15
Quantifying Exposure

15.1 METHODS FOR QUANTIFYING EXPOSURE

15.1.1 Overview

This chapter will present the various methods used to quantify exposure and other utilisation components, focusing mainly on the most common and generic approach of Monte Carlo simulation. The methodology for exposure quantification, including a discussion of the approaches to modelling risk factors in different asset classes and their co-dependencies, and also the computational aspects, will be explained. By way of illustration, a number of examples will be shown, looking in particular at the impact of aspects such as model choice and calibration, as well as the effect of netting and margin.

In general, counterparty risk is defined by future exposure, which is usually expressed in the form of potential future exposure (PFE), expected positive exposure (EPE), and expected negative exposure (ENE). These components are determined directly from knowing the distribution of the future value of the relevant portfolio. More generally, xVA calculations require the quantification of a ‘utilisation’ component, as discussed in Section 5.4.6 (Equation 5.6), which may represent aspects such as funding or capital. All such components can be seen as being directly, or indirectly, linked to the same distribution of future value (Table 15.1). For credit, funding, and margin, the utilisation component is directly linked to the value, with potential consideration of issues such as rehypothecation. For capital and initial margin, the expected capital profile (ECP) and expected initial margin profile (EIM) are more complex calculations but are also functions of the future value of the relevant portfolio. Hence, the problem of quantifying exposure can be broadened into the problem of quantifying the utilisation components (although we will refer to it as quantifying exposure).

Exposure quantification generally involves a long-term portfolio calculation and cannot be tackled by more short-term approximations. There are many complexities in exposure quantification that must also be dealt with, such as:

• Model choice. It is necessary to choose a model for each underlying risk factor, usually across multiple asset classes. Pragmatically, it is important to use relatively simple and tractable models to avoid the implementation and performance being inadequate. Note that it is necessary to use a generic model definition as xVA is generally a portfolio-level problem, and the underlying transactions must be treated in a self-consistent way.
• Calibration. The practical calibration of the chosen model to market and/or historical data is generally challenging given the large number of risk factors, lack of data for some components (e.g. volatility), and large number of correlation parameters. Numerical methods may be required for calibration to market prices.
• Numerical implementation. The choice of implementation of the model is important as it will impact performance and accuracy. Furthermore, the optimal implementation for one xVA component may not be the case for another.

  Table 15.1 Utilisation components of different xVA terms. See Glossary for definitions.

  Area	xVA term	Utilisation component	Calculation
  Credit	CVA	EPE
  	DVA	ENE
  	n/a	PFE	Quantile of
  Funding (and margin)	FVA	EFV
  	FCA	EPE
  	FBA	ENE
  Capital	KVA	ECP	Function of
  Initial margin	MVA	EIM

• Ageing. In a long-term portfolio calculation, it is necessary to correctly ‘age’ a portfolio by taking into account cash flow payments, contract expiration, and cancellation features.
• Path dependency. Path dependencies can also be important, most commonly when it is necessary to treat termination clauses, margin, and physically-settled options.

As a result of the above, the practical calculation of exposure inevitably involves choosing a balance between sophistication and resource considerations.

15.1.2 Parametric Approaches

These approaches are not model-based but instead aim to parameterise exposure based on a number of simple parameters that have potentially been calibrated to more complex approaches. Their advantage is that they are simple, but they are not particularly risk-sensitive and often represent more complex features poorly.

The simplest such approach is the current exposure method (CEM), used historically in counterparty risk capital calculations, as discussed in Section 8.2. The new standardised approach for counterparty credit risk (SA-CCR) for capital (Section 13.4.3) is more sophisticated, although a bit more complicated. These approaches approximate future exposure as the current positive exposure plus an ‘add-on’ component that represents the uncertainty of the PFE. At the transaction level, the add-on component should account for:

• the specifics of the transaction in question (moneyness, currency, nature of cash flows, and maturity); and
• the volatility of the underlying asset class.

For example, longer time horizons will require larger add-ons, and volatile asset classes – such as foreign exchange (FX) and commodities – should attract larger add-ons. At the portfolio level, such approaches should also aim to capture:

• netting; and
• margin (both variation and initial margin).

Such effects are often difficult to incorporate accurately and often use fairly crude rules, although more sophisticated methodologies have been proposed (e.g. Berrahoui et al. 2019). These may still be attractive to small banks and market participants, who need a basic and easy-to-implement methodology for PFE, credit value adjustment (CVA), or xVA.

15.1.3 Semianalytical Methods

Semianalytical methods are generally more sophisticated than the simple parameter approaches as they are model-based. Their advantage lies in avoiding the time-consuming process of Monte Carlo simulation. A semianalytical method will generally be based on:

• making some simple assumptions regarding the risk factor(s) driving the exposure;
• finding the distribution of the exposure as defined by the above risk factor(s);
• calculating a semianalytical approximation to a risk metric for that exposure distribution.

Some very simple and general semianalytical expressions were described in Section 11.1.5 and formulas can be found in Appendix 11A. One product-specific and well-known semianalytical formula can be found in Sorensen and Bollier (1994), who show that the exposure of an interest rate swap (IRS) can be defined in terms of a series of interest rate swaptions.¹ The intuition is that the counterparty might default at any time in the future and, hence, effectively cancel the non-recovered value of the swap, economically equivalent to exercising a swaption.

The swaption analogy is illustrated in Figure 15.1, with more mathematical details given in Appendix 15A. The EPE of the swap will be defined by the interaction between two factors: the swaption payoff and the underlying swap duration (these are the two components in the simple approach given in Section 11.2.2, Equation 11.4). These quantities respectively increase and decrease monotonically over time. The overall swaption value, therefore, peaks at an intermediate point.

Figure 15.1 Illustration of interest rate swap EPE as defined by swaption values, which are given by the product of the swaption payoff and the risky duration value (shown on the secondary y-axis).

---

Spreadsheet 15.1 Semianalytical calculation of the exposure for a swap.

---

This approach naturally captures effects such as the asymmetry between payer and receiver swaps (Figure 15.2) and unequal payment frequencies, such as in a basis swap. In the former case, the underlying swaptions are in-the-money (ITM) or out-of-the-money (OTM) for payer and receiver swaps respectively. In the latter case, the strike of the swaptions moves significantly OTM when an institution receives a quarterly cash flow whilst not needing (yet) to make a semiannual one.

The Sorensen and Bollier formula gives a useful insight into exposure quantification: specifically that the exposure calculation of terms such as EPE will be at least as complex as pricing the underlying product itself. To quantify the EPE of a swap, one needs to know about swaption volatility (across time and strike), components far beyond those needed to price the swap itself. The value of the swap does not depend significantly on volatility, and yet the EPE for the swap does. The above intuition will also be useful when discussing the calibration of volatility in exposure models (Section 15.4).

The above analogy can be extended to other products, since any non-path-dependent transaction can be represented as a series of European options. A semianalytical approach would clearly be the method of choice for evaluating the exposure of a single transaction, although some cases such as cross-currency swaps, and more exotic products, will still be problematic. In some circumstances, semianalytical approximations can also be extended beyond the single transaction level to, for example, a portfolio of IRSs in a single currency, as discussed by Brigo and Masetti (2005b). The ability to do this may often be useful, as some end user counterparties may trade a rather narrow range of underlying products, the exposure of which may be modelled semianalytically. However, multidimensional portfolios will typically need to be treated in a more generic approach.

Figure 15.2 Illustration of the EPE for a payer interest rate swap as defined by the swaption values and the ENE for the same swap (or the EPE for the equivalent receiver swap).

Obvious drawbacks of such semianalytical approaches are:

• They depend on simplifying assumptions made with respect to the risk factors involved. Hence, complicated distributional assumptions cannot typically be incorporated. Put another way, the model(s) chosen must allow analytical tractability.
• Path-dependent aspects (exercises, breaks) will be hard to capture, as will margin, which is path-dependent, although approximations for collateralised exposures may be easier to formulate, as discussed in Section 15.5.
• It may be possible to incorporate netting or, more generally, portfolio effects at some level. For example, Brigo and Masetti (2005b) consider an approach for a portfolio in a single currency. Note that the above approaches are risk neutral by their nature and therefore would not naturally support the use of real-world calibrations (although this may not be a major concern).
• Such approaches may not be future-proof. In particular, the need to incorporate new products and changes in market practice (e.g. the increasing use of initial margins) may be hard to capture.

Whilst there will be situations where analytical calculations may be possible (e.g. quantifying the CVA of a counterparty trading in only a single currency), the fact that such calculations are not generic is difficult from an operational point of view.

Note also that certain cases where the EPE is equal, or approximately equal, to the expected future value (EFV) can be treated more easily, potentially without the need for complicated calculations. Two examples are:

• ITM portfolios. Where the EPE is close to the EFV (see Figure 15.3).
• Options with upfront premiums. Where EPE is equal to EFV.

Figure 15.3 EPE, ENE, and EFV for an ITM portfolio.

15.1.4 Monte Carlo Simulation

Monte Carlo simulation, whilst the most complex and time-consuming method to quantify exposure, is generic and copes with many of the complexities, such as transaction specifics, path dependency, portfolio effects, and collateralisation. It is the only method that in the case of a high-dimensionality portfolio can realistically capture the relatively large number of risk factors and their correlations.

A generic approach also provides flexibility as market practice and regulation changes. For example, initial margins have been rare in the past but are becoming increasingly important (although incorporating dynamic initial margins into Monte Carlo simulation is complex, as discussed in Section 15.6.6).

Whilst add-on and analytical approaches still sometimes exist, Monte Carlo simulation of exposure has been considered state-of-the-art for some time. Banks generally use Monte Carlo implementation across all products and counterparties, even when analytical approximations may be achievable in some cases.

It is important to note that, whilst Monte Carlo simulation is generic, it will still pose a challenge in terms of implementation and computational time. It is therefore important to consider the choice of model, calibration, and framework to avoid unnecessary workload. Additionally, whilst it is possible to include aspects such as initial margin, these often create even higher workloads due to their inherent complexity.

Monte Carlo simulation is also a challenging technique to use when it is required to calculate sensitivities (Greeks) for calculating hedges and explaining profit and loss (P&L) changes. This is because the inherent noise may cause real problems operationally. This will be discussed later in Section 21.2.2.

15.2 EXPOSURE ALLOCATION

15.2.1 Overview

Since xVA is generally a portfolio-level calculation, it will be necessary to have methods for allocating exposure back to individual transactions. This may be required for pricing new trades or for making accounting adjustments. There are generally two methods that could be used for exposure allocation, depending on the purpose. Incremental allocation looks at the incremental effect of transactions sequentially and is forward looking. Marginal allocation is backwards looking and allocates back the total exposure to each transaction. Each of these methods will be illustrated with a simple example below and then used later in the examples in Section 15.6.

It is clear from the above examples that netting benefits can be substantial. It is not clear, though, how to allocate these benefits to the individual transactions. If the EPE (and ENE) of the transactions are considered on a standalone basis, then this will overstate the actual risk. However, there is no unique way to distribute the netted exposure amongst the transactions.

15.2.2 Incremental and Marginal Exposure

Consider two exposures defined by normal (Gaussian) distributions with different means and standard deviations, as illustrated in Figure 15.4. The distributions have different moments: the first has a positive mean and a smaller standard deviation, and the second has a negative mean but a larger standard deviation. The result of this is that the EPEs are similar: 7.69 and 7.63 for transaction 1 and transaction 2, respectively.² Assuming zero correlation, the total EPE of both transactions would be 10.72.³

Figure 15.4 A simple example of two different normal distributions with similar EPEs.

Now the question is how to allocate the EPE of 10.72 between the two transactions. The most obvious way to do this is to ask simply in which order they arose. If transaction 1 came first, then by definition it would contribute an EPE of 7.69 at that time. By simple subtraction, transaction 2 would then have to represent only 3.03. If the order were reversed, then the numbers would be almost the opposite. This will be referred to as ‘incremental exposure’, since it depends on the incremental effect, which in turn depends on the ordering. Incremental allocation is usually most relevant because the nature of trading is sequential. This is consistent with pricing transactions at their inception.⁴

The incremental exposure is defined via:

(15.1)

In other words, it is the exposure of the netting set with the new transaction added , minus that of the netting set alone . A similar formula applies for other metrics such as ENE and PFE, and for calculations requiring another type of aggregation. Moreover, this does not have to represent the addition of a transaction but could also be removal (e.g. unwind) or other restructuring.

In risk management, it is common and natural to ask the question of from where the underlying risk arises. Risk managers and xVA desks may find it useful to be able to ‘drill down’ from a number representing total exposure and understand which transactions are contributing most to the overall risk. This can be important information when considering whether to unwind transactions or enter into more business. Marginal exposure is useful in this context since, just because transaction-level (standalone) exposures are similar, it does not mean that the contributions to the total netted exposure are also similar. Marginal exposure (e.g. EPE) will naturally lead to the definition of marginal xVAs.

Instead of looking at EPEs sequentially, it might be desirable to find a fair breakdown of exposure, irrespective of the order in which transactions were originated. This could be relevant if two transactions are initiated at the same time (e.g. two transactions with the same counterparty on the same day) or for analysing exposure to find the largest contribution. One could simply pro rata the values in line with the standalone EPEs. Whilst this may seem reasonable, it is not theoretically rigorous. A more robust way to do this is via marginal exposure.

Marginal risk contributions are well-studied concepts due to the need to allocate risk measures back to individual constituents. For example, they have been described by Arvanitis and Gregory (2001) for credit portfolios, and a discussion on marginal value-at-risk can be found in Jorion (2007). In most situations, a marginal EPE contribution can readily be calculated as the derivative of the risk measure with respect to its weight (Rosen and Pykhtin 2010). Hence, it is necessary to numerically calculate the derivative of the total EPE with respect to each constituent exposure in order to know the marginal EPEs. This will be described more intuitively below. The marginal EPEs will, under most circumstances,⁵ sum to the total EPE as required. More mathematical details are given in Appendix 15B.

The marginal EPEs under the assumption of independence between the two exposure distributions are summarised in Table 15.2 and compared also to incremental exposure and the crude pro rata approach.⁶ We can see that the marginal EPE of transaction 2 is actually quite significantly higher than that of transaction 1, even though the standard EPE is lower. The distribution with a smaller expected value and a larger standard deviation is more ‘risky’ than the one with the opposite characteristics.

---

Spreadsheet 15.2 Example marginal exposure calculation.

---

Table 15.2 Summary of different EPE decompositions for the simple example in Figure 15.4, assuming independence between exposures.

	Incremental (1 first)	Incremental (2 first)	Pro rata	Marginal
Transaction 1	7.69	3.09	5.38	3.95
Transaction 2	3.03	7.63	5.34	6.77
Total	10.72	10.72	10.72	10.72

There are two obvious reasons why marginal exposure might be a meaningful measure. Firstly, the situation where two or more transactions happen simultaneously (or within a short time interval) and incremental allocation would, therefore, be inappropriate (and arbitrary); and secondly, when it is important to allocate the exposure back to constituents in order to make some decision, such as which transactions to terminate or restructure. So, whilst incremental exposure is clearly useful for quantifying new individual transactions, marginal exposure can also be a useful measure in other situations.

In summary, incremental exposure is relevant when exposure is built up sequentially, which is usually the case in practice. It is potentially unfair, in that incremental exposure depends on the timing as well as individual characteristics. Marginal allocation is fair but changes each time a new exposure is added, which is not appropriate for charging to the originator of the risk (directly via xVA, or indirectly via PFE and credit limits). In order to illustrate this, consider adding a third exposure based on a normal distribution with mean 7 and standard deviation 7 (again, with a similar standalone EPE of 7.58). The numbers change, as shown in Table 15.3. Whilst the marginal EPEs seem fairer, the impact of the third exposure is to change the magnitude of the first two (indeed, the first is increased and the second is reduced). By construction, the incremental exposures do not change.

Section 15.6 will give some more real examples of incremental and marginal exposure and discuss how they are both useful in practice.

15.2.3 Impact of Dependency

An important aspect in relation to allocation is the underlying dependency – or correlation – in the portfolio. It will be useful to define broadly three possible types of portfolio:

• Directional. This is a situation of similar trades with high underlying dependency (high positive correlation). End users will often be in this situation, as they may be executing a narrow range of transactions for hedging purposes (e.g. swapping floating to fixed rates in a single currency). A regional bank trading with a global bank may also lead to a directional portfolio.
• Balanced. A portfolio with different types of transaction, either of different underlying (e.g. currencies) or asset classes, would tend to be more balanced. This may be the case for an end user with broader hedging needs or the relationship between two banks.
• Offsetting. In this situation, a portfolio would have trades in both directions (e.g. paying and receiving fixed). This may be the case between two banks, or for more active clients (such as hedge funds) who may execute certain transactions in both directions (e.g. buy or sell a commodity) and put on offsetting trades to (partially) lock in profits or losses.

Table 15.3 As Table 15.2, with a third transaction added.

	Incremental (1 first)	Incremental (2 first)	Pro rata	Marginal
Transaction 1	7.69	3.09	4.86	4.45
Transaction 2	3.03	7.63	4.82	5.67
Transaction 3	3.76	3.76	4.79	4.36
Total	14.48	14.48	14.48	14.48

Figure 15.5 Marginal EPEs for the simple two-transaction example shown in Figure 15.4 as a function of the correlation between the normal distributions. The total is the sum of the marginal EPEs. The total standalone EPE is also shown for reference.

In order to understand the impact of the type of portfolio on exposure allocation, the simple exercise in Figure 15.4 is repeated for a range of correlation values, as shown in Figure 15.5 (these calculations can be seen in the aforementioned Spreadsheet 15.2). The total EPE is smallest at -100% correlation and increases with the correlation as the overall netting benefit is reduced. The breakdown of total EPE into marginal components depends very much on the correlation. At zero correlation, as already shown above, transaction 2 has a larger contribution to the overall EPE. With a negative correlation, the more ‘risky’ transaction 2 has a positive marginal EPE that is partly cancelled out by transaction 1 having a negative marginal EPE. At high correlations, the marginal EPEs are both positive and of almost equal magnitude (since there is little or no netting benefit).

The point being emphasised here is that for portfolios with low or negative correlation, marginal EPEs are particularly important to understand. The marginal EPE of a transaction depends on the relationship of that transaction to others in the netting set. A transaction which is risk reducing (negative marginal EPE) in one netting set might not have the same characteristic in a different netting set.

With respect to the consideration of new transactions, we can again characterise three different situations:

• Risk increasing. A directional transaction would be in the same direction as the existing (directional) portfolio. In this situation, exposure allocation is of limited interest because the exposures will be largely additive, as shown in Figure 15.5 (high positive correlation).
• Neutral. In this situation, a new transaction has a relatively neutral impact on the existing portfolio (e.g. it is an FX trade in a portfolio dominated by interest rate trades). The incremental contribution of this new transaction would be expected to be small.
• Offsetting. If a new transaction is in the opposite direction to (dominant risk factors in) the existing portfolio, then it will tend to reduce exposure. The incremental contribution of this new transaction would be expected to be of the opposite sign (i.e. negative for EPE), indicating a benefit (EPE) or cost (ENE).

Risk-increasing transactions will tend to increase the overall xVA, and risk-reducing transactions will reduce xVAs. Neutral transactions will have a relatively small impact on xVA terms. In this context, it is important to keep in mind that some xVA terms can represent costs and benefits: a risk-reducing transaction is advantageous in the former case and disadvantageous in the latter.

15.3 MONTE CARLO METHODOLOGY

15.3.1 Basic Framework

Despite being a relatively expensive technique from a computational standpoint, Monte Carlo simulation is a completely general methodology which can, therefore, cope with high dimensionality (e.g. multiple currencies and FX pairs) and path dependency (e.g. margin modelling). Monte Carlo involves generating scenarios based on the underlying risk factors in the portfolio, with each scenario being a joint realisation of risk factors at a given point in time. Scenarios need to be self-consistent for a given portfolio, since it must be possible to see portfolio effects such as netting. Strictly speaking, this ‘scenario consistency’ is not required across different portfolios, but for various reasons, most implementations will effectively consider only one portfolio.

Exposure simulations are normally implemented via a path-wise generation of exposure on a fixed grid (Figure 15.6), which generates an entire possible trajectory. The calculation of xVA would then be done as a one-dimensional integral over this grid. A path-wise approach has certain advantages, such as more efficient generation of risk factors and treatment of path-dependent features such as margin. It is also more natural when calculating and visualising PFE. However, this approach is less efficient from a convergence point of view and also leads to additional P&L noise (and difficulties in ‘P&L explain’) due to the grid moving across discrete features such as cash flows for calculations on sequential days.

In a ‘direct’ simulation approach, the time grid need not be fixed and default times are not bucketed. For xVA calculations, where only a single value – and not a full profile, as in the case of PFE – is required, a direct approach will require more simulation effort but fewer portfolio revaluations. This can lead to a better convergence (Section 17.2.2), greater stability of CVA over time, and theta P&L explain (Section 21.2.7). It is also more naturally linked to the implementation of wrong-way risk (WWR) models (Section 17.6).

Figure 15.6 Illustration of the difference between path-wise and direct simulation. Note that the direct simulation may be more time consuming and be problematic with respect to capturing path dependence because adjacent points are independent. In the direct approach, the time points are not fixed and can be chosen arbitrarily.

Both path-wise and direct methods should converge to the same underlying result, notwithstanding numerical errors in the path-wise discretisation. The path-wise method is more suitable for PFE calculations and also for path-dependent derivatives, derivatives with Bermudan features, and margin modelling. For pure xVA calculations of vanilla products, a direct approach may be preferable.

Typically, in the order of 10,000 simulations are used, with quasi-random sequences used for variance reduction. It may be necessary – or at least desirable – to use more simulations for the calculation of Greeks.

In a path-wise implementation, it is first necessary to use a grid for the simulation, as illustrated in Figure 15.7. The number of grid points must be reasonably large to capture the main details of the exposure, but not so large as to make the computations unfeasible. A typical value is in the region of 50–200.

Note that the spacing of the above dates need not be uniform for reasons such as roll-off (discussed below) and identifying settlement risk. In addition, since intervals between simulation points are often greater than the length of the margin period of risk (MPoR), it may be necessary to include additional ‘look-back’ points for the purposes of simulating the impact of margin (Section 15.5.3).

When exposure is calculated at only discrete points, as in a path-wise simulation, it is possible to miss key areas of risk. Profiles can be highly discontinuous over time due to maturity dates, option exercise, cash flow payments, and break clauses. The risk of missing jumps in exposure caused by these aspects is called ‘roll-off risk’. Such jumps may be small in duration but large in magnitude. These components are typically considered by regulators to be counterparty risk (and not settlement risk) (for example, see BCBS 2015b). The impact of roll-off risk is shown in Figure 15.8, which shows a PFE calculation using simulations with different levels of granularity. Whilst roll-off risk manifests itself in PFE terms via the profile having an incorrect and rapidly-changing structure, for xVA calculations it will lead to additional noise in day-to-day calculations.

Roll-off risk can be controlled by using time-heterogeneous time grids (as in Figure 15.7), at least providing a better definition as discrepancies become closer. However, this can mean that xVA and PFE quantities can change significantly from day to day due to exposure jumps gradually becoming engulfed within the more granular short-term grid. A better approach is to incorporate the critical points where exposure changes significantly (e.g. due to maturity dates, settlement dates, and cash flow payment dates) into the time grid. However, this would need to be done separately for each portfolio. This may also be beneficial to provide the ability to change grids to reflect different maximum maturity dates, margin terms, and underlying product type. However, in a generic xVA framework, this may be impractical due to the definition of ‘portfolio’ being specific to the value adjustment (VA) being calculated.

Figure 15.7 Illustration of time grid for exposure simulation.

Figure 15.8 PFE for a counterparty calculated at different levels of granularity. In the normal case, the time intervals are spaced by 10 business days, whilst in the less granular case, the interval is five times greater.

15.3.2 Revaluation, Cash Flow Bucketing, and Scaling

The calculation of xVA is computationally demanding due to the use of Monte Carlo simulation and the need to evolve the portfolio across the entire lifetime of trades (which in turn requires full revaluation). Suppose the total population of transactions and exposure calculation involves:

• 10,000 simulations;
• 100 time steps (path-wise simulation);
• 250 counterparties; and
• 40 transactions with each counterparty (on average).

Then the total number of trade revaluations would be (10 billion). This has very significant implications for the computation speed of pricing models, as this step usually represents the bottleneck of a PFE or xVA calculation. On top of the above, Greeks calculated via ‘bump-and-run’ approaches which calculate finite-difference sensitivities will add a potentially linear computational burden with respect to computation time and the total number of sensitivities required.

The above will also be sensitive to the total portfolio under consideration. In the past, a CVA calculation may have been done on only the uncollateralised counterparties, with no other xVAs considered. This reduced scope meant that the problem was more manageable, as the potentially large number of collateralised transactions would not need to be captured and simulated. However, it is now common to consider collateralised counterparties for the calculation of CVA (see Section 3.1.4), which – even ignoring the additional complexity in modelling margin discussed in Section 15.5 – requires the simulation of a much larger universe, including interbank counterparties. Furthermore, other xVA terms are not only related (solely) to uncollateralised transactions (e.g. capital value adjustment – KVA, margin value adjustment – MVA) and therefore require a much larger portfolio to be handled.

The first implication of the above is that pricing functions must be highly optimised, with any common functionality (such as calculating fixings) stripped out to optimise the calculation as much as possible. Whilst such pricing functions are usually relatively fast, the sheer volume of vanilla products makes this optimisation important.

For more complex products, whilst there may be fewer of them in a typical xVA portfolio, pricing often involves lattice-based or Monte Carlo methods that may be too slow. There are various ways to get around this problem, for example:

• Approximations. Sometimes crude, ad hoc approximations may be deemed of sufficient accuracy – for example, approximating a Bermudan swaption as a European swaption (which allows an analytical formula).
• Grids. Grids giving the value of a transaction can be used as long as the dimensionality of the problem is not too large. Such grids may be populated by front-office systems and therefore be in line with trading desk valuation models. The PFE/xVA calculation will look up (and possibly interpolate) the relevant values on the grid rather than performing costly pricing.
• American Monte Carlo methods. This is a generic approach to utilising future Monte Carlo simulations to provide good approximations of the exposure at a given point in time, in a path-wise simulation approach.⁷ Examples of this and related approaches can be found in Longstaff and Schwartz (2001), Glasserman and Yu (2002), and Cesari et al. (2009). This may be the best solution for xVA quantification, but may not be as relevant for PFE and risk management.

For both vanilla and more complex transactions, it may be useful to use techniques such as machine learning to evaluate such pricing functions (and their Greeks) very rapidly, as discussed later in Section 21.3.3.

It is usually reasonable that xVA calculations do not perfectly represent the underlying value of transactions. This is different from, for example, incoming market risk capital requirements, which for internal model approval require an exact, or very close, alignment of front-office and risk models due to the so-called ‘P&L attribution test’ (BCBS 2019a), which compares theoretical predicted P&L with actual P&L changes. The potential for slight misalignment in xVA means that if a pricing function can be speeded up significantly with only a small loss of accuracy, then this will be worthwhile.

Another important optimisation is to bucket cash flows in a single currency so as to reduce the number of pricing evaluations (Figure 15.9). Given the inherent approximation of using a relatively large discrete time step, this should not represent a further significant approximation. In a path-wise approach, cash flow bucketing may not represent any further approximation since the integral has already been discretised.

Inevitably, even for simple products, there are valuation differences between the exposure calculation and the official valuation. These differences are sometimes corrected by applying shifts, as illustrated in Figure 15.10. For example, multiplying all paths by (where t represents the current time and T the final maturity) would affect an amortising scaling. For forward contracts and cross-currency swaps (or contractual mismatches, such as the misspecification of a cash flow), a fixed scaling is more appropriate. Ideally, since different transactions vary significantly in terms of their contractual features and valuation methodologies, shifts should be applied at the transaction level.

Figure 15.9 Illustration of cash flow bucketing in a single currency. The cash flows between the dates shown (top) can be realistically combined into a single payment (bottom) whilst preserving the overall valuation and potentially other aspects such as the sensitivity.

Figure 15.10 Illustration of the use of a proportion shift to correct an exposure simulation to the official valuation.

15.3.3 Risk-neutral or Physical Measure

Scenario generation for risk-management purposes and arbitrage pricing theory tend to use different ‘measures’. Arbitrage-based pricing uses the so-called risk-neutral measure, which is justified through hedging and arbitrage considerations. For example, in a risk-neutral framework, interest rate volatilities (and associated parameters such as mean reversion) would be derived from the prices of interest rate swaptions, caps, and floors, rather than estimated via historical time series. In addition, the drift of the underlying variables (such as interest rates and FX rates) will need to be calibrated to forward rates, rather than coming from some historical or other real-world analysis.⁸ Parameters (and therefore probability distributions) such as drifts and volatilities are market implied and need not correspond to the real distributions (or even comply with common sense).

On the other hand, in a risk-management application, one does not need to use the risk-neutral measure and may focus rather on the real-world or physical measure, estimated using, for example, historical data. Models calibrated using historical data predict future scenarios based on statistical patterns observed in the past and assume that this previous behaviour is a good indicator of the future; such models are sometimes slow to react to changes in market conditions. Models calibrated to market prices tend to be more forward looking, but contain components such as risk premiums and storage costs that introduce bias. Furthermore, they may produce exposures that jump dramatically – for example, during a period of high volatility.

The question of risk-neutral or physical measures is quite easy to answer in the case of CVA. Risk-neutral approaches have become standard for quantification of CVA (and FVA). This has been catalysed by the more active management of such components and accounting standards. Parameters are generally risk neutral, to the extent that this is possible. The estimation of non-observable parameters in this context is discussed in Section 15.4.5.

The need for CVA to be based on a risk-neutral simulation is summarised by BCBS (2017) and relates to using the most advanced approach under the Fundamental Review of the Trading Book (FRTB) CVA capital charge (Section 13.3.5). BCBS (2017) makes the following statements:

• Drifts of risk factors must be consistent with a risk-neutral probability measure. Historical calibration of drifts is not allowed.
• The volatilities and correlations of market risk factors must be calibrated to market data whenever sufficient data exists in a given market. Otherwise, historical calibration is permissible.

Where a difference in market practice exists, this relates more to PFE (credit limits) and regulatory capital calculations using the internal model method (IMM) (Section 13.4.5). Here it is probably best to consider the calibration of drift, volatilities, and correlations as separate topics.

Drift (or trend) is an increasingly-important parameter as the exposure simulation time horizon becomes longer since it has an approximately linear effect, whereas volatility has more of a square root impact. In other words, the drift of an underlying variable can be just as important as its uncertainty. Futures (or equivalently forward) prices have long been an important mechanism of price discovery in financial markets, as they represent the intersection of expected supply and demand at some future point in time. Forward rates can sometimes be very far from spot rates, and it is important to understand whether or not this is truly the ‘view of the market’. Some important technical factors are:

• Interest rates. Yield curves may be upwards sloping or downwards sloping (and a variety of other shapes) due to the risk appetite for short-, medium-, and long-term interest rate risk and the view that rates may increase or decrease.
• FX rates. Forward FX rates are determined from an arbitrage relationship between the interest rate curves for the relevant currency pair. The expectation of future FX rates may have an influence on the current interest rate curves in the corresponding currencies. However, there has long been doubt regarding the ability of long-term forward FX rates to predict future spot FX rates. See, for example, Meese and Rogoff (1983) and a review by Sarno and Taylor (2002).
• Commodity prices. In addition to market participants' view of the direction of commodity prices, storage costs (or lack of storage), inventory, and seasonal effects can move commodities futures apart from spot rates. For high inventories, the futures price is higher than the spot price (contango). When inventories are low, commodity spot prices can be higher than futures prices (backwardation). However, non-storable commodities (e.g. electricity) do not have an arbitrage relationship between spot and forward prices, and therefore the forward rates might be argued to contain relevant information about future expected prices.
• Credit spreads. Credit curves may be increasing or decreasing either due to demand for credit risk at certain maturities or the view that default probability will be increasing or decreasing over time. Historically, the shape of credit curves has not been clearly seen to be a good predictor of future credit spread levels.

There has been much empirical testing of the relationship between spot and futures prices across different markets. It is a generally held belief that the futures price is a biased forecast of the future spot price. On the other hand, choosing a historical drift may be inappropriate: for example, in a falling interest rate environment, it will produce more simulations where rates decline, which may be precisely opposite to the expected economic behaviour. Using a drift of zero may, in certain situations, be argued to be most appropriate and subject to the least bias.

Using a different drift tends to offset the exposure distribution, as illustrated in Figure 15.11. In this example, since the interest rate curve is upwards sloping (long-term interest rates are higher than short-term rates), the risk-neutral drift is positive, leading to the 99% PFE being higher than the 1% PFE, as explained in Section 11.2.3.

Figure 15.11 Illustration of the PFE for a five-year interest rate swap computed with both real-world (RW) and risk-neutral (RN) simulations for the drift. Note that, in order to isolate the drift impact, historical volatility has been used in both cases.

Regarding volatility, historical estimates make the implicit assumption that the past will be a good indication of the future. It is also necessary to decide what history of data to use; a short history will give poor statistics, whereas a long history will give weight to ‘old’, meaningless data. In quiet markets, the lack of variability in historical time series will give misleadingly low volatility. Using historical data from stress periods (as required in IMM approaches, see Section 13.4.5) can alleviate this, but then yet more subjectivity is introduced in choosing the correct period of stress.

In many markets, there is also implied volatility information, potentially as a function of the strike and the maturity of the option. However, implied volatility will not be observed for OTM options and long maturities. The use of implied volatility (which will react quickly when the market becomes more uncertain) may be justified via the ‘market knows best’ view (or at least, the market knows better than historical data). However, risk premiums embedded in market-implied volatilities will lead to a systematic overestimation of the overall risk. It has been argued that implied volatility is a superior estimator of future volatility (Jorion 2007, Chapter 9) compared with historical estimation via time-series approaches. The stability of the volatility risk premium and the fact that an overestimation of volatility will always lead to a more conservative risk number give greater credence to this idea.⁹ One example of risk-neutral volatility calibrations being preferable is in the case of pegged currencies where the historical FX volatility is small, but option prices can provide information on the likelihood of the peg breaking, leading to a much higher volatility regime (Clark 2016).

Larger volatility will increase EPE, ENE, and PFE in absolute terms. Figure 15.12 shows the PFE of a cross-currency swap under both real-world and risk-neutral volatility assumptions. Here the main impact is simply that risk-neutral volatilities tend to be higher than real-world ones and hence both the EPE and ENE are bigger.

When using risk-neutral volatility, the term structure is also important, as shown in Figure 15.13. An upwards-sloping volatility term structure leads to a higher exposure at longer maturities.

Another difference between historical and implied volatility is the term structure impact (including aspects such as mean reversion in models). A volatility skew across time (e.g. long-dated volatility being higher than short-dated volatility) means that forward volatility is higher than spot volatility. However, empirical evidence does not always support forward volatility being predictive of actual future volatility.

Whilst it is at least conservative to assume volatilities are high, the same is not true of correlation inputs. When estimating a correlation for modelling exposure, there may not be an obvious way of knowing whether a high or low (or positive or negative) value is more conservative. Indeed, in a complex portfolio it may even be that the behaviour of the exposure with respect to correlation is not monotonic.¹⁰ Implied correlations are sometimes available in the market. For example, a quanto option has a payoff in a different currency and thus gives information on the implied correlation between the relevant FX rate and the underlying asset. One key aspect of correlation is to determine WWR: a quanto credit default swap (CDS) – a CDS where the premium and default legs are in different currencies – potentially gives information about the correlation between the relevant FX rate and the credit quality of the reference entity in the CDS (Section 17.6.4). Whilst implied correlation can sometimes be calculated, for most quantities no market prices will be available and so historical data will typically be used.

Figure 15.12 Illustration of the PFE for a 10-year cross-currency swap computed with both real-world (RW) and risk-neutral (RN) simulations for the volatility. Note that, in order to isolate this impact, risk-neutral drift (forward rates) has been used in both cases.

Figure 15.13 Illustration of the PFE for a 10-year cross-currency swap computed with flat and upwards-sloping volatility term structures.

BCBS (2015b) reports that, out of 19 participating banks, only one used a risk-neutral (market-implied) calibration for all risk factors in its IMM model. A further five banks were reported to use a combination of risk-neutral and historical data for calibration (e.g. drifts calibrated to market-implied data and volatilities to historical data). The remaining banks used only historical data. Many large banks have separate implementations for xVA quantification and counterparty risk assessment (PFE and IMM) across the front office, and risk functions which will usually follow broadly risk-neutral and real-world approaches, respectively.

Using a risk-neutral calibration for an IMM does have various advantages. It potentially allows alignment with xVA calculations, which may provide better internal assessment of trades and hedging performance. However, this is not without problems: regulatory capital requirements under a risk-neutral IMM are still typically required to use stressed calibrations (e.g. stressed-implied volatility from a period in the past), which will be misaligned with the xVA desk view of current market-implied volatility. Furthermore, the use of risk-neutral calibrations may make backtesting results (Section 13.4.5) harder to interpret.

A more subtle problem with the separation between physical and risk-neutral approaches is illustrated in Figure 15.14. Revaluation of transactions at future dates needs to be done using risk-neutral valuation (the so-called ‘Q-measure’), as this is what happens in reality. Simulating using the physical (‘P-measure’) can lead to inconsistency and implementation difficulties (e.g. in using American Monte Carlo methods, as discussed above). On the other hand, also simulating in the Q-measure, as is the case for CVA (and typically also FVA), makes the setup self-consistent. Using a risk-neutral calibration within an IMM approval would also create self-consistency.

However, valuation adjustments that look at the capital (KVA) and initial margin (MVA) are not solely based on risk-neutral valuations but also on historical data:

• Capital. Internal model methodologies typically (although not always, as discussed above) use historical data to calibrate, and standardised methodologies (e.g. SA-CCR) use weights based on historical calibration.

  

  Figure 15.14 Illustration of the different requirements from scenario models for risk management and pricing/valuation. For risk management, simulations tend to be done under the real-world (P-) measure, whilst revaluations must be risk neutral (Q-measure). For pricing purposes, both scenarios and revaluations are risk neutral. For capital and initial margin (KVA and MVA), the underlying calculation is based on the P-measure.

• Initial margin. Central counterparty (CCP) methodologies for over-the-counter (OTC) derivatives use historical simulation (Section 9.3) based on historical data, Standard Portfolio Analysis of Risk (SPAN) (Section 9.2.2) uses spanning ranges based on historical experience, and the International Swaps and Derivatives Association's Standard Initial Margin Model (ISDA SIMM^TM) (Section 9.4.4) uses parameters calibrated to historical data.

The above implies that KVA and MVA may need to be calculated in a region more like the two shown at the bottom of Figure 15.14. For example, the initial margin that a CCP would charge at some point in the future would, strictly speaking, require a simulation under the physical measure, as CCPs use historical and not market-implied parameters in their calculations. There are therefore questions as to whether a KVA or MVA calculation should use the physical or risk-neutral measure. The former would be consistent with assessing the likely real-world costs of capital and initial margin, but would require a separate calculation. The latter would be consistent with CVA/FVA and could be included in the same simulation methodology. An example of this for KVA will be shown in Section 19.2.4.

Some of the above problems have been discussed in the recent literature. For example, Hull et al. (2014) discuss a ‘joint measure model’, representing the single forward-looking movement of interest rates under both the P- and Q-measures. Sokol (2014) discusses the challenges of long-term simulation in general, and Kenyon and Green (2014b) discuss the problem with regulatory methodologies being defined in the P-measure, meaning that pricing of components such as MVA and KVA can no longer be considered risk neutral. Kenyon, Green, and Berrahoui (2015) discuss the appropriate measure to use for PFE computation in more detail.

15.3.4 Aggregation Level

It is worth emphasising that xVA is, in general, a portfolio-level calculation (Section 5.2.6). Whilst this will be discussed in more detail in the relevant Chapters (17–20), Figure 15.15 illustrates a possible hierarchy. The main point to emphasise at this point is that it may be necessary to aggregate xVA components at different levels. For example, CVA – for obvious reasons – is a counterparty-level (or netting-set level) quantity. However, other components such as funding and capital are not necessarily counterparty specific and may need to be considered at a different level and aggregated accordingly. This, in turn, requires large computational power and data science methods to cope with running calculations at the correct aggregation level (e.g. for pre-deal pricing). However, realistic shortcuts are often acceptable.

In general, it is possible to represent exposure simulation data in three dimensions corresponding to the transaction , simulation , and time step , leading to a future value represented as . Assume that transactions need to be aggregated. The future aggregate value is characterised by the matrix:

(15.2)

Figure 15.15 Illustration of the portfolio hierarchy for xVA calculations which will be covered in more detail in later chapters. Note that the margin agreement and netting set may be the same, although it is possible to have more than one margin agreement covered by the same netting agreement. See Glossary for definitions.

It may not be necessary to store all of the individual transaction information, , although this might be needed for calculating certain quantities (see the discussion on marginal exposure in Section 15.2.2).

Related to the aggregation level is the need to treat margin differently. From a counterparty risk (CVA) perspective, margin only needs to be available in a default scenario, whereas for funding (FVA), it has to be available in all situations and rehypothecable. This can require different exposure simulations (discussed later in Section 18.3.2).

15.4 CHOICE OF MODELS

15.4.1 Overview

This section will give an overview of the commonly-used models in exposure simulation and xVA quantification. Given that this is a cross-asset class problem, the number of different model choices and total number of risk factors can be substantial. It is ultimately important to choose relatively simple models for reasons of tractability and stability. There are also 80/20-rule-type considerations. For example, it may be necessary to build a model to cover a relatively small number of products in some asset classes (e.g. some banks have only a limited number of inflation or commodity products). Additionally, some asset classes – typically equity and credit – often have short-dated and/or mainly collateralised transactions for which xVA is less important. The discussion below will reflect this, with interest rate and FX models discussed in more detail since they tend to be most used by uncollateralised counterparties such as corporates.

It is important to strike a balance between a realistic risk factor model and one that is parsimonious. For example, there are 50–60 or more risk factors defining an interest rate curve, whereas the simplest interest rate models involve only one factor. A model involving two or three factors may represent a compromise. Such an approach will capture more of the possible curve movements than would a single-factor model, but without producing the unrealistic curve shapes and arbitrageable prices that a model for each individual risk factor might generate. A more advanced model will also be able to calibrate more accurately and to a broad range of market prices (e.g. swaption volatilities). Models must obviously not be too complex, as it must also be possible and practical to simulate discrete scenarios of the risk factors using the model. Typically, many thousands of scenarios will be required at many points in time, and hence there must be an efficient way in which to generate these many scenarios.

It is important to emphasise that the simulation model must be generic to support consistent simulation of the many risk factors that would be required to value a quite complex netting set. Calibrations also tend to be generic, rather than netting set or transaction specific. This is very different from classical modelling of derivatives, where many models (and individual calibrations thereof) are typically used. This leads to unavoidable differences (hopefully small) between current valuation, as seen from the xVA system, and the relevant front-office systems, as discussed previously in Section 15.3.2. The need to use a generic approach across asset classes creates inevitable cross-dependencies: for example, all other asset classes will exist in a model where interest rates are stochastic.

Another reason for simpler underlying models for risk factors is the need to incorporate dependencies (correlations) in order to capture the correct multidimensional behaviour of the netting sets to be simulated.¹¹ The correct description of the underlying risk factors and correlations leads to a significant number of model parameters. A balance is important when considering the modelling of a given set of risk factors (such as an interest rate curve) and the correlation between this and another set of risk factors (such as an interest rate curve in another currency). There is no point in having sophisticated univariate modelling and naïve multivariate modelling.

Ultimately, exposure simulation modelling choices will depend on the nature of the problem at hand, important considerations being:

• Complexity of portfolio. As noted above, a simple portfolio with mainly vanilla transactions will not warrant a more sophisticated modelling approach, and it is more appropriate to focus on the correct treatment of vanilla products (e.g. a reasonable volatility calibration).
• Collateralisation. Collateralised positions also suggest more simple models can be used, since the nature of margin is to shorten time horizons and neutralise positive and negative portfolio values. Furthermore, the choice of MPoR dominates in importance over the choice of exposure simulation model. Margin (collateral) modelling is discussed in Section 15.5.
• Dimensionality. High dimensionality (e.g. several interest rate and/or FX risk factors for a given counterparty) may suggest simpler models due to the importance of considering the modelling of the underlying co-dependencies.
• Wrong-way risk. If WWR models (Section 17.6) are to be implemented, then it is even more important to keep the basic setup simple.
• Computational workload. Due to the likely need for rapid pricing, Greeks, and scenario analysis, it is important that the modelling framework chosen does not lead to excessive hardware requirements.

By necessity, exposure simulations may not incorporate complex features such as stochastic volatility and stochastic basis, and may not calibrate fully to features such as volatility skews.

There are different ways to implement Monte Carlo simulation on a grid. The first is the so-called Euler scheme, which generates step-wise diffusions sequentially and is subject to ‘discretisation bias’. An alternative approach that aims to eliminate this bias is to use ‘integrated diffusions’. However, such a technique requires integral expressions to be calculated. Using relatively simple models means that such expressions are closed form.

The discussion below is concerned primarily with risk-neutral exposure models which would be most relevant for xVA calculations. This is an important point since real-world and risk-neutral approaches may lead to different modelling choices. For example, when modelling interest rates using historical data, an approach such as principal component analysis (PCA) may be preferred as a means to capture historical yield curve evolution. However, a risk-neutral interest rate model would more naturally be an arbitrage-free term structure approach. For PFE/IMM implementations, other considerations also come into play. For example, specifications of mean reversion, together with other aspects such as models allowing negative rates, can be particularly problematic for PFE, as a high quantile may be viewed as defining an economically unreasonable event. Sokol (2014) discusses these topics in more detail.

For more mathematical details on the implementation of risk-neutral exposure simulation, the reader is referred to books by Cesari et al. (2009) and Brigo et al. (2013) for details on the underlying mathematical framework, and Green (2015), who discusses both this and specific modelling choices for each asset class.

15.4.2 Interest Rates

For most xVA applications, the interest rate is the most important risk factor simulation. This is because interest rate products often dominate other asset classes in terms of importance, and also because other asset classes will have a sensitivity to interest rates through discounting. The simplest interest rate model used is the one-factor Hull–White (HW1F) model, which is usually formulated as a short-rate Gaussian model (Hull and White 1990). Such models have a large amount of analytical tractability due to being Markovian (no memory), allowing closed-form calculations of discount factors and volatility products (caps, floors, and swaptions). This allows an efficient calibration to the initial yield curve and volatility term structure, and also allows a fast determination of the yield curve in future states. Together with the Markovian property – which means that the simulation is not path dependent – this makes Monte Carlo simulation more straightforward. Short-rate models also have the advantage that they are typically easier to combine with models for other asset classes.

The most obvious drawbacks of the HW1F model are:

• the inability to calibrate to all implied volatilities (caps/floors and swaptions);
• the strong correlation between rates of different tenors; and
• the fact that the model allows negative rates.

Whilst negative interest rates may not be a concern as long as there is a reasonable fit to the yield curve and volatility term structure, they can create technical challenges (e.g. if a Black model is used for pricing caplets). In such cases, using the simulation model (e.g. HW1F) for pricing caplets and implementing a scaling approach may be the most practical solution. The inability of a single-factor model to realistically model the relationship between tenors is problematic if there exist significant positions of this type (e.g. pay and receive swaps or constant maturity products), and would support the use of a two- or multifactor HW model, in which there is still reasonable analytical tractability.

---

Spreadsheet 15.3 Simple simulation of an interest rate swap exposure.

---

When pricing exotic interest rate instruments such as Bermudan swaptions, so-called ‘market models’ have become a standard for pricing and allow calibration to the entire at-the-money (ATM) volatility term structure. Methods such as SABR (stochastic alpha beta rho) (Hagan et al. 2002) can be used to introduce a ‘skew’ and therefore price swaptions of different strikes. However, market models, whilst more flexible in terms of calibration to instruments such as caplets and swaptions, are less analytically tractable, non-Markovian, and require simulation with a large number of state variables and complicated drift terms. For xVA applications – which are already by their nature computationally demanding and also require the modelling of other asset classes – market models are often (but not always) deemed too complex, and a HW1F model is more common.

Where it exists, the overnight indexed spread (OIS) or risk-free rate (RFR) curve of a given currency is usually the curve chosen for diffusion purposes. However, in most currencies there exist multiple interest rate curves which can be characterised by their tenor (e.g. OIS, 1M LIBOR, 3M LIBOR, 6M LIBOR, and 12M LIBOR). The divergence between these curves in 2008 (often known as the OIS-LIBOR basis, discussed in Section 16.1.3) suggested a need to model the basis between these curves reasonably. Such ‘dual curve’ pricing would then use different curves for discounting (e.g. OIS) and projection (e.g. LIBOR). Ideally, an interest rate model would individually calibrate to all observable interest rate curves in a given currency and model the relationship between these curves in an arbitrage-free setting. Achieving the latter is challenging due to the fact that different basis curves will likely be highly, but not perfectly, correlated and that basis spreads will likely, but not always, be positive (in the sense that longer tenors will have higher interest rates). There is also the problem that the instruments that would be needed to calibrate a stochastic basis model – such as options on basis swaps – are very illiquid. These points may be simplified after the IBOR transition (Section 14.3.4).

Given the difficulties in modelling stochastic basis, exposure simulation models normally assume a deterministic basis. In such a model, the different rates are correctly calibrated using fixed spreads, but are assumed to be driven by the same random component and are therefore perfectly correlated. This will miss some potential optionality, the materiality of which will depend on the underlying portfolio (e.g. the number of basis swaps).

The interest rate swaption market is large, due to the fact that many fixed-income securities and structured products embed swaption-like optionality, and the fact that there are many market participants (e.g. corporations swapping callable debt). European swaptions are common calibration instruments for exposure simulation models.

The European swaption market is defined by three dimensions:

• the maturity of the underlying swap (e.g. 2, 5, 10, 20, and 30 years);
• the expiry of the option (e.g. 1, 3, 6, and 9 months and 1, 2, 5, and 10 years); and
• the strike (e.g. ATM plus or minus 25, 50, 100, 150, 200, 300, and 400 basis points).

The US dollar and Euro markets are by the far the most liquid, but many other major currencies have a reasonable range of observable swaption prices.

The terms ‘diagonal’ or ‘co-terminal’ refer to a group of swaptions where the option and swap maturities sum to a constant number. For example, the set of swaptions with option and swap maturities defined by 2Yx8Y, 4Yx6Y, 5Yx5Y, 6Yx4Y, and 8Yx2Y are the 10-year co-terminal swaptions. Bermudan swaptions are typically hedged with the underlying co-terminal European swaptions.

Co-terminal swaptions are also interesting from an exposure simulation point of view. For example, the work of Sorensen and Bollier (1994) discussed in Section 15.1.3 is helpful since it shows that the EPE profile of an IRS can be defined as a series of European co-terminal swaptions corresponding to the swap maturity. This is logical since the exposure from the potential default on the swap is economically driven by the exercising of a European swaption on a swap with the remaining life of the original swap at the default time. For example, the four-year discounted EPE on a 10-year swap is equivalent to the price of a 4Yx6Y swaption today.

Whilst the Sorenson–Bollier analogy would seem to suggest that a co-terminal calibration strategy to the ATM T-year co-terminal swaptions would accurately represent the EPE of a T-year IRS, this is not true due to moneyness. The swaptions traded in the market would have to be all at the prevailing swap rate whilst, in reality, the ATM swaptions will trade at the forward swap rate. There is also, therefore, the question of which co-terminal swaptions to use for a portfolio of swaps of different maturities and the calibration of skew.

In the HW1F model, the volatility and mean reversion parameters will determine the volatility structure produced by the model. Both of these parameters can be made time dependent without any loss in tractability. Swaption-implied volatility¹² within the model generally decreases monotonically with tenor, although it can increase if mean reversion is negative. By extension, it is possible to produce a hump in volatility – as is sometimes seen in the market – if a term structure of mean reversion is used.

The mean reversion parameter in the HW1F model strongly affects the correlation between rates of different tenors. This can have a large impact for portfolios where there is a big sensitivity to the movement between such tenors (e.g. a portfolio with paying and receiving fixed at different tenors). This is similar to the impact of mean reversion on the pricing of exotics such as Bermudan swaptions.

A common strategy when calibrating the HW1F model for exposure simulation is to use a fixed mean reversion parameter and calibrate to a set of co-terminal swaptions (e.g. 20 years), which is relatively straightforward via a ‘bootstrap’ procedure. Mean reversion can be very unstable over time when calibrated to swaptions of one maturity (likely due to the model being overspecified), but is more stable if two or more maturities (e.g. 10-year and 20-year co-terminals) are used. Calibrating mean reversion can also more easily adapt to large changes in the volatility surface that are sometimes observed through time. Calibrating to all swaptions with time-dependent mean reversion may be considered more accurate than the co-terminal approach, although such a calibration is necessarily a global process.

Many implementations fix mean reversion at a level such as 0.01 or 0.03 (depending on the currency) due to the possible stability issues otherwise, and calibrate to a chosen co-terminal swaption maturity. This is fairly robust when using a piece-wise constant volatility and bootstrap procedure. As noted above, such an approach does not fit the full ATM volatility surface especially well. For example, Figure 15.16 shows swaption-implied volatilities for a HW1F model with fixed mean reversion calibrated to the 20-year co-terminal swaptions. Whilst the co-terminals (e.g. 5Yx15Y and 10Yx10Y) are matched closely, the model underestimates the left side of the diagonal (e.g. 1Yx5Y) and overestimates the right side (e.g. 10Yx20Y). This will lead the exposure of swaps – especially those not of 20-year maturity – to be incorrect, as will be shown below (Section 15.6.1).

As discussed above, market models allow a better calibration to the ATM volatility surface, and with methods such as SABR can produce a better representation of the skew. However, this comes at the expense of additional computational costs and potential instabilities. Cheyette models (Cheyette 2001; Andersen and Piterbarg 2010b) can also be useful for xVA purposes as they are Markovian and have analytical tractabilities, whilst allowing volatility skew to be calibrated.

15.4.3 Foreign Exchange

The standard FX model used for exposure simulation tends to be a relatively simple lognormal approach with (piece-wise constant) time-dependent volatility. Whilst this may seem overly simplistic, it is important to note that, due to the necessary interest rate modelling, this will already constitute a stochastic interest rate environment. This effectively means that the overall FX model is this assumption for FX together with a HW1F model (for example) for each of the (domestic and foreign) interest rate processes. The relatively simple choice keeps the multidimensional exposure simulation (i.e. n interest rates and n-1 FX rates) reasonably straightforward. Forward FX models are more convenient for option calibration, but spot FX models are more convenient for (historical) correlation estimation.

Figure 15.16 Model versus actual implied volatility for swaptions in a HW1F model calibrated to the 20-year co-terminal swaptions.

Given that an exposure simulation model is multicurrency, it is necessary to define a base or ‘domestic’ currency, which is most obviously the main and/or reporting currency of the bank in question. Only FX pairs involving the base currency need be simulated, since other pairs can be derived via ‘currency triangles’.

Whilst FX spot processes have no term structure as such, the appropriate construction of market FX forwards is essentially a drift calibration. Within the joint hybrid simulation, there are conceptually two types of FX forwards:

• Model-implied, ‘risk-free’ forwards that are constructed from the relevant spot FX and interest rate curves and are internally consistent. However, FX forwards constructed in this manner are not consistent with market FX forwards, since they will not incorporate the corresponding cross-currency basis spreads observed in FX forwards or cross-currency swap basis swaps. The risk-free FX forwards would be used solely internally within the model for model-consistent, no-arbitrage conversion of units of value from one currency to another.
• Risky FX forwards that are consistent with market-observed FX forwards which, for example, correctly reprice cross-currency basis swaps. Similar to interest rate modelling, this is typically achieved through a constant basis. As with the single currency basis, cross-currency basis spreads are derived from market instruments (FX forwards and cross-currency swaps) during construction of the respective cross-currency discount curves.

Pegged currencies are also challenging to model as volatilities will typically be small, but there will be a risk of the peg breaking, accompanied by a substantial jump and/or increase in volatility. Jump-diffusion processes have often been used to characterise emerging markets or pegged currencies. The shorter the time horizon, the greater the importance of capturing such jumps (e.g. Das and Sundaram 1999). As noted in Section 15.3.3, the options market may provide interesting information regarding such probabilities, although it will be important to consider the calibration scheme carefully (the likelihood of the exchange rate regime changing may be seen first in the price of OTM options).

A typical volatility calibration would calibrate across a set of ATM FX option prices within the underlying three-factor (i.e. stochastic interest rate) model. Using a piece-wise constant volatility function in a lognormal FX model, together with two HW1F models, is a simple bootstrap procedure with analytical pricing formulas. Using more sophisticated interest rate and FX models will make calibration and simulation more difficult.

In general, FX options on ‘cross-FX rates’ (not involving the base currency) will not be correctly reproduced. This is because, for example, the USD/GBP FX process will be uniquely defined by the USD/EUR and USD/GBP processes. In some cases, this may not be material due to the nature of the underlying portfolio. Otherwise, a potential solution is to use the correlations between the FX rates to calibrate to the correct volatility term structure for the cross-FX rate (as discussed in Section 15.4.5, such correlations would otherwise typically be defined by historical estimates). Strictly speaking, the correlations would need to be time dependent to allow an accurate fit. This could either be a bespoke process to calibrate to important FX rates or a global calibration to attempt to fit all FX option volatilities. In reality, the complexity that this introduces may not be worthwhile.

15.4.4 Other Asset Classes

Exposure simulation models for asset classes other than interest rate and FX will generally be of lower importance depending on the portfolio in question. For example, credit products are typically always collateralised, and many equity, commodity, and inflation derivatives will be transacted by financial counterparties on a collateralised basis and not used much by uncollateralised counterparties such as corporates. The following approaches are generally taken:

• Inflation. A common approach here is a three-factor representation of the inflation process (lognormal) and the real and nominal rates (HW1F), sometimes known as the Jarrow-Yildirim model (Jarrow and Yildirim 2003). This approach is based on a foreign-currency analogy, where real and nominal rates are modelled, and the inflation rate is seen as an ‘FX rate’ connecting these rates. The real rate volatility may be set as a multiplier to the nominal rate volatility with fixed mean reversion and a high correlation between real and nominal rates. The drift of the inflation rate can be calibrated to zero-coupon inflation swaps, and the volatility to inflation caplets for the most liquid strike (no skew). Given that most xVA implementations will already use a multicurrency framework, this is a natural approach to fit within the same framework.
• Commodities. Some commodities tend to be highly mean reverting around a level, which represents the marginal cost of production (see Geman and Nguyen 2005 and Pindyck 2001). Furthermore, many commodities exhibit seasonality in prices due to harvesting cycles and changing consumption throughout the year. Commodities trading as spot (e.g. precious metals) may use a lognormal assumption similar to FX or equity. Forward-based commodities may use what is sometimes referred to as a Gabillon (1992) model, which is an analytically-tractable two-factor model for the (positively-correlated) spot and long-term prices. Most common calibration instruments are commodity swaps and swaptions (see Geman 2009).
• Equity. Whilst less important due to being short-dated and/or collateralised/exchange-traded, a common equity approach is a lognormal (Black–Scholes) with dividends. A factor model may be used to avoid simulation of a large number of equity underlyings.
• Credit. Credit models are required for two reasons: to represent the exposure of credit derivatives and also to model WWR. A CIR++ model (Brigo and Mercurio 2001) is quite common for credit derivatives, although it should be noted that all such counterparties will probably be collateralised. WWR is discussed in more detail in Section 17.6.

15.4.5 Correlations, Proxies, and Extrapolation

A typical exposure simulation will require a large number of correlation estimates between underlying risk factors. For example, assuming a one-factor interest rate representation, simulating 20 interest rates would lead to 19 FX rates (involving the base currency) and then correlations. Even for a single transaction, this dependency can be important: a cross-currency swap has risk to the FX rate and the two interest rates, and hence, at a minimum, three risk factors, and the three correlations between them must be accounted for. However, individual correlations will have very different importance in defining future exposure. For two IRSs in different currencies, the correlation between the interest rates may be a very important parameter. However, the correlation between, for example, the price of oil and an FX rate may be unimportant or completely irrelevant. It will be informative to make a distinction between intra- and inter-asset class correlations.

The nature of a lot of client business for banks is asset class-specific by counterparty. For example, interest rate products may be traded with one counterparty, and commodities with another. Intra-asset class dependencies will be important components. For example, to specify the future exposure of two interest rate transactions in different currencies, the correlation between the interest rates is important. Indeed, it may be a more important factor than the impact of subtle yield curve movements, which justifies the use of a relatively parsimonious (e.g. HW1F) interest rate model. Intra-asset correlations can be estimated from time series and may also be observed via the traded prices of products such as spread options, baskets, and quantos.

In some cases, the population of transactions with a given counterparty will cover two or more asset classes or contain cross-asset-class transactions such as cross-currency swaps. The inter-asset class correlation between the risk factors must then be considered carefully. Inter-asset-class correlations are harder to estimate from historical time series as the correlations are more likely to be unstable due to the more subtle relationship across asset classes. Furthermore, inter-asset correlations are less likely to be able to be implied from the prices of market instruments. However, such correlations, especially for uncollateralised counterparties, can often be less important due to single-asset-class transactions. Having said that, even a relatively simple end user of derivatives, such as an airline, could in theory trade across commodities, FX, and interest rate products, creating a future exposure dependent on many inter- and intra-asset-class correlation parameters.

In general, it is not possible to imply such correlations from market prices (with the exception of the FX volatility case mentioned in Section 15.4.3), and so they are typically estimated historically. The estimation generally uses a reasonably large data history and may avoid estimation methods such as exponentially-weighted moving average (EWMA, Section 9.3.4), which could lead to instability in correlation numbers. The historical correlations may only be updated periodically (e.g. quarterly).

When using an integrated diffusion scheme, it is necessary to generate integrated correlation matrices for propagating the simulation, so as to maintain the correct joint multidimensional correlation structure. These integrated correlation matrices are generated from the incremental co-variances and variances associated with each time step period via numerical integrals of the deterministic (piece-wise constant) volatility term structure functions for each respective process, and the corresponding instantaneous pairwise correlation value.

Where risk factors have no underlying developed option market, it may be appropriate to use proxies to define the volatility term structure. An alternative is to use historical estimates on volatility, although in such cases it is important to consider the impact of mean reversion. In situations such as interest rates, a constant (e.g. historical) volatility together with mean reversion can create low volatility for long-term horizons. On the other hand, where there is no mean reversion (e.g. FX), the risk factor distribution over long-term horizons may be excessively broad.

Related to the above point, volatilities often need to be extrapolated beyond the last observable point, which will obviously require relatively simple assumptions. However, applying flat extrapolation to FX and interest rate model volatility parameters leads to monotonically-increasing implied FX forward volatilities, potentially leading to excessively high long-dated volatility, which would impact instruments with long-dated FX risk such as cross-currency swaps. It may be relevant to provide some ad hoc adjustment, such as reducing volatilities beyond a certain point, to control such problems.

15.5 MODELLING MARGIN (COLLATERAL)

15.5.1 Overview

In order to deal with collateralised counterparties, exposure simulation models require the modelling of the future risk-mitigation effect of margin on the underlying credit exposure to a particular counterparty. Since metrics such as EPE depend on the entire exposure profile, it is necessary to quantify the impact of the future margin that would be held in each given scenario at a future date. Margin modelling is important in the following situations:

• bilateral derivatives with margin agreements;
• exchange-traded and centrally-cleared trades; and
• repos and securities financing trades.

Bilateral derivatives represent the most important category above. However, it is also important to model margin in cases such as centrally-cleared trades, both to assess the capital requirements to the CCP (Section 13.6) and also to determine the risk when acting as a clearing member. Repos and securities financing trades are generally low risk due to being short-dated.

Uncollateralised exposure should be considered over the full time horizon of the transaction(s) in question. Long-term distributional assumptions, such as mean reversion and drift, are important and the specifics of the transactions, such as cash flow dates and exercise times, must be considered. In general, margin partially changes this by transforming a risk that should be considered usually over many years into one that is primarily relevant over a much shorter period. As discussed previously (Section 7.5.2), this period is commonly known as the MPoR. The impact of the MPoR is illustrated in Figure 15.17, which shows some future point at which a position is strongly collateralised (e.g. no threshold in the margin agreement), and hence the main concern is the relatively small amount of risk over the MPoR. Note that, due to the length of the MPoR, aspects such as drift and the precise details of the transaction may not be important. Indeed, some of the intricacies of modelling exposure can often be ignored, as long as the counterparty is well collateralised. The problem now becomes a short-term market risk issue and therefore shares some commonalities with market risk value-at-risk (VAR) methodologies (Section 2.6.1).

Figure 15.17 Schematic illustration of the impact of collateralisation on future exposure.

Whilst the above is generally true, the overall impact of margin is not always straightforward and may not reduce the exposure as much as might be expected. Furthermore, certain contractual terms can be more difficult to assess and quantify. Note also that, whilst margin may leave residual market risk that is only a fraction of the uncollateralised risk, it will be more difficult and subjective to quantify and indeed hedge.

It could be argued that, in some situations, modelling margin over the entire life of a portfolio is unnecessary, and that the current risk for a short period equal to the MPoR can be used as a proxy for this. Indeed, in certain situations it might be misleading to model the entire life of the portfolio: for example, a client portfolio of short-dated derivatives that are being constantly hedged and replaced. Regulatory capital methodologies have historically recognised this through the use of the so-called ‘shortcut’ method (Section 13.4.1).

However, going forward, such simple approaches do not appear to be recognised. Regulators generally require internal models to make full multiperiod calculations of margined exposures,¹³ rather than use single-period approximations (e.g. the shortcut method – is being removed as a possible regulatory capital calculation, as discussed in Section 13.4.1). The need to fully model margin in regulatory capital calculations is emphasised by the incoming rules for standardised CVA (SA-CVA) (Section 13.3.5), with BCBS (2017) stating:

For margined counterparties, the exposure simulation must capture the effects of margining collateral that is recognised as a risk mitigant along each exposure path. All the relevant contractual features such as the nature of the margin agreement (unilateral vs bilateral), the frequency of margin calls, the type of collateral, thresholds, independent amounts, initial margins and minimum transfer amounts must be appropriately captured by the exposure model.

With respect to different transactions and trading relationships, regulation only treats them differently in the choice of the MPoR, where the following rules generally apply:¹⁴

• the MPoR should be five business days for repurchase transactions (repos), securities- or commodities-lending or -borrowing transactions, and margin-lending transactions;
• the MPoR should be five business days for centrally-cleared transactions;
• the MPoR should be 10 business days for all other (generally bilateral OTC) derivatives;
• the MPoR shall be at least 20 business days for netting sets involving either illiquid margin or an OTC derivative that cannot be easily replaced; and
• the MPoR must be increased by N – 1 days for a contractual remargining period of N which is greater than one day.

There is, therefore, a general need to capture margining in any exposure simulation.

15.5.2 Margin Period of Risk

The close-out process in the aftermath of a default will not occur instantaneously, but rather will be a gradual process involving a combination of:

• delay before the counterparty is deemed or assumed to be in default;
• macro-hedging, replacement/unwinding of hedges; and
• liquidation of non-cash margin.

As noted in Section 7.5.2, it is important to realise that the MPoR is a model parameter and – since the exposure simulation modelling will be necessarily simplistic – will not align with the actual time taken in a default scenario. There are a number of effects that exposure simulations will ignore or simplify and it is important that the chosen MPoR accounts for these. Some examples are:

• Risk reduction. At some point during the default process, risk will be gradually reduced as a result of hedging, replacing, and unwinding positions. This would tend to shorten the risk horizon, as discussed in Section 7.5.2 and shown in Appendix 7A.
• Conditionality. Exposure quantification generally assumes implicitly that after a counterparty default, the economic conditions will remain unchanged. In reality, market conditions in the aftermath of a default may be different; most obviously, there may be increased volatility. This will be particularly significant for a large financial counterparty. Indeed, in the aftermath of the Lehman Brothers bankruptcy, the volatility in the CDS market was five times greater than in the preceding period (Pykhtin and Sokol 2013). Counterparties that post margin generally tend to be larger financial institutions and have larger OTC derivatives exposures. The impact of their default, notwithstanding any margin considerations, may be considered to be significant. This would tend to increase the required MPoR unless modelled directly. Following the ‘square root of time’ rule (Section 7.5.2), in order to mimic the impact of volatility doubling in the aftermath of a default, it would be necessary to quadruple the MPoR.
• Disputes. In the case of a dispute, the undisputed amount should be transferred, and then the parties involved should enter into negotiations to agree on the disputed amount. The latter procedure may take significant time and would suggest an increase in the MPoR. Indeed, this is one of the aspects under Basel III that can lead to an increase in the regulatory MPoR (Section 13.4.5).
• Cash flows and margin flows. Although the MPoR is a relatively short period, for some portfolios there may be cash flows that would occur during this interval. There is also the question of whether both parties or merely the defaulting party will cease to make margin payments during the MPoR.

The discussion above should illustrate that the MPoR is a parameter that may incorporate many effects and it should not be interpreted as being precisely related to the actual time to close out a portfolio in a counterparty default scenario. A typical exposure simulation model effectively assumes that the whole portfolio will be closed out and replaced at the end of the MPoR, and ignores many or all of the above effects.

Very little attention has been given to the precise modelling of the events during the MPoR. One recent exception looking at the impact of cash flow payments is Andersen et al. (2015). It is important to balance the incremental benefit of more advanced margin modelling against the fact that the MPoR is by nature very hard to quantify. The MPoR is a ubiquitous parameter that captures the general uncertainty and delay inherent in the margin process. However, it does express many uncertain aspects, such as margin disputes and increased market volatility in a single variable.

Note finally that the MPoR is used for modelling credit exposure and for the purpose of quantifying components such as PFE and CVA. It is not relevant for modelling margin for the purpose of calculating collateral and funding value adjustments. However, such adjustments may still need to quantify a collateralised exposure, as will be discussed in Section 15.5.3.

15.5.3 Modelling Approach

The obvious output of an exposure simulation is the value of the portfolio at each point in time and in each scenario, which will be denoted as . This allows quantities such as EPE to be calculated easily and there is, therefore, no need to store any intermediate information, such as cash flows or calculations that led to the evaluation of this value at a given point in time.

Knowing will allow various other features to be included after the original calculation of the uncollateralised exposure. For example, a mandatory break clause would be assumed to lead to the portfolio value being zero at all points in time after the break. A resettable transaction can be treated by resetting the value to zero and changing the contractual rate at each of the reset dates. Since clauses such as resets and mandatory breaks are transaction level, this would need to be done prior to aggregation of the portfolio value.

Margin, at first glance, is fairly easy to include. Since margin will typically apply at the portfolio level, it is not necessary to retain information on individual transactions. The uncollateralised portfolio value along a given simulation path can simply be used together with the logic to determine, at each point, how much margin would be posted. This amount, in turn, would determine the collateralised portfolio value. Examples of how to do this were presented in Section 7.3.6. Traditional margin agreements are usually treated separately, independently, and after the quantification of uncollateralised exposure, since they generally depend on the value of the underlying portfolio and portfolio-specific fixed contractual terms (such as thresholds). However, there are caveats to this which will be discussed below.

---

Spreadsheet 15.4 Simple margin simulation based on portfolio value.

---

In order to deal with collateralised counterparties, exposure simulation models must incorporate the implicit delay caused by the MPoR (Figure 15.18). To quantify the collateralised exposure at a given time , it is necessary to know information regarding how much margin would have been requested previously up to time .

The simplest way to modify an exposure simulation to include margin is, therefore, to include additional points to determine the amount of margin that would be called prior to the start of the MPoR. This could use either a look-back approach with single points, or a more continuous approach¹⁵ simulating at all points, with a time step equal to the MPoR (Figure 15.19). The former approach is clearly simpler and allows an easily-configurable MPoR without changing the main time step of the simulation. The latter approach, whilst more expensive, does allow any path dependency in margining to be represented.

Generalising the previous definition of credit exposure (Section 11.1.1, Equation 11.1) to include margin, but ignoring aspects such as segregation for now, gives:

Figure 15.18 Schematic illustration of the use of MPoR to model the impact of margin.

Figure 15.19 Illustration of time grid for exposure simulation, with additional points included for margin calculations.

(15.3)

This – seemingly reasonably – defines the collateralised exposure as being determined by the portfolio value at the current time less the amount of margin required at a previous time . In this situation, a positive value for would indicate margin received, whilst a negative value would indicate margin posted.

One possible problem with the above is that it assumes that the amount of margin at a given time can be calculated knowing only information available at that time. This is generally true but not always, one example being minimum transfer amounts (MTAs). Take the following example: the current portfolio value is and the value at a point in the past corresponding to the MPoR is . It may, therefore, be reasonable to assume that , resulting in a positive exposure of However, this would assume that whatever margin was required at time () was above any minimum transfer amount. Table 7.7 in Section 7.3.6 illustrates that the value could be either lower or higher than this depending on the size of the MTA and the values prior to the point . To model this path dependency properly would require more time steps, as in the ‘continuous’ sampling shown in Figure 15.19. Given that MTAs are typically small, this may not be considered important, and unless there is any strong path dependency (e.g. a large minimum transfer amount), then a look-back approach is a reasonable approximation.

Equivalently, there is the question of whether or not there is a difference between margin held that needs to be returned and margin that needs to be posted outright. Again, a distinction does not often have to be made, but there are situations where this is important, such as in the consideration of ‘cheapest-to-deliver’ optionality (Section 16.2.3).

Assuming a simple modelling of a collateralised exposure via Equation 15.3 makes the following assumptions:

• the defaulting party will make any contractual margin payments up to the time , but cease doing so after this point;
• the surviving party will also stop making margin payments at the same point; and
• both counterparties will continue to pay cash flows during the MPoR period (which will be captured by the change between and ).

The above is probably unreasonable as the (defaulting) counterparty will likely cease any cash flow payments at the same time as stopping the posting of margin. Furthermore, the surviving party may at some point cease to make contractual payments (cash flows and margin), but only when it is aware of the default and/or is confident in the legal basis for stopping payments. Using the previous definitions of pre- and post-default, it is possible to propose more realistic assumptions (‘improved model’) compared to the ‘classical model’ (Table 15.4).

The improved model – or some variation of it – would be necessarily more complex than the classical model. In particular, it would require an extra time point to be included that would represent the point at which the counterparty is known to be in default, and so the surviving counterparty may cease to make payments. It would also require the (non-netted) cash flows to be known precisely, rather than being represented implicitly via the evolution of the term in Equation 15.1.

Note that regulation only specifies the length of the MPoR and does not define what precise assumptions should be made during this period. Regulation does not, therefore, prescribe a more complex modelling approach to be taken. It is also clear that banks are using different modelling assumptions within their internal models (BCBS 2015b). Clearly, it is important to consider whether improved assumptions, with their additional complexity, are required.

Table 15.4 Assumptions used in classical and improved models for modelling margin within the MPoR.

		Classical model		Improved model
		Pre-default	Post-default	Pre-default	Post-default
Defaulting party	Cash flows	Continue		Stops
	Margin	Stops		Stops
Surviving party	Cash flows	Continue		Continue	Stops
	Margin	Stops		Continue	Stops

It is also important to distinguish between MPoR modelling for capital purposes and for CVA calculations. In the former case, it could be argued that a better – and possibly more prudent – modelling of the MPoR is advantageous to avoid portfolios being undercapitalised. For CVA calculations, this may be seen as less relevant since it may not be possible to charge CVA to many collateralised counterparties, and the accounting CVA is not supposed to represent prudent – but rather fair – assumptions. However, given that future regulation (BCBS 2017) is aligning regulatory capital with accounting CVA, this point may not apply at some point in the future.¹⁶

15.5.4 Initial Margin

The historical approach to modelling margin was relatively simple because – assuming a classical model for the MPoR – the approach only relied on the portfolio value at each time step in the simulation. This is because variation margin is directly related to the portfolio value and the contractual terms in the margin agreement. The only complexity in modelling margin would have been in relation to an improved modelling of the MPoR, as discussed in Section 15.5.3.

However, the growing presence of initial margin being held against bilateral OTC derivatives due to the ‘bilateral margin requirements’ (Section 7.4) is requiring more advanced methods for modelling the amount of future margin held to quantify correctly the risk-reducing impact of initial margin. Whilst such additional margin is not completely uncommon historically, its impact prior to the bilateral margin requirements has been of less concern for the following reasons:

• In bilateral derivatives, the additional margin was often based on simple metrics such as percentage of notional, which are easy to model.
• In centrally-cleared derivatives, where a party was acting as a clearing member (and therefore facing clients bilaterally), there may have been less concern over residual risk since the portfolios would typically be short-dated.
• Traditional non-internal model methods for quantifying capital, notably CEM described in Section 13.4.2,¹⁷ did not treat initial margin or other overcollateralisation specifically. This actually led to a rather favourable treatment and would often lead to capital charges (via the exposure at default) being zero. This is no longer the case due to the replacement of CEM with SA-CCR, which does treat such components (known as the ‘net independent collateral amount’) directly but in a rather conservative fashion, as described in Section 13.4.3. So, for non-IMM banks, the regulatory capital treatment of initial margin has become less favourable.

Considering the modelling of collateralised exposure in a general way, the calculation of the future amount of margin held can be considered – in order of increasing complexity – to be a function of the following:

• deterministic characteristics of the portfolio (e.g. the notional or time to maturity of a transaction) that do not depend on the market conditions;
• quantities dependent on the market state at the time point in question (e.g. variation margin linked to the portfolio-level, or possibly transaction-level, value, and contractual quantities such as thresholds);
• as above, but requiring additional calculations (e.g. sensitivities and aggregation rules as required for the ISDA SIMM, as described in Section 9.4.4); and
• as above, but also requiring the time series of variables up to the future time point in question (e.g. CCP initial margin methodologies using historical simulation).

The first two points above are relatively easy since – in addition to the appropriate time-step modelling for the MPoR mentioned above – it requires only basic information and calculations. This applies mainly to the general discussion in Section 15.5.3. The third and fourth situations above introduce dramatically more difficult calculations and data requirements, such as:

• reproduction of the necessary calculations that may only be usually done at time zero (e.g. sensitivities and VAR);
• the reconstruction and generation of all the required data (e.g. historical time series); and
• the provision of adequate computational resources in order to accomplish the above.

Due to the number of simulations and time steps in a typical exposure calculation, the above represents a significant computational requirement, especially since it likely covers a very large universe of transactions. For example, to include, without approximation, the risk-mitigation benefits of initial margin calculated under ISDA SIMM in an exposure simulation would require:

• the ability to determine which trades are subject (including partially)¹⁸ to bilateral initial margin requirements and to allocate the €50m threshold to each portfolio;
• the ability to calculate dynamically the trade sensitivities (in addition to trade values) in all future scenarios (this is both an implementation and computation challenge);
• the incorporation of SIMM parameters and aggregation rules into the exposure simulation (and the provision of subsequent updates of these on an annual basis); and
• the ability to reconstruct the time series of returns for each risk factor so as to be able to determine how SIMM would recalibrate at a given future date.

Typically, there are a number of different methods that have been proposed to overcome such challenges. These approaches are similar to the need to calculate future initial margin for the calculation of MVA, discussed later in Section 20.2.4, and include:

• Simple approximations. Approaches based on normal distribution assumptions have been described by Andersen et al. (2017) and Gregory (2016).
• Proxies. The use of approximate pricing formulas (and their sensitivities) that are much faster to compute (e.g. Zeron and Ruiz 2018).
• Adjoint algorithmic differentiation (AAD). For fast calculation of sensitivities, as required by ISDA SIMM.
• Regression techniques. The use of information on risk factor evolution within the exposure simulation to approximate the initial margin (e.g. Anfuso et al. 2016 and Caspers et al. 2017).

It is important to balance carefully the effort in building a complex and accurate methodology for initial margin calculation against using a simpler but more conservative approach (that may produce, for example, less capital relief but be much easier to put into production). Not surprisingly, banks have not found it easy to get approval to incorporate the risk-reducing benefit of future initial margin into their IMM capital calculations (Figure 15.20).

Figure 15.20 Incorporation of initial margin into IMM capital approaches. Source: McKinsey/Solum Survey (2017).

15.6 EXAMPLES

15.6.1 Interest Rate Swap Example

---

Spreadsheet 15.5 HW1F model for exposure for interest rate products.

---

The first example will show the standalone exposure profiles for a pay fixed IRS with a notional of 1,000 using a HW1F model which can be reproduced with Spreadsheet 15.5. Figure 15.21 shows the interest rate curve to which the model is calibrated (in terms of the ‘zero rates’). Also shown are the values of co-terminal forward starting swaps. Note that the value of the forward starting swaps is equal to the discounted EFV profile since, by arbitrage, the discounted expected value of the swap at some point in the future must be equal to the current price of a swap starting at that future point. This calculation of EFV is relatively simple and – unlike EPE or ENE – can be calculated without simulation. Note that the EFV is positive when paying the fixed rate, as discussed in Section 11.2.3. The equivalent receiver swap would have an equal and opposite EFV.

It is possible to use Monte Carlo simulation to calculate the exposure distribution of the swap.¹⁹ First, it is necessary to simulate the interest rate paths, which are approximately centred around the forward rates as shown in Figure 15.22. A flat volatility of 1% and mean reversion parameter of 0.01 were used. Since the swap cash flows are paid on a quarterly basis, this same discretisation is used for the path-wise simulation of interest rates.

Given the interest rate simulations, it is possible to revalue the swap at each future date via the analytical formulas for discount factors. The resulting future value paths of the swap are shown in Figure 15.23. This expands into an envelope, but then contracts due to the ageing effect of cash flows (Section 11.2.2). Also shown are PFEs at the 5% and 95% confidence levels and the EPE and ENE.

Figure 15.21 Interest rate curve and forward values of co-terminal forward starting swaps.

Figure 15.22 Interest rate paths with the forward rates shown (solid line).

Figure 15.23 Future values of the swap. The PFEs (5% and 95%) are shown by the dotted black lines and the EPE and ENE are shown by the solid black lines.

Finally, Figure 15.24 shows the EPE, ENE, and EFV of the swap. Note that the sum of the EPE and ENE is approximately equal to the EFV, which is shown. The exact EPE and ENE are also shown via the valuation of the appropriate co-terminal swaptions (Section 15.4.2), which in this interest rate model can be evaluated analytically.

Figure 15.25 shows profiles for four variations on the above swap. First, a swap with unequal cash flow payment dates (receive quarterly, pay semiannually) which shows a jagged profile due to the unequal payment dates. Second, an ITM swap receiving a higher fixed rate. Note that the fact that the swap is ITM means that the exposure profile is more predictable: in the swaption analogy (Section 15.1.3), this is because the underlying swaption is either ITM (EPE) or OTM (ENE) and therefore less sensitive to volatility. Third, a three-year into seven-year (3Yx7Y) forward starting swap that shows a slightly different structure due to the absence of cash flows during the forward start period. Finally, a long physically-settled swaption based on the same forward starting swap. Note that the swaption is – like the ITM swap – reasonably predictable but does have some negative exposure after the exercise date, as discussed previously in Section 11.2.6.

Figure 15.24 EPE, ENE, and EFV of the swap from Monte Carlo simulation. Also marked are the analytical results from valuing the forward start swaption (EFV) and appropriate co-terminal swaptions (EPE and ENE).

The above example used a constant volatility assumption. As discussed in Section 15.4.2, volatility calibration is a challenge, especially when using relatively simple models. Figure 15.26 shows the EPE of a 20-year swap with a rate (not par rate) equal to that of the 10Yx10Y forward starting swap and a volatility function calibrated to the 20-year co-terminal swaptions. By construction, the 10Yx10Y swaption is reproduced almost exactly, but the EPE does not match precisely the other co-terminal swaptions due to the different market rates.

15.6.2 Trade-level Exposures

The examples in this section will use a number of real transactions, which are as follows:²⁰

• 7Y payer IRS. Seven-year pay fixed USD interest rate swap.
• 5Y payer IRS. Five-year pay fixed USD interest rate swap.
• 5Y receiver IRS. The opposite (i.e. receiver) of the above.

  

  Figure 15.25 EPE, ENE, and EFV for (top to bottom) a swap with unequal cash flows, an ITM swap, a forward starting swap (3Yx7Y), and a physically-settled swaption (3Yx7Y).

  

  Figure 15.26 EPE for a 20-year swap with a fixed rate equal to the 10Yx10Y forward starting swap compared to 20-year co-terminal swaption values.

• 5Yx5Y long payer swaption. Five-year long swaption on five-year pay fixed USD interest rate swap with physical delivery.
• 5Y USDJPY XCCY. Five-year pay USD, receive JPY cross-currency swap.

All transactions have a notional of $100, except the cross-currency swap, which has a smaller notional of ¥10,000.²¹ The exposures have been simulated at time intervals of three months, with a total of 2,000 simulations. The transaction-level EPE/ENE and PFE profiles (both positive and negative) are shown in Figure 15.27. All the results below will be reported in US dollars.

The interest rate and cross-currency swaps show the characteristic shapes discussed in Sections 11.2.2 and 11.2.1, with drift effects determined by the shape of the yield and interest rate differential respectively. The swaption exposure profile is more complex due to the fact that it is physically settled. During the first five years, the exposure can only be positive due to the long optionality, but there is some chance of negative exposure for the last five years where a swap may be present. However, on many paths where the swap would be OTM, the swaption would not have been exercised, and so the negative exposure (ENE or 5% PFE) is relatively small. The exposure profile of options was discussed previously in Section 11.2.6.

15.6.3 Portfolio Exposures

---

Spreadsheet 15.6 Illustration of the impact of netting.

---

Figure 15.27 EPE, ENE, and PFE profiles for the 7Y payer IRS (top left), 5Yx5Y long payer swaption (top right) and 5Y USDJPY XCCY (bottom).

Portfolio effects cause exposures to be non-additive such that each transaction contributes less than its standalone exposure to the total exposure. One way to see this impact is by calculating the incremental exposure of a new transaction with respect to the existing portfolio. As discussed in Section 15.2.2, the impact of the portfolio effect depends on the interaction between the new transaction and the existing portfolio. Figure 15.28 shows the standalone and incremental exposure (EPE and ENE) for a directional trade.²² In this case, there is only a relatively small reduction in EPE and ENE due to the fact that the exposure is almost additive.

Figure 15.29 shows a similar example, but where the new trade can be considered to be neutral or offsetting.²³ In this case, the reduction in EPE and ENE is large as a result of the strong cancellation effect between the portfolio and the new trade. Note that the effect is relatively complex: although the absolute reduction in EPE and ENE is similar, the ENE profile actually changes sign. Note also that there is no incremental effect for the EFV (which is the sum of the EPE and ENE): the incremental EFV is the same as the standalone EFV.

Figure 15.28 Standalone and incremental exposures for the case of a directional trade.

Figure 15.29 Standalone and incremental exposures for the case of a risk-reducing trade.

We now compare incremental and marginal exposures for two different trades: a cross-currency swap (XCCY Swap) and an IRS.²⁴ Figure 15.30 shows the EPE allocated incrementally (in both ways) and marginally. Note that the top line is the same in all cases, as this represents the total EPE. The second transaction allocated incrementally has a relatively small exposure, to the detriment of the first transaction. The marginal allocation is more balanced and would be appropriate if the transactions occurred at the same time.²⁵ The respective CVA and FVA numbers for this example will be shown later in Sections 17.4.2 and 18.3.3.

Figure 15.30 Illustration of the breakdown of the EPE of the interest rate and cross-currency swaps via incremental (interest rate swap first), incremental (cross currency swap first), and marginal allocation.

Marginal exposures can be materially different from standalone exposures. The marginal allocation of EPEs for the five transactions²⁶ is shown in Figure 15.31 and compared to the standalone contributions (which are significantly larger due to the lack of portfolio effects). The marginal allocation is very different compared to the standalone contributions. Most obviously, the receiver IRS has a negative marginal EPE contribution due to being risk reducing with respect to the payer IRS (which has an equal and opposite marginal EPE) and the swaption.

---

Spreadsheet 15.7 Marginal EPEs.

---

Marginal EPEs will translate directly into marginal xVA (and PFE) contributions, which will be revisited in Sections 17.4 and 18.3.3. Marginal allocations may be useful to analyse when attempting to reduce the overall exposure of the transactions in order to comply with the credit limit²⁷ or to optimise xVA. All other things being equal, the transaction with the higher marginal EPE (or PFE) is the most relevant to look at. There are two points of interest here. Firstly, the marginal allocation is not homogeneous with time and so, depending on the horizon of interest, the highest contributor will be different. Secondly, it is not always easy to predict, a priori, which transaction will be the major contributor.

Figure 15.31 Illustration of the standalone (top) and marginal (bottom) EPE of the five transactions. Note that in the latter case the total reflects the subtraction of the negative contribution for the receiver interest rate swap.

15.6.4 Notional Resets

Notional resets are similar to collateralisation, with the following differences:

• they are typically defined at the transaction rather than portfolio level;
• the reset is a settlement paid in cash; and
• the reset is periodic (e.g. quarterly), and therefore the risk reduction is weaker compared to the shorter MPoR (e.g. 10 days).

A common example of a reset is in a cross-currency swap. In a standard cross-currency swap, as the FX rate moves, one party will be ITM and the other will be OTM with respect to the exchange of notional at the end of the transaction. This can create a large exposure, as shown in Figure 15.27. In a notional resetting cross-currency swap, the notional of one leg resets on every coupon period to rebalance the FX exposures in the transaction. The mark-to-market difference at each reset is settled from the OTM to the ITM party.

---

Spreadsheet 15.8 Notional resetting cross-currency swap.

---

Figure 15.32 shows the exposure of standard and notional resetting cross-currency swaps (quarterly coupons). The exposure is clearly reduced by the reset, which brings the value back to zero at each quarterly date.

15.6.5 Impact of Variation Margin

As discussed in Section 15.5.3, modelling variation margin is relatively straightforward as it is generally related directly to the value of the underlying portfolio, together with any relevant parameters such as thresholds. This means that the impact of margining can be accounted for after the simulation of exposure, under the assumption that the margin agreement depends only on the net portfolio value and not on other market variables.²⁸ This has been described previously in Sections 11.3.3 and 15.5.3.

Figure 15.32 Exposure of standard and notional resetting cross-currency swaps.

---

Spreadsheet 15.9 Quantifying the impact of margin on exposure.

---

The examples below use a portfolio of three interest rate swaps and one cross-currency swap with a total notional of £325. More details can be found in Spreadsheet 15.9.²⁹ The exposure simulation is done with a ‘continuous’ grid (Figure 15.9) with a time step of 10 days and uses a ‘classical’ model (Table 15.4). This means that the MPoR can be assumed to be an integer multiple of this amount. The thresholds and MTAs are assumed to be zero. There is assumed to be no market risk on the margin itself (i.e. it is paid in cash in the portfolio base currency).

In general, for a strongly-collateralised position (variation margin, zero threshold, and assumed MPoR of 10 days), the amount of margin will track the value of the portfolio with an inherent delay due to the modelled MPoR. This is illustrated in Figure 15.33 which shows a single simulation path. Note that there is material noise due to the MPoR and that this noise is present even when the portfolio value is negative (Section 11.3.3). This is due to the need to post margin against a negative portfolio value, which creates risk in case the portfolio value increases and the counterparty defaults.³⁰

The overall impact of margin on EPE and PFE (the latter at the 95% confidence level) is shown in Figure 15.34. There is still a significant exposure due to the MPoR: although the 10-day MPoR is significantly shorter than the portfolio maturity (125 times),³¹ the risk reduction will be more approximated by the square root of time (approximately 11 times).

Figure 15.33 Illustration of an individual simulation path and the impact of margin on that path.

Figure 15.34 Illustration of EPE (top) and 95% PFE (bottom) calculated with and without collateralisation.

Furthermore, the collateralised exposure also shows ‘collateral spikes’ (Section 7.3.6) due to the payment of cash flows (for which margin is not received immediately). These spikes represent an increase in the exposure for a length of time corresponding to the MPoR. The collateral spikes are not as pronounced for the PFE since this represents an extreme scenario, whereas the cash flow payment is the same in all scenarios. Note also that the relative reduction in the PFE is better, since not all simulations contribute equally to the PFE, and those with the most contribution (when the exposure is high) are precisely the ones where the most margin is taken. Note that margin spikes are considered by regulators to be part of counterparty risk and should not be treated separately as settlement risk (see BCBS 2015b).

Figure 15.35 Illustration of EPE calculated with different threshold assumptions.

It is possible to approximate the collateralised EPE and PFE quite well, as shown in Figure 15.34 (see Appendix 15C for more details). The PFE approximation is better since it is less sensitive to effects such as margin spikes.

The discretisation of the exposure simulation is clearly important in terms of correctly capturing the impact of cash flows. Furthermore, the precise modelling assumptions within the MPoR can have a significant impact on quantities such as EPE. This has been discussed by Andersen et al. (2017).

The above example of a strongly-collateralised portfolio is relevant for the consideration of EPE and PFE and also later CVA (and possibly DVA). It is probably not relevant for funding (FVA) considerations where the MPoR would be assumed to be negligible since this does not correspond to a default scenario (Section 11.4.2).

One situation where funding would be a consideration is if there is a significant threshold in the margin agreement, although this is becoming increasingly uncommon. Figure 15.35 shows the EPE in a high-threshold margin agreement compared to the case of a zero-threshold or no margin agreement (i.e. infinite threshold). As expected, the exposure with a high threshold is intermediate between the other cases.

15.6.6 Impact of Initial Margin

Although initial margin is historically quite rare, receiving it against bilateral derivatives portfolios is becoming increasingly common due to the bilateral margin requirements (Section 7.4). Clearly, received initial margin can potentially reduce exposure to negligible levels if it is substantial enough. Note that the initial margin posted should not increase the exposure, as long as this is appropriately segregated.

One way of looking at an initial margin is that it converts counterparty risk into ‘gap risk’. The gap risk is defined in this case by the chance of the exposure ‘gapping’ through the initial margin during the MPoR. Quantifying the residual exposure now becomes more difficult since it is driven by more extreme events and will be very sensitive to modelling assumptions. When assessing gap risk, one should be more concerned about distributional assumptions such as fat tails, jumps, and extreme co-dependency.

The modelling of received initial margin can be a challenge depending on the underlying methodology used to define the amount. Traditional ‘independent amounts’ based on simple metrics such as percentage of notional or the standardised margin schedule (Section 7.4.3) are easy to capture since they depend on known quantities within the exposure simulation (such as the remaining maturity for a given transaction). Whilst this does require transaction-specific information to be retained for a portfolio exposure calculation, this is relatively straightforward and not computationally demanding. More sophisticated initial margin methodologies, such as ISDA SIMM (Section 9.4.4), represent a more difficult challenge because (in addition to transaction-level information) they are model-based calculations. This means that (ideally) the full model-implied initial margin will be replicated within the exposure calculation. Put another way, as discussed in Section 15.5.4, SIMM-based initial margin is scenario dependent and will be different in each exposure simulation at the same point in time.

Since the initial margin is taken to a high confidence level (e.g. 99%), there is the question of whether a full modelling of exposure is actually required. At such high coverage, it would be expected that the residual exposure would be immaterial. Indeed, in a simple example (Appendix 15D), it is possible to show that 99% initial margin would reduce the exposure by over two orders of magnitude. However, this type of stylised analysis ignores features such as collateral spikes, which are typically not captured by an initial margin methodology (as such methodologies typically only consider market risk and not quantities such as cash flows). Furthermore, there are several reasons why the initial margin held against a portfolio will not equate to a true 99% coverage:

• the requirement to post initial margin (IM) - where it applies - only impacts new transactions, and so legacy trades will not have initial margin held against them (although this effect will fade over time until all legacy trades have matured);
• there is a threshold of up to €50m that can be applied to the IM amount; and
• there are exempt transactions (e.g. some FX).

Figure 15.36 illustrates some of the above points by showing the EPE in the presence of a 99% IM and compared to the case with only the variation margin (shown previously in Figure 15.34). Also shown is the case of partial coverage where IM is not taken against all of the underlying transactions.³²

Figure 15.36 shows that even with a full 99% IM, the residual risk is material, which is mainly due to the collateral spikes. When IM provides only partial coverage, the residual exposure is larger still.

Figure 15.36 EPE with and without IM.

It will clearly be increasingly important in the future to model accurately the impact of dynamic IM. This is a very challenging computational task since it requires a calculation of the future IM that would be held at every single scenario in the exposure calculation.

NOTES

1. 1 We note that these semianalytical formulas are generally concerned with calculating risk-neutral exposures using underlyings such as traded swaption prices. Such approaches can also be used for real-world calculations (as is usual for PFE), but this is not as straightforward.
2. 2 These EPE numbers can be computed using the formula in Appendix 11A. This effect is similar to an ITM option having a similar value to an OTM option with a greater underlying volatility.
3. 3 Since the distributions are independent, we can calculate the combined mean and variance as and , respectively, and then use the formula in Appendix 11A.
4. 4 This leads to a transfer pricing concept for xVA at trade inception, where a hard payment is made to the xVA desk.
5. 5 One obviously difficult case is where there is a margin agreement with a non-zero threshold or an initial margin. This is discussed in more detail by Rosen and Pykhtin (2010).
6. 6 In the case of normal distributions, the analytical expression makes the calculation of marginal EPE quite easy without the need for simulation, as shown in Spreadsheet 15.2.
7. 7 This can be used for both xVA and PFE applications, although the accuracy in the latter case is typically worse. Furthermore, being a generic approach, this will not match front-office valuations exactly.
8. 8 As described below, risk-neutral drift may often be used anyway for calculating exposure for risk-management purposes.
9. 9 Using implied volatility might be expected to produce an upwards bias due to a risk premium, leading to higher (more conservative) risk numbers.
10. 10 Meaning, for example, that the worse correlation may not be equal to 100% or -100%, but somewhere in between.
11. 11 Correlation is often the specific way in which dependency is represented, and it is very commonly used. We will use correlation from now on, but note that there are other ways to model dependency, as discussed further in Chapter 17.
12. 12 This typically refers to the implied volatility of the forward swap rate in a Black model.
13. 13 European Banking Authority (EBA) Capital Requirements Regulation states ‘The model shall estimate EE [EPE] at a series of future dates t1, t2, t3, etc.’ EBA Capital Requirements Regulation > Part Three > Title II > Chapter 6 > Section 6 > Article 284. www.eba.europa.eu.
14. 14 EBA Capital Requirements Regulation > Part Three > Title II > Chapter 6 > Section 6 > Article 285. www.eba.europa.eu.
15. 15 Continuous with respect to the MPoR unit.
16. 16 The alignment of accounting and regulatory CVA is part of the SA-CVA approach described in Section 13.3.5.
17. 17 This is also the case for the so-called ‘standardised method’ that was mentioned briefly during the explanation of CEM.
18. 18 For example, the FX component of cross-currency swaps is currently exempt from initial margin.
19. 19 A total of 5,000 simulations were used for all the examples.
20. 20 I am grateful to IHS Markit for providing the simulation data for these examples.
21. 21 This is to prevent the inherently riskier cross-currency swap dominating the results. The USD notional was $43.50.
22. 22 This trade is the 7Y payer IRS described in Section 15.6.2, with the existing portfolio assumed to be made up of the 5Yx5Y long payer swaption. This is directional since both the swap and swaption are in the same currency and paying the fixed rate.
23. 23 This trade is the same 7Y payer IRS, but the existing portfolio is assumed to contain the 5Y USDJPY XCCY. There is no definitive definition of neutral or offsetting. In this case, the EPE effect is more neutral, since it reduces but stays mainly the same sign, whereas the ENE changes sign and so would be better defined as offsetting.
24. 24 This corresponds to the 5Y USDJPY XCCY and 7Y payer IRS.
25. 25 There is still some problem of allocation here since, for trades occurring at the same time, we wish to allocate marginally, whilst the total impact of the trades should be allocated incrementally with respect to existing trades in the netting set. An obvious way to get around this problem is to scale the marginal contributions of the trades so that they match the total incremental effect.
26. 26 Defined at the start of Section 15.6.2.
27. 27 In such a case, PFE, rather than expected positive exposure (EPE), would be the appropriate metric to consider. Whilst the marginal PFE numbers are systemically higher than the EPEs shown, there is no change in the qualitative behaviour.
28. 28 There are situations where this assumption may not be entirely appropriate – for example, margin parameters may be defined in different currencies to the deals to which they apply. In practice, this means some FX translations may be required when the margin parameters are applied within the simulation. However, in the majority of situations the assumptions made will be valid and will greatly simplify the analysis of collateralised exposures.
29. 29 I am grateful to IBM for providing the underlying portfolio simulations which were first used in the second edition of this book.
30. 30 This is because variation margin is not segregated (Section 11.4.3).
31. 31 Five years divided by 10 days, assuming 250 days in a year.
32. 32 Since the portfolio in question contains a cross-currency swap for which the FX portion is exempt from initial margin (Section 7.4.2).