To mitigate credit risk in trading, one or both parties may be required to post both initial and variation margin. While initial margin is posted by both counterparties at the inception of the trade, variation margin is periodically exchanged (daily or even intra-daily) during the life of the trade due to changes in the mark to market value of the position.

The margin protects against a counterparty defaulting if the trade doesn’t move in their favour. In the case of centrally cleared derivatives, an intermediary manages the margining requirements of the counterparties. For OTC derivatives which are reasonably standardized, central clearing may still be a regulatory requirement. However, for OTC derivatives that are bespoke and illiquid, determining appropriate margining requirements may be more challenging due to the difficulty in regular valuation of the contracts. Margining requirements have become more stringent since the GFC.

While one does not want to be inadequately protected against non-payment by a counterparty, excessive margining also imposes costs on the derivatives markets. The immediate questions is, how does one estimate the amount of margin required by a counterparty to ensure that, at some chosen statistical confidence level, they can meet their obligations? This question is the subject of this article.

While margining is often a regulatory requirement, financial services firms including proprietary trading firms and brokers/exchanges, may wish to conduct their own margin or collateral modelling to ensure they are protected against default by counterparties due to sudden changes in valuation of their trades. An example of this in the context of cryptocurrency exchanges is provided later in the article.

## Margin modelling methodologies

Approaches to margin modelling may have much in common with methodologies used for market risk calculations. In particular, value at risk (VAR) and expected shortfall methods. The VAR method assume a distribution type with which to model risk factor (market data) shifts (typically a normal distribution), calibrates the distributions to the market somehow (typically historical simulation, which analyses the shifts in market variables over some specified time interval), and then calculates the 99% quantile worst market move (or some other specified confidence level) of the derivative or portfolio valuation over some given time horizon. Expected shortfall methods are a slight modification on this approach which look at the mean of all outcomes above the 99% quantile.

More frequent margin calls will reduce the credit risk (due to the shorter time horizon available for the market to move), at the expense of more operational inconvenience.

These methods require careful choice of a historical window for calibration. If only the most recent data is used, rather than a historically stressed period, it may cause over leveraging during more stable market periods, leading to higher exposure when the markets become more tumultuous.

Another question that arises is how to calculate margin requirements for an entire portfolio instead of a single trade. A portfolio may have offsetting trades, and a decision must be made about what correlations to assume and what level of diversification benefit to allow.

## The ISDA standard initial margin model (SIMM)

To avoid disputes over margining amounts, it’s important to have a single, transparent and standardized approach that all parties must adhere to. The International Swaps and Derivatives Association prescribes an approach known as the standard initial margin model. The ISDA documentation for this approach can be found here.

The ISDA includes four product classes: Interest rates and FX (ratesFX), credit, equity and commodity.

Six “risk classes” are specified. These are market data categories on which trade valuations depend. They are interest rate, credit (qualifying), credit (non qualifying), equity, commodity, and FX. Each trade depends on one or more “risk factors” from each category, such as interest rate curves, equity prices and exchange rates. The model postulates that the moves in any given market data or “risk factor” are normally distributed, which is also the approach commonly taken for market risk. The changes in trade valuations are also assumed to be normally distributed, something which is not actually true for nonlinear products like options even when the risk factor shifts are normal, and which becomes less true the larger the shifts are. The valuation shifts depend on the changes in the risk factors through trade sensitivities.

We first discuss how to calculate the initial margin for an individual trade. The approach is to calculate the delta, vega and interest rate sensitivities of the trade. The ISDA has calculated numbers representing how much they believe each risk factor will move at 99% confidence, which then multiply the sensitivities. The process by which the user finds the correct number is fairly intricate, and we won’t attempt to reproduce every detail here. The approach also takes into account the ISDA’s view on correlations between risk factors.

For options, a “curvature” term is also added. It is actually the vega contribution scaled by a “scaling factor”, which represent’s the ISDA’s view on how gamma and vega are related for vanilla options.

Having calculated the initial margin for an individual trade, we now discuss how to aggregate trades to calculate the initial margin of the entire portfolio. A fantastic simplifying feature of normal distributions, is that the sum of two normal distributions is another normal distribution with some new standard deviation. The new standard deviation is a function of the old ones, calculated using what I will call the “square root rule”: \(\)

\[ \sigma_{agg} = \sqrt{\sum_i \sigma_i + \sum_r \sum_{s \neq r} \psi_{rs} \sigma_r \sigma_s},\]

where \(\psi\) is a matrix of correlations between the distributions. One can aggregate the normally distributed valuation shifts for different risk factors for any given trade, giving a total normal distribution for each trade. One can then aggregate all these normal distributions across all trades in the same way, yielding a new normal distribution corresponding to the total valuation shift in the whole portfolio..

The ISDA methodology has three differences from the procedure described above. Firstly, it applies the square root rule to the initial margins, rather than the standard deviations of the distributions as described above. Since an initial margin is just some quantile of a normal distribution distribution, it is proportional to the standard deviation, so that the square root rule applies to the initial margins just as like it does to the standard deviations.

Secondly, It’s easy to verify that for the square root rule, the order in which we aggregate all the normal distributions doesn’t matter. The ISDA approach aggregates the normal distributions in a specific order. For each product class, and within each risk class \(X\) within the product class, it first aggregates the delta margin across all trades, and similarly the vega margin, curvature margin and base correlation margin. It then aggregates these four distributions as a simple sum:

\[ IM_X = DeltaMargin_X + VegaMargin_X + CurvatureMargin_X + BaseCorrMargin_X \]

This is the third difference. This sum is more conservative than that obtained by using the square root rule, and assumes that there is no diversification benefit between these four types of moves. The ISDA may believe that in stressed conditions, these different classes of risk factors may move together.

The ISDA then uses the square root rule to aggregate these initial margins across all risk classes within the product class. Just like with the four margin types, the ISDA stipulates that no diversification benefit may be assumed between the product classes, so that the total amount of initial margin required is the sum of those for each of the four product classes:

\[ SIMM_{Product} = SIMM_{RatesFX}+SIMM_{Credit}+SIMM_{Equity}+SIMM_{Commodity}\]

That is, above the product class level, we don’t apply the square root rule but a straight sum. I’m not sure whether this is based in part on some analysis the ISDA has done concerning correlations between product classes, or is simply conservatism.

In any event, we’ve now calculated the total initial margin across our entire portfolio, as stipulated by the ISDA’s standard initial margin model.

## FX sensitivities

Consider an FX trade with underlying \(S = CCY1CCY2\) and present value \(PV(S)\) expressed in CCY2. When calculating sensitivities, SIMM considers a 1% shift \(dS=.01S\). The sensitivity with respect to \(CCY1\) is defined to be the corresponding shift

\[sen_{CCY1} = dPV.\]

The sensitivity with respect to \(CCY2\) is defined to be

\[sen_{CCY1} = -Sd(PV/S).\]

In other words, we convert \(PV\) to \(CCY1\), find the shift, and then convert back. Using the chain rule for differentiation, it’s easy to show that

\[sen_{CCY1} + sen_{CCY2} = .01PV.\]

Once one sensitivity is calculated, the other can be easily found using this formula.

## SIMM contribution of option premium settlement

For options, the option premium is typically settled two business days after the trade date. When a SIMM calculation is performed for an option trade before the option premium has settled, the premium cashflow is included in the calculation of the sensitivities. Since the option premium is equal and opposite to the usual PV of the trade, this means that the “total PV” including the premium payment is zero at the inception of the trade. Using the above formula, this means that the two sensitivities are equal and opposite.

The contribution that the premium payment makes to \(sen_{CCY1}\) and \(sen_{CCY2}\) depends on which currency settlement occurs in. Either way, since it is a fixed and known payment, its sensitivity to one of the currencies will be zero, and its sensitivity to the other currency will be \(.01Premium\).

## Bitmex’s insurance fund

Given the high volatilities of cryptocurrencies, market risk and margin modelling is particularly challenging. When some major cryptocurrency exchanges combine this with very high leverages, managing their risk becomes particularly challenging.

On a traditional exchange, if the loss on a trade exceeds the posted margin, the exchange will insist that the trader put forward additional margin amount to cover the loss, and chase the debt if necessary.

Bitmex have taken a margining approach different to that used in traditional finance. They call this their insurance fund. In essence it means that Bitmex will liquidate your position at half margin, and appropriate the amount remaining after liquidating your position at the best available price. This money is then used to compensate for other trades by other participants which lost more than the margin amount. This means that traders which lost more than their margin are compensated, at the expense of traders who didn’t lose all of their margin.

To new participants, it seems surprising that the exchange could “steal” half your margin and keep it. However, Bitmex point out that, on the upside, traders are guaranteed to never lose more than their margin amount.