30) | (0.25) | |||||
| DAX | 83 | 2.92 | 83 | 3.07 | 116 | 3.30 | 0.33 |
| (0.32) | (0.34) | (0.31) | |||||
| MIF30 | 55 | 3.31 | 55 | 3.66 | 77 | 3.41 | 0.53 |
| (0.45) | (0.49) | (0.39) | |||||
| OBX | 71 | 2.04 | 70 | 2.89 | 113 | 2.47 | 2.02• |
| (0.24) | (0.35) | (0.23) | |||||
| PSI20 | 41 | 1.91 | 43 | 2.38 | 57 | 2.32 | 0.99 |
| (0.30) | (0.36) | (0.31) | |||||
| IBEX35 | 74 | 2.62 | 78 | 3.35 | 118 | 3.21 | 1.50 |
| (0.30) | (0.38) | (0.30) | |||||
| OMX | 88 | 2.59 | 98 | 2.77 | 177 | 2.67 | 0.46 |
| (0.28) | (0.28) | (0.20) | |||||
| FTSE100 | 126 | 2.99 | 134 | 3.36 | 204 | 3.21 | 0.93 |
| (0.27) | (0.29) | (0.22) | |||||
| SWISS | 72 | 2.81 | 84 | 2.89 | 103 | 3.12 | 0.17 |
| (0.33) | (0.32) | (0.31) |
Hill tail estimates γ are calculated for lower tail, the upper tail, and both tails for each stock index future. The symbols −, +, * represent the lower tail, upper tail, and both tails, respectively. The optimal number of values in the respective tails, m, is calculated following the method proposed by Phillips et al. (1996). Standard errors are presented in parentheses for each tail value. Tail stability is calculated in the last column, with the symbol • representing significant different upper and lower tail values at the 5% level.
All the tail estimates range between 2 and 4 with the exception of the lower tail estimate for the Portuguese PSI20 contract, indicating their fat-tailed characteristic. We find that all values are significantly positive, corresponding to the requirement that 
. Testing the stability in tails from lower and upper values, findings indicate that while the right tail estimators are always greater than their left tail counterparts, a common margin requirement is sufficient based on a 5% significance level in each case with the exception of the OBX index. This implies that common initial margin for long and short traders would be sufficient based on the contracts analyzed.
We now look at the VaR or quantile, which would be of interest to the clearinghouse in setting initial margins, and compare the results for different distributions. What are the margin levels, or quantiles, required to cover a range of extreme price changes under the assumptions of a normal and Fréchet distribution? The latter method explicitly assumes the existence of fat-tail returns for stock index futures. The results are presented in Table 16.3, which incorporate the common margin requirement, so 98% covers all eventualities with the exception of a 2% default, that is, 1% long and short, respectively.
Table 16.3 Common margin requirements to cover extreme price movements
| Contract | Method | 98% | 99% | 99.80% | 99.90% | 99.98% |
| BEL20 | Normal | 2.15 | 2.38 | 2.86 | 3.04 | 3.45 |
| Student-t | 4.20 | 5.40 | 9.44 | 11.95 | 20.52 | |
| Extreme | 3.10 | 3.88 | 6.52 | 8.16 | 13.72 | |
| KFX | Normal | 2.47 | 2.73 | 3.27 | 3.48 | 3.95 |
| Student-t | 7.37 | 10.51 | 23.63 | 33.45 | 74.84 | |
| Extreme | 3.93 | 4.96 | 8.53 | 10.76 | 18.48 | |
| CAC40 | Normal | 2.93 | 3.25 | 3.89 | 4.14 | 4.70 |
| Student-t | 5.72 | 7.36 | 12.86 | 16.28 | 27.96 | |
| Extreme | 3.92 | 4.69 | 7.09 | 8.48 | 12.83 | |
| AEX | Normal | 2.61 | 2.89 | 3.46 | 3.69 | 4.18 |
| Student-t | 7.80 | 11.12 | 25.01 | 35.40 | 79.20 | |
| Extreme | 3.95 | 5.00 | 8.62 | 10.91 | 18.82 | |
| DAX | Normal | 2.97 | 3.28 | 3.93 | 4.19 | 4.75 |
| Student-t | 5.78 | 7.44 | 13.00 | 16.45 | 28.26 | |
| Extreme | 4.20 | 5.19 | 8.45 | 10.43 | 16.98 | |
| MIF30 | Normal | 3.88 | 4.30 | 5.15 | 5.48 | 6.22 |
| Student-t | 7.57 | 9.73 | 17.02 | 21.54 | 37.00 | |
| Extreme | 5.34 | 6.54 | 10.48 | 12.83 | 20.56 | |
| OBX | Normal | 2.83 | 3.14 | 3.76 | 4.00 | 4.54 |
| Student-t | 8.47 | 12.08 | 27.17 | 38.45 | 86.03 | |
| Extreme | 3.83 | 5.07 | 9.72 | 12.86 | 24.66 | |
| PSI20 | Normal | 3.78 | 4.19 | 5.02 | 5.34 | 6.06 |
| Student-t | 11.31 | 16.11 | 36.25 | 51.31 | 114.80 | |
| Extreme | 6.55 | 8.83 | 17.68 | 23.85 | 47.74 | |
| IBEX35 | Normal | 3.46 | 3.83 | 4.58 | 4.88 | 5.53 |
| Student-t | 6.73 | 8.66 | 15.15 | 19.17 | 32.93 | |
| Extreme | 4.79 | 5.94 | 9.81 | 12.17 | 20.09 | |
| OMX | Normal | 3.67 | 4.07 | 4.87 | 5.19 | 5.88 |
| Student-t | 10.53 | 15.00 | 33.74 | 47.76 | 106.86 | |
| Extreme | 5.35 | 6.93 | 12.65 | 16.40 | 29.95 | |
| FTSE100 | Normal | 2.56 | 2.83 | 3.39 | 3.61 | 4.09 |
| Student-t | 4.98 | 6.41 | 11.20 | 14.17 | 24.35 | |
| Extreme | 3.41 | 4.23 | 6.99 | 8.67 | 14.31 | |
| SWISS | Normal | 2.13 | 2.36 | 2.83 | 3.01 | 3.42 |
| Student-t | 4.16 | 5.35 | 9.36 | 11.84 | 20.34 | |
| Extreme | 2.99 | 3.73 | 6.24 | 7.79 | 13.04 |
The values in this table represent the margin requirements needed to cover a range of extreme price movements for each contract, for example, 98% of all movements. The associated margin requirements are calculated relying on extreme value theory, normal, and Student-t distributions. Student-t degrees of freedom are given by the Hill tail estimates γ*, which are also incorporated in the extreme value estimates. Values are expressed in percentages.
Dealing with a normal and extreme value comparison, it is clear that the former method underestimates the true margin requirement for any price movement and that this becomes more pronounced as you try to cover the larger price movements. This indicates that the fat-tailed characteristic has greater implications for margin setting as you move to greater extremes. The inclusion of the Student-t distribution allows for some degree of fat tails (the respective Hill tail estimates are used to proxy for the degrees of freedom for the different futures). These Student-t results are larger than the other measures because of estimated degrees of freedom.
16.5 Conclusions
In this chapter, we provided an overview of margin setting for futures contracts using EVT. We discussed the margin account and its composition. We detailed how margins are set in practice. Margins are important. Each exchange's clearinghouse must impose margins given the relationship of securing the safety of the exchange against large price movements for contracts and encouraging investor participation in trading. Low margins discourage (encourage) the former (latter), whereas high margins discourage (encourage) the latter (former).
We examined the potential role of EVT in margin setting. Here, a statistical analysis was pursued to calculate margin levels focusing on extreme price movements of futures. These returns are located in the tails rather than the entire distribution, as it is the violation of these that margin requirements are meant to combat against. Given previous findings of futures price changes being associated with fat tails, EVT and the limiting Fréchet distribution were applied in the calculation of the risk characteristics for the futures analyzed. The tail indexes were measured using the nonparametric Hill estimates, and this is appropriate given the semiparametric nature of the relationship of a set of futures price changes and the Fréchet distribution.
We illustrated the use of the Hill tail index by estimating VaRs, that is, the margin requirement that would be imposed to protect investors from a range of extreme price movements for different confidence levels. For a selection of stock index futures, we found that common margin requirements are generally sufficient. We illustrated that assuming normality results in underestimation and smaller margins than using EVT, and this becomes more pronounced as you try to protect against returns further out in the tail of a distribution.
This is an area that is ripe for further work. Issues that might be considered include looking at setting margins for other assets such as options. Here you would have to take the nonlinear payoffs into account when modeling potential tail behavior. Further, the discussion here was focused on univariate modeling of individual contracts or for a collection of contracts. We have not commented on the potential for using EVT in a multivariate context where you would account for diversification effects in a portfolio context. Diversification allows for the netting-off of some of the individual contracts risk and would require assessing the modeling of dependence between tails of multiple assets.
Acknowledgment
Cotter acknowledges the support of the Science Foundation, Ireland, under Grant Number 08/SRC/FM1389. We thank the editor, Francois Longin, for helpful comments on this chapter.
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