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Modeling Assumptions behind the ISDA CDS Standard Model
October 26, 2014

In February 2009 the International Swaps and Derivatives Association (ISDA) released the ISDA CDS Standard Model as an open source project as part of an effort to standardize global CDS markets. The model is intended to standardize the way in which the running spread can be converted to an upfront fee, as well as how the cash settlement amount is calculated for a CDS. Also as part of this effort, several changes were made to both North American and European CDS contract specifications.

FINCAD has independently replicated the ISDA pricing methodology, and this functionality is included in both FINCAD Analytics Suite 2010 and the new F3 product line. Through extensive testing we have observed excellent agreement between the ISDA open source code and the FINCAD implementation. There are several benefits to using the FINCAD implementation rather than calling the open source ISDA code directly; perhaps the most significant is the provision of all the risk exposures of the CDS, information which is not provided by the ISDA code. In addition, the FINCAD implementation calculates a number of other useful quantities which the open source code does not. This implementation project involved gaining a detailed understanding of roughly 30,000 lines of open source code. In this article we summarize the main modeling assumptions made in the ISDA codebase, and explain why the ISDA model will give slightly different results to other modeling approaches, without using mathematical formulae.

## The Zero Curve and the Survival Curve

The most important assumption is that the discount factor curve and the survival curve are both assumed to be piecewise exponential functions. What does this mean? The discount factor curve (or "zero curve") describes how the price of risk-free zero coupon bonds, of various maturities, decrease when they are sorted by their maturity dates. The price of a zero coupon bond is 1 if the maturity date is today, and approaches 0 if the maturity date is far in the future, due to the decreasing time-value of money. These zero-coupon bonds are risk-free in the sense that the probability of default is negligible. This zero curve is implied and built from quotes for cash deposits, futures and swaps; these are financial instruments for which either the probability of default is negligible (e.g., Government Treasury bills), or their payments depend on supposedly risk-free reference rates and have mechanisms to neutralize the effects of counterparty default risk (e.g., margin accounts, or collateral agreements). The quotes for these liquid instruments, along with an interpolation rule for dates that are in between the maturities of these instruments, provide enough constraints to uniquely determine the discount factor on any arbitrary maturity date. The interpolation rule assumed in the ISDA model is that the zero curve is a series of segments, each of which joins smoothly (without jumps) with its neighbour, and whose shape is that of exponential decay. This shape of the zero curve is equivalent to saying that the instantaneous forward curve (i.e., the forward rates of infinitesimally short tenor) is a staircase of horizontal steps. The steps occur on dates that correspond to the maturities of the instruments used to build the curve.

Likewise, the survival curve shows how the probability that the reference entity (e.g, a corporation) will not have defaulted at a given point in time in the future, changes as a function of that time. This probability is 1 today, assuming that the reference entity has not already defaulted, and decreases to zero as longer and longer time horizons are considered. Typically, this survival curve is implied and built from quotes for CDS spreads for the reference entity of various maturities. The ISDA model assumes, as before, that this survival curve is a series of exponentially decaying segments. This shape is equivalent to saying that the hazard rate (i.e., the rate of expected loss divided by the expected remaining amount at risk) is a staircase of horizontal steps. The steps occur on dates that correspond to the maturities of the CDS spreads used to derive the curve.

The advantage of making these assumptions for the shapes of the curves is that it greatly simplifies, and speeds up, the calculation of the mathematical integrals that are needed for CDS pricing, as will be seen in the following discussion.

## CDS Pricing Overview

Let's start with the protection leg (or "contingent", or "default" leg), which for pricing purposes can be thought of as a single payment in the event of a default. This payment is made by the protection seller as compensation for the protection buyer's net loss, and is calculated as the notional amount minus the amount that is assumed can be recovered (i.e., the recovery rate times the notional amount). In other words, the absolute payment is the product of the notional amount, and 1 minus the expected recovery rate. However, this payment is only made in the event of default, and can occur at any point in time up to the maturity date. To illustrate, imagine that default could only occur on one of two special dates with a known probabilities P and Q, otherwise the reference entity will not default before maturity. In this imaginary situation, the survival curve is 1 up to the first special date, jumps down to 1-P after it, and jumps down further to 1-P-Q after the second date. For pricing purposes the expected payment (in the statistical sense) on the first date is the absolute payment times its probability P, and the expected payment on the second date is the absolute payment times its probability Q. The value of the protection leg is the sum of the Net Present Values (NPV) of the expected payments, which means multiplying the expected payments by the discount factor on each of the two payment dates, and adding them up.

However, in reality, default could occur on any date, so it is necessary to consider every possible default date and weight each possibility according to the probability of default on that date. The value of the protection leg then becomes a sum over a continuum of possible times, rather than a sum over two dates. Computationally, it is easier to make the time increment infinitesimally small, rather than daily, and in the limit the sum becomes a mathematical integral, or an area under a curve. This function to be integrated is the probability of default (for each infinitesimally small period of time) times the discount factor that applies during that small time interval. Under the assumptions made in the ISDA model, both of these quantities are piecewise exponentially decaying functions of time, so their product is also a piecewise exponentially decaying function. The decay constant of the product of two exponentials is the sum of the decay constants of each, which in this case are the instantaneous forward rate (for the discount factor) and the hazard rate (for the default probability). Furthermore, the exponential function can be integrated using elementary calculus, and there is a simple formula for the result, which can be rapidly computed.

Let us now consider the premium leg. This is a regular series of payments, made by the protection buyer, that continues as long as the reference entity has not defaulted, or until the maturity of the CDS, whichever is the sooner. Each payment is the product of the notional amount, the premium rate (or spread), and the accrual factor for that period (e.g., around 0.25 for quarterly payments). However, this absolute payment is only made if the reference entity has not defaulted, so for pricing purposes the expected payment (in the statistical sense) is the absolute payment times the probability of survival at the end of the period. This probability needs to be interpolated from the survival curve using the assumption of the piecewise exponential shape. The contribution of this payment to the CDS price is the NPV of the expected payment, which means multiplying further by the discount factor on the payment date. This factor needs to be interpolated from the zero curve, again using the piecewise exponential assumption. The value of the premium leg is then calculated by summing the contribution from each payment.

A major extra complication when calculating the overall price of the premium leg is caused by the payment of accrued premium upon default. This feature was prescribed as standard in the specification of the ISDA Standard North American Corporate CDS Contract (SNAC). It means that in the event of default, no further regular premium payments would be made, although the premium for the current period would be paid in proportion to the fraction of the period that elapsed before the default event. The absolute amount of the accrued premium would be the full-period premium pro-rated by the accrual fraction to the default date. So the NPV of the expected payment on each possible default date would be the product of the full-period premium, the accrual fraction to that date, the probability of default on that date, and the discount factor on that date. Adding these contributions up for each possible default date gives the contribution of this feature to the CDS price. As before, this summation is simplified in the ISDA method by summing over a continuum of infinitesimally small time periods, rather than over each possible default date. The calculation of this integral is very similar to the one for the protection leg, except that there is now an extra factor multiplying the exponential function to be integrated; this factor is the accrual fraction, which is a linear function of time, and therefore easily handled again by elementary calculus.

## Other Approaches

It can be seen from the above that the assumption of exponential functions for the shape of the zero curve and of the survival curve help significantly in the calculation of the CDS price; especially in the summation over all possible default times for the protection leg and for the accrued premium. However, other modeling assumptions are also consistent with the data used to construct the two curves. For example, other ways of summing over possible default times include:

• Sum over days: the summations are performed over each possible default date; the discount factor and survival probability on each date are interpolated from the zero curve and survival curve respectively, and the probability of default on each date is calculated as the change in survival probability from previous date to the current date.
• Period averaging: here the integral is approximated by a series of rectangles whose width corresponds to a premium period; during each period the discount factor is taken to be a constant (equal to its average value at the start and end of the period), and the probability of default for the period equals the change in the survival probability from the start to the end of the period.
• Trapezoid rule or Simpson's Rule: these are standard numerical methods to evaluate integrals; the time axis can be sliced into an arbitrary number of periods, and the discount factor and survival probability at the boundaries between each period are interpolated.
• Discount from end: in this approximation, it is assumed that default can only occur at the end of each period of the premium leg; the discount factor is taken as the value at the end of each period, and the probability of default for the period again equals the change in the survival probability from the start to the end of the period; the accrual factor for the accrued premium is taken on average to be one-half of the period length.

These different approaches will lead to different results for the CDS price; the magnitude of the difference will generally be small, but will depend on the details of the CDS contract, on the shape of the zero curve and the survival curve, and on the interpolation method used. In general it will not be possible to arbitrage these differences using the liquid instruments used to build the zero curve and survival curve because the precise structure of these curves cannot be known; assumptions on how to interpolate between maturities are necessary to fill in the whole timeline.

## Other Details of the ISDA Model

The ISDA Model also differs from other possible approaches in the way that it handles dates, accrual fractions, the construction of payment schedules, and the bootstrapping of the zero curve. For example:

• The "time", in years, used in the calculation of the expected accrued premium is calculated as the actual/365f accrual factor between the protection start date and the period end date, plus an extra 0.5/365, presumably to account for the transition from the business world of discrete dates to the mathematical universe of continuous time.
• When bootstrapping the zero curve from swap rates, it is assumed that the floating leg (including principal payment at maturity) prices to par, thereby ignoring any known fixing of the reference rate.
• The expected recovery rate is treated as constant for all possible times of default.

## Summary

This document has shown that the ISDA CDS Standard Model incorporates certain modeling assumptions that would lead to slight pricing differences from other approaches. These other approaches would be equally consistent with the price of the liquid instruments used to build the zero curve and survival curve. However, the standardization of the methodology allows market participants to have certainty around, for example, the conversion of a running spread to an upfront fee, or upon Mark-To-Market values.

The FINCAD implementation of the ISDA model offers a comprehensive and enhanced set of outputs, including:

• CDS value and leg values
• Accrued interest