Credit Default Swap Valuation

Risk Management Institute∗

National University of Singapore

First version: September 1, 2018, this version: September 2, 2018

1 Introduction

This documentation describes the valuation of credit default swap (CDS) in Risk ManagementInstitute.

The credit default swap, or CDS, is a financial swap agreement that the buyer makes a seriesof payments to the seller in exchange of the potential compensation from the seller upon thecredit default event. The counterparties of the a CDS contract are termed as the protectionbuyer or seller. The protection buyer enters the contract to receive the potential compensatedamount incurred when the predefined default event happens. As the obligation, they are re-quired to pay the premium, measured by premium rate times the length of each payment period,to the sellers regularly, either before the default event takes place or when the contract expires(if there is no default occurrence). This mechanism provides protection for the CDS buyers asit compensates the loss of the protection buyers once there are default events.

Prior to the disclosure of the “Big Bang” protocol in the North America in 2009, the CDSwas traded at the par rate that ensured the zero entry cost of the contract. In other words,

Figure 1: The payments between the CDS buyer and seller

under this agreement, the CDS premium c might vary from contract to contract to produce thezero contract value. Also, the premium and maturity payment dates from different contractswere of great variety. This prevented the cash flows of different contracts from being smoothlyoffset. In addition, there was ambiguity to determine the credit events and thus gave rise tomultiple lawsuits. This outmoded mechanism hindered the process of liquidation and lackedtransparency, and brought about significant systematic risk to the whole economy.

In 2009, the “Big Bang” protocol was introduced by ISDA (International Swaps and Deriva-tives Association) to resolve this problem. This new protocol standardizes the CDS contractsby fixing the premiums of all the contracts to specific values and the standard payment datesto certain calendar days. Also, the swap execution facilities and center clearing houses wereestablished to reduce the overall risk in the markets. Netting the exposures under ISDA, insteadof trading at zero net present value at the inception of CDS as an unstandardized CDS does,the protection buyer may pay or receive from their counterparty the upfront payment minusthe accrued interest. The upfront payment is the “clean value” resembling that in a bond.The (undiscounted) accrued interest in the first premium payment measures the value in thepremium leg across the unprotected period before the effective date. It is vital to determinethe upfront fee and accrued interest, rather than the par spread, in the CDS pricing. We willdiscuss it further in the later sections.

∗This script, based on the presentation slides of WU Chen in the RMI Project Team meeting and the frameworkof Richard White (2013) from OpenGamma Quantitative Research, is authored by ZHU Xianhao.

1

In the late 2010, China unveiled its onshore credit derivatives market with the launch oftwo Credit Risk Mitigation (CRM) products: the CRM Agreement (CRMA) and the CRMWarrant (CRMW). However, the market had remained languid since then. In mid-2016, theNational Association of Financial Markets Institutional Investors ratified a major revision ofits credit derivatives guidelines with the introduction of two new products, CDS and CLN, byincorporating international CDS standard features. In 31st Oct 2016, some banks (Bank OfChina, China Construction Bank, etc.) started their first trade in CDS with total notional300M CNY.

In the following section, some features in the standard CDS will be introduced, whereafterthe methodologies of the CDS pricing, including the definitions of two curves, the expression ofthe model and the implementation details, will be presented. The calibration of the curves willbe expounded in last section.

2 The Standard Credit Default Swap

The standard CDS has multiple differences from the legacy contracts. First, the legal effectivedate changes from T+1 to T-60/T-3 (credit event in North America and European/China) or T-90 (succession event in North America and European). Second, the pattern of accrued interestand coupon payments are different. This brings about the upfront payment in a standard CDScontract.

In this section, some contractual issues, such as dates and their conventions, and the coupon,accrued interest and upfront payments, will be discussed. The definitions are the same in theNorth America, European and China, unless otherwise stated.

2.1 Dates and conventions

Following the definitions in [2], the list below specifies the dates, as illustrated in Figure 2 and3, pertaining to the valuation of a CDS contract.

• The trade date, denoted as t, is the date the trade takes place. t ± i may represents thei-th date before (-) or after (+) the trade date in the following definitions.

• The protection effective date or step-in date, expressed as te, is the date the protectiontakes effect, usually being t+1. It has the different meaning from the legal effective date.1

• The cash-settlement date, denoted as tc, is the date the upfront fee is paid, usually t + 3in North American and Europe, and t+ 1 in China.

• The valuation date is the date to which the cash flows are discounted. The market valueis reported if the valuation date is the trade date, and the cash settlement is reported ifthe valuation date is the cash-settle date.

• The first accrual begin date or prior coupon date, represented as s1, as its name implies,is the date from which the coupon calculation starts counting. If the trade date is not aIMM date2, it is the previous IMM date before the trade date.

• The accrual start and end dates are the dates from and to which each coupon is calculated.Except the last accrual end dates, they are adjusted IMM dates by the following businessdate convention.

1See also [1].2In the context of CDS, the IMM dates are 20th March, June, September and December.

2

• The payment date is the following business day adjusted IMM date3 when each premiumis paid.

• The maturity date, shown as T or eN , is an unadjusted IMM date that is also the finalaccrual end date. The contract expires and the protection ends on that date.

s1First accrual

begin date(Adjusted IMM date)

t

Trade

date

t + 1 day

te = tcStep-in/

settlementdate

First accrual end

or payment date

(Adjusted IMM date)

T = eNMaturity

date

The first coupon accrual period

Accured interest calculation period

Figure 2: Some critical dates of a standard CDS contract in China

s1First accrual

begin date(Adjusted IMM date)

t

Trade

date

t + 1 day

teStep-in

date

t + 3 days

tcSettlement

date

First accrual end

or payment date

(Adjusted IMM date)

T = eNMaturity

date

The first coupon accrual period

Accured interest calculation period

Figure 3: Some critical dates of a standard CDS contract in North America and Europe

We now clarify the day count conventions associated with the pricing. The time betweentwo days are defined as

∆t =days

365 (or 360),

be it ACT/365F or ACT/360.The day count conventions used for discounting are curve specific. In Chinese markets,

the convention of 7-day repo cash rate is ACT/365F and adjusted with the following businessday convention; SHIBOR cash rate follows ACT/360 and adjusted with the modified followingconvention. The cash rates accrue from the quote date. Accordingly, the 7-day repo IRScontract floating leg resets and compounds weekly while SHIBOR3M IRS floating leg quarterly.Both IRS contracts exchange cash flows quarterly. As for the swap rates, both 7-day repo andSHIBOR3M apply ACT/365F with modified following business day convention. The swap ratewill take effect on the next business day.

When it comes to the CDS premium payment, the annualized periods adopted to evaluatethe coupons are usually ACT/360, while the discount curve or credit curve may be exhibited inthe same convention, namely ACT/360, or a different convention, say ACT/365F. For example,the notation ∆(t, s) may stand for the length of (t, s] measured by ACT/360, but t and sthemselves can be ACT/365F. Due to the complexity of the day count issues, we define aconversion ratio θ(·, ·) to handle this potential inconsistency:

θ(t, s) =∆(t, s)

s− t3In [2], it is described as “the IMM dates adjusted to the next business dates”.

3

If ∆(t, s) is measured by the same convention as the denominator, the ratio is always one; if∆(t, s) is ACT/360 (ACT/365F) but t and s is ACT/365F (ACT/360), the ratio is constantly365/360 (360/365); if the numerator or the denominator is ACT/ACT, the ratio is not fixed.The application of the ratio above is in Section 4 and 5. In the later sections, we assume theseconventions are properly taken into account in the algorithm.

2.2 Coupon, accrued interest and upfront payments

After the standardization, the coupon or premium is fixed to some specific value. The CDSspread is set to 100 or 500 bps in the North America, 25, 100, 500 or 1000 bps in Europe, and100 or 250 bps in China.

Besides, the payments are made only on the IMM dates (adjusted if it is not the lastpayment) in the standardized contract, unlike the customized payment dates in a legacy deal.This facilitates the hedging and netting, as the variations in the patterns of payment datesacross the existing contacts have been eliminated.

In addition, as for the first coupon payment, in contrast with the legacy short/long stubarrangement, in a standard CDS contract the first premium is fully-paid. In other words, it isaccrued from the first accrual begin date to the first accrual end date. The first coupon paymentvaries little from the remaining coupons and the cash flows are thus smoothed.

This setting leads to an “extra” payment in the first coupon for the unprotected periodbefore the protection effective date. The accrued interest is therefore introduced. It equals thepremium from the first accrual start date to the step-in date.

All the changes elaborated above increase the fungibility of the standard CDS contracts,even of those issued between two IMM dates. For example, a one-year CDS contract on certainreference obligation, issued on March 20, 2018, has the same future cash flows as the contractson the same reference obligation issued up to June 19 2018. The main difference between themare the accrued interest on the unprotected period and the upfront fee. The upfront fee is theclean value of the CDS, being the dirty entry cost (net present value of the future cash flows)plus the accrued interest. It is usually paid on the cash-settle date.

In a conventional contract, the CDS spread is adjusted to the zero entry cost, and the CDSpricing means determination of the par spread. However, in a standard CDS contract, as thespread is predefined, apart from determining the par spread, the calculation of the upfrontpayment, along with the accrued interest, is essential in the standard CDS contract pricing.

3 Two Curves

In the reduced form model, the value of a credit default swap is determined by the discountcurve, pertinent to the risk free interest rate, and the credit curve, relevant to the hazard rate.

3.1 The discount curve

The discount curve in this section is defined as the term structure of the non-defaultable zerocoupon bond value. The zero bond value at time t with maturity T , also termed as the discountfactor, is defined as

D(t, T ) = EQ(e−

∫ Tt rsds|Ft

), (3.1)

where rs is the instantaneous spot rate.The discount factor of D(t, T ) has two alternative expressions, defined by forward rate

f(t, T ) and zero cash rate y(t, T ), respectively. The forward rate is a deterministic variablewith f(t, s) ∈ Ft and D(t, T ) satisfies

D(t, T ) = e−∫ Tt f(t,s)ds ⇔ f(t, T ) = −∂ lnD(t, T )

∂T= − 1

D(t, T )

∂D(t, T )

∂T. (3.2)

4

Remark. If the instantaneous spot rate {rt} is deterministic, one can conclude rT = f(t, T ) ∈Ft. For the random instantaneous spot rate {rt}, it becomes rT = f(T, T ) ∈ FT .

The term structure of y(t, T ) with respect to T is known as the yield curve. It is the zerointerest rate observed at and accruing from time t to time T :

D(t, T ) = e−y(t,T )(T−t) ⇔ y(t, T ) = − 1

T − tlnD(t, T ). (3.3)

Note that in Equation (3.2) and (3.3), the discount factors are characterized by continuouslycompounded rates, while the expressions in terms of the simple rates also make sense:

D(t, T ) =1

1 +R(t, T )(T − t)=

1M∏i=1

(1 + F (t, Ti−1, Ti)(Ti − Ti−1)

)where R(t, T ) is the simple zero cash rate and F (t, Ti−1, Ti) is the forward rate between Ti−1

and Ti with T0 = t and TM = T . R(t, T ) and F (t, Ti−1, Ti) satisfy:

R(t, T ) =1

T − t

(e∫ Tt rsds − 1

)and F (t, Ti−1, Ti) =

1

Ti − Ti−1

(e∫ TiTi−1

f(t,s)ds − 1

).

There are other definitions of the discount curve. It can also be defined in terms of f(t, Ti)or y(t, Ti). In practice, it can be inputted as a mixture of zero cash rates and swap rates (seeSection 4.5 and Section 5.4). All these expressions are equivalent and mutually convertible.

3.2 The credit curve

The credit curve is of great similarity to the discount curve, with the hazard rate (or defaultintensity) resembling the interest rate and the survival probability resembling the discountfactor.

We use τ to denote the time of default. The default event {τ ≥ t} happens after time t.The reduced form valuation of CDS assumes the default event follows a Poisson process

NT = 1{t ≤ τ ≤ T},

The default intensity is {λt}, be it stochastic or deterministic. The default time is defined tobe

τ = inf

{s :

∫ T

tλudu ≥ E1

}where E1 is a exponential random variable with parameter 1.

Remark. The compensated process

MT = NT −∫ T

tλu1{τ > u}du

is a martingale.

For ease of clarity, we introduce function bivariate function S(t, T ) such that

S(t, T ) :=

{P (τ > T |τ > t) for T ≥ t1 for T < t

5

is the survival probability at time t. Define filtration F λT = σ ({λT : t ≤ λ ≤ T}), the probabilityof default is

P (τ ≤ T |τ > t)=P(E1 ≤

∫ T

tλudu

∣∣∣∣τ > t

)=E

(P(E1 ≤

∫ T

tλudu

∣∣∣∣FλT)∣∣∣∣τ > t

)=1− E

(exp

{−∫ T

tλudu

}∣∣∣∣τ > t

)(3.4)

with the second term in (3.4) being the survival probability S(t, T ). Analogous to the for-ward interest rate, the forward hazard rate (forward default intensity) h(t, T ) is a deterministicvariable

P (T ≤ τ < T + dT |τ ≥ T ) ≈ h(t, T )dT, with dT → 0

from which we deduce the survival probability S(t, T ) satisfies

h(t, T ) = − 1

S(t, T )limdT→0

S(t, T + dT )− S(t, T )

dT= −∂ lnS(t, T )

∂T, (3.5)

or equivalently,

S(t, T ) = e−∫ Tt h(t,s)ds.

Similar to (3.3), the continuous zero hazard rate are as follows:

S(t, T ) = e−Λ(t,T )(T−t) ⇔ Λ(t, T ) = − 1

T − tlnS(t, T ) (3.6)

and the simple zero hazard rate has a definition analogous to the simple zero interest rate.Because h(t, ·), Λ(t, ·), and S(t, ·) are equivalent, this document refers to them interchange-

ably as the credit curves. Sometimes the term structure of the par spreads, defined in Section 4,is also referred to as the credit curve.

4 The CDS Valuation

The CDS valuation in this document is based on two assumptions: 1)the recovery rate isindependent of the interest rate {rt}; 2) the time of default τ is independent of {rt}. Theseassumptions enable us to split the expectation of the product into product of expectations.

As in [2], only the value of CDS at the cash-settle date is considered. The cash-settle day ina standard CDS contract, is the next date in China and the third business day after the tradeday in North America and Europe. Suppose the trade date is t and the valuation date (or thecash-settle date) is tc, the CDS value with respect to the CDS buyer is

V (t) = E(V (tc)

∣∣Ft). (4.1)

The financial interpretation to (4.1) is that the CDS value represents the expected cash-settleamount at time tc evaluated with the information available at time t. Otherwise speaking, allthe potential cash flows are discounted to time tc with the most recently updated curves on thetrade date.

6

4.1 The premium leg

The premium leg represents the value paid from the protection buyer to the seller. It can besegmented into multiple regular coupon (premium) payments until the default event or the CDSexpiry date, and a single payment of the accrued premium in the event of default.

Let ∆(t, s) denotes the annualized time span between time t and s (t ≤ s) under theconvention of coupon calculation. The convention of the coupon payment can be kept abreastof the discount/credit curve by defining the ratio θ(t, s) := ∆(t, s)/(s − t). As an example,suppose the coupon payment is annualized with ACT/360 (most widely used), and the curveswith ACT/365, the value of θ(·, ·) is constantly 365/360.

Denote the accrual start and end times as (s1, e1], (s2, e2], . . . , (sN , eN ], and the step-in datebelongs to (s1, e1]. The dates of premium payment are t1, . . . , tN . The time si, ei, and ti, alongwith t and tc, are algebricaically asscociated with the convention of the discount curve. As forChinese 7-day repo discount curve, for examle, it is ACT/365. One may equate ti with ei (andsi+1), or make ei (and si+1) slightly previous to ti

4. With the notional P and the spread c, theaggregated CDS coupon payments are calculated as

V Dc (t; c) =PE

(c

N∑i=1

∆(ti−1, ti)e−

∫ titcrsds1{τ > ei}

∣∣∣∣∣Ft)

=Pc

D(t, tc)

N∑i=1

∆(ti−1, ti)D(t, ti)S(t, ti). (4.2)

where c∆(ti−1, ti) is the premium payment of the i-th period. For constant θ := θ(·, ·), 365/360for instance, the last result on the right hand side is

V Dc (t; c) =

Pc

D(t, tc)θ

N∑i=1

(ti − ti−1)D(t, ti)S(t, ti). (4.3)

The probability density of the default event 1{t < τ < T} can be deduced from the survival

probability S(t, T). Condition on Ft, it is −∂S(t,T )∂T . Therefore, the accrual upon default follows

Va(t; c) =PE

(c

N∑i=1

∆(si, τ)e−∫ τtcrsds

1{si < τ < ei}

∣∣∣∣∣Ft)

=− Pc

D(t, tc)

N∑i=1

∫ ei

si

∆(si, τ)D(t, τ)∂S(t, τ)

∂τdτ. (4.4)

and with constant θ, it will be

Va(t; c) =− Pc

D(t, tc)θ

N∑i=1

∫ ei

si

(τ − si)D(t, τ)∂S(t, τ)

∂τdτ. (4.5)

Note the integration over [s1, t] is zero as ∂S(t,τ)∂τ = 0 when τ < t.

The dirty value of the premium leg is thus the composition of (4.2) and (4.4) (or (4.3) and(4.5)):

V Dprem(t; c) = V D

c (t; c) + Va(t; c) (4.6)

4Various treatments of the date issue lead to different models as in Section 5.4, but the variations in thenumerical results are trivial.

7

4.2 The protection leg

The recovery rate RR(τ) is the ratio of the notional P that can be free of loss at default timeτ . To simplify the modeling, we assume RR = E (RR(τ)|Ft). In Chinese markets, the CFETSmodel assumes RR = 25% ,and in many other cases RR = 40% is widely applied.

The value of the protection leg, from the protection seller to the buyer, is

Vprot(t) =PE(

(1−RR)e−∫ τtcrsds

1{t < τ ≤ T}∣∣∣Ft)

=−P (1−RR)

D(t, tc)

∫ T

te

D(t, τ)∂S(t, τ)

∂τdτ. (4.7)

The expectation above is computed based on the aforementioned two assumptions: the inde-pendence of RR(t) versus {rt}, and {rt} versus τ .

4.3 The upfront payment

From the perspective of the protection buyer, the dirty present value of the potential receivedcash flows from the CDS with coupon rate c is the premium leg V D

prot(t) minus the protectionleg Vprem(t; c):

V D(t; c) =Vprot(t)− V Dprem(t; c)

=P

D(t, tc)

(cN∑i=1

(∫ ei

si

∆(si, τ)D(t, τ)∂S(t, τ)

∂τdτ −∆(ti−1, ti)D(t, ti)S(t, ti)

)

−(1−RR)

∫ T

te

D(t, τ)∂S(t, τ)

∂τdτ

). (4.8)

V D(t; c) is the de facto cash amount paid by the protection buyer to the seller. Anotherrelated notion is the clean amount. This concept considers the influence of the accrued interestand is reflected in the quoted price. Express the accrual as Va(t; c) = Pc∆(s1, te). The cleanpremium leg value at time tc, excluding the accrued interest between the first accrual begindate s1 and the step-in date te, is

V Cprem(t; c) = V D

prem(t; c)− Va(t; c).

The clean CDS value, or the upfront payment from the protection buyer to the seller, is thusthe dirty value plus the accrued interest:

V C(t; c) =Vprot(t)− V Cprem(t; c) = V D(t; c) + Va(t; c).

More specifically, the upfront fee is

V C(t; c) =P

D(t, tc)

(cN∑i=1

(∫ ei

si

∆(si, τ)D(t, τ)∂S(t, τ)

∂τdτ −∆(ti−1, ti)D(t, ti)S(t, ti)

)

−(1−RR)

∫ T

te

D(t, τ)∂S(t, τ)

∂τdτ

)+ Pc∆(s1, te). (4.9)

It is by convention that the market quote a CDS by Points Upfront : Up(t; c) := V C(t; c)/P ,and the clean price is defined as 100(1− Up).

8

4.4 The spread PV 01 and par coupon

The PV 01 measures the sensibility of the CDS value to its coupon spread in terms of 1bpschange in c. Mathematically, the dirty PV01s is defined as

PV 01D(t; c) =−∂VD(t; c)

∂c× 0.0001

=0.0001P

D(t, tc)

N∑i=1

(∆(ti−1, ti)D(t, ti)S(t, ti)−

∫ ei

si

∆(si, τ)D(t, τ)∂S(t, τ)

∂τdτ

), (4.10)

and clean PV01, or the Risky PV01, is computed as

PV 01C(t; c) =−∂VC(t; c)

∂c× 0.0001

=0.0001P

D(t, tc)

N∑i=1

(∆(ti−1, ti)D(t, ti)S(t, ti)

−∫ ei

si

∆(si, τ)D(t, τ)∂S(t, τ)

∂τdτ

)− 0.0001P∆(s1, te) (4.11)

=PV 01D(t; c)− 0.0001P∆(s1, te). (4.12)

Another notion is the CDS par spread corresponding to the legacy trade. It is the couponrate making the zero CDS clean value. Defining the par coupon (spread) as cP (t, T ) in lieu ofc in (4.9) and setting V C

(t; cP (t, T )

)= 0, the value of cp(t, T ) is solved as

cp(t, T ) =(1−RR)

∫ Tt D(t, τ)∂S(t,τ)

∂τ dτ

D(t, tc)∆(s1, te) +N∑i=1

(∫ eisi

∆(si, τ)D(t, τ)∂S(t,τ)∂τ dτ −∆(ti−1, ti)D(t, ti)S(t, ti)

) .(4.13)

From (4.9), (4.11) and (4.13), the mark-to-market clean CDS value can be alternatively ex-pressed as

V C(t; c) = 10000× PV 01C(t; c)(cP (t, T )− c

). (4.14)

Note that it is consistent with the result:

V C(t; c) =V Cprem

(t; cP (t, T )

)− V C

prem(t; c) (4.15)

= (cP (t, T )− c)P

(E( N∑i=1

∆(ti−1, ti)e−

∫ titcrsds1{τ > ei}|Ft

)

+E( N∑i=1

∆(si, τ)e−∫ τtcrsds

1{si < τ < ei})∣∣∣Ft)−∆(s1, te)

)

One observation is that as PV 01C(t; c) tends to be positive (ignoring the accrual term).This implies that when c > cp(t, T ) (or c < cP (t, T )), the positive upfront payment should beprovided by the protection seller (or buyer).

Remark. One may get the similar result of the dirty value V D(t; c):

V D(t; c) = 10000× PV 01D(t; c)(cP (t, T )− c

)− PcP (t, T )∆(s1, te),

orV D(t; c) = 10000× PV 01C(t; c)

(cP (t, T )− c

)− Pc∆(s1, te).

9

4.5 Other risk measures

There are other results of interest. The first is the Spread DV01. Express the credit curve interms of the par spread cp(t, Ti) with i = 1, ..n. If the curve is flat with value cp(t, Ti) ≡ cp,from (4.14), it is calculated as

DV 01S(t; c) = 0.0001× ∂V C(t; c)

∂cP= PV 01C(t; c) +

∂PV 01C(t; c)

∂cP(cP (t, T )− c

)The explicit expression of the above equation is not immediately available. As for the non-flat credit curve, it may be of greater difficulty. Instead, we apply difference method throughshifting the curve up by 1 bp and computing the value change.

By the same token, we calculate the Interest Rate DV01, denoted as DV 01IR(t; c), throughmoving the interest rate curve up by 1 bps. To put it another way, we evaluate it by adding 1 bpto all the zero cash rates and swap rates, and calculating the value variation in the CDS contract.One interesting observation is that DV 01IR(t; c) is usually mush smaller than DV 01S(t; c). Thisimplies the CDS valuation is more sensitive to the credit spread than to the risk-free rate. Thefluctuation associated with the credit healthiness may have greater effects than the interest ratemovement.

The Recovery Risk is mathematically characterized as −0.01 × ∂V C(t;c)∂RR . It measures the

sensitivity of CDS value to 1% change in the recovery rate.The notion Value on Default, also named as Default Exposure, is the clean value to the CDS

protection buyer if the default event occurs immediately. Upon default, the protected valuethe protection buyer can receive is P (1 − RR), and the CDS is cleared instantly. The buyer’sDefault Exposure is thus

DE(t; c) =P (1−RR)− V C(t; c) = P (1−RR)− V D(t; c)− Pc∆(s1, te).

Arguably, if the market witness a high credit spread curve, the occurrence of the default eventis expected. In this case, DE(t; c) will be small.

4.6 The approximations

As the byproduct of the above results, [2] offers multiple approximations associated with thecomputation of the CDS contract value, the par spread and the quick determination of hazardrate.

To approximate the first integral that represents the accrued interest in (4.9), one may applythe trapezoidal rule over [s1, eN ]:

N∑i=1

∫ ei

si

∆(si, τ)D(t, τ)∂S(t, τ)

∂τdτ

=

N∑i=1

∫ ei

si

∆(si, τ)D(t, τ)dS(t, τ)

≈−1

2

N∑i=1

∆(si, ei)D(t, ei)(S(t, si)− S(t, ei))

The last equation is made by removing the zero term ∆(si, si)D(t, si)(S(t, si)− S(t, ei)) in thequadrature. Similarly, the second integral in (4.9), given t ≈ te, is

(1−RR)

∫ T

te

D(t, τ)∂S(t, τ)

∂τdτ

≈−1−RR2

((1 +D(t, e1))(1− S(t, e1)) +

N∑i=2

(D(t, si) +D(t, ei))(S(t, si)− S(t, ei))

).

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The par coupon is approximately

cP (t, T )≈(1−RR)

((1 +D(t, t1)

)(1− S(t, t1)

)+

N∑i=2

(D(t,ti−1)+D(t,ti)

)(S(t,ti−1)−S(t,ti)

))N∑i=1

(∆(ti−1, ti)D(t, ti)

(S(t, ti−1) + S(t, ti)

))− 2D(t, tc)∆(s1, te)

.

with ti ≈ ei for i = 1, . . . , N .For the purpose of determining the hazard rate h(t, τ) (or Λ(t, τ)), we assume λTt := h(t, τ) =

Λ(t, τ) is a constant for all t ≤ τ ≤ T . A further assumption that the premium is paid on acontinuous basis eliminates the accrual interest. In this case, there is no difference between thedirty value and the clean value. The premium leg is

Vprem(t; c) =PE(c

∫ T

te

e∫ utcrsds

1{τ > u}du∣∣∣∣Ft)

=Pc

D(t, tc)

∫ T

te

D(t, u)S(t, u)du,

=Pc

D(t, tc)

∫ T

te

D(t, u)e−λTt (u−t)du, (4.16)

and the protection leg is identical to (4.7). Suppose the CDS par spread is available. Replacingc with cp(t, T ) and equating Vprem(t) and Vprot(t), the value of the constant (forward and zero)hazard rate is

λTt =cp(t, T )

1−RR. (4.17)

Given the par spread, with due recovery rate RR, the constant hazard rate is attainable; con-versely, if the hazard rate is calibrated, the par spread is accessible immediately.

Moreover, suppose the interest rate f(t, T ) and y(t, T ) are also flat with constant levelrTt := f(t, T ) = y(t, T ) for all T . From (4.15), (4.16) and (4.17), the Points Upfront is

Up(t) =cP (t, T )− cD(t, tc)

∫ T

te

e−(rTt +λTt )(u−t)du.

Applying Taylor expansion ex = 1 + x+O(x2) and ignoring the difference between time t andtc, the hazard rate λTt is approximately

λTt ≈c+ Up(t; c)/(T − t)

1−RR.

5 The Implementation

In practice, the construction of two curves, the discount and the credit curves, plays the keyrole in the CDS pricing. To evaluate the values and integrals associated with D(t, ·) and S(t, ·)in (4.9), one must specify the interpolation method between the available data observationnodes. After obtaining the numerical expression of (4.9), we can calibrate the hazard rate, orthe according credit curve.

5.1 The log-linear interpolation

Suppose some individual values along the discount curve and credit curve are available withtenors T f = {τ f1 , . . . , τ

fnf } and T h = {τh1 , . . . , τhnh}, respectively. Given the values, either from

observation or calibration, of D(t, τi) and S(t, τi) at the node τ ∈ T = T f ∪ T h = {τ1, . . . , τn}

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(τi−1 < τi for i > 1), one may apply log-linear interpolation to attain D(t, τ) and S(t, τ) withτi−1 ≤ τ ≤ τi. Namely, with ∆τi = τi − τi−1, Di = D(t, Ti) and Si = S(t, Ti), one may obtain

D(t, τ) = exp

{τi − τ∆τi

lnDi−1 +τ − τi−1

∆τilnDi

}= Di−1 exp

{τ − τi−1

∆τi(lnDi − lnDi−1)

},

S(t, τ) = exp

{τi − τ∆τi

lnSi−1 +τ − τi−1

∆τilnSi

}= Si−1 exp

{τ − τi−1

∆τi(lnSi − lnSi−1)

}.

By definition (3.2) and (3.5), we can obtain the piecewise constant forward interest rate andforward hazard rate with τi−1 ≤ τ ≤ τi:

fi ≡ f(t, τ) = − 1

∆τi(lnDi − lnDi−1) ,

hi ≡h(t, τ) = − 1

∆τi(lnSi − lnSi−1) .

The discount factor and survival probability is thus captured in a conciser manner:

D(t, τ) =Di−1e−fi(τ−τi−1), (5.1)

S(t, τ) =Si−1e−hi(τ−τi−1). (5.2)

for τi−1 ≤ τ ≤ τi.

Remark. The piecewise constant forward interest rate and forward hazard rate generally pro-duce nonlinear zero interest rate and zero hazard rate curves respectively, unless the term struc-ture of the curves are flat. The relations are

y(t, T ) =fi(T − Ti−1) + fi−1(Ti−1 − t)

T − tor

Λ(t, T ) =hi(T − Ti−1) + hi−1(Ti−1 − t)

T − t.

The figure below illustrates one pattern of the forward rate and the zero rate versus time-to-maturity T :

Figure 4: An example of the forward rate curve and its zero rates

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5.2 The numerical evaluation

We start the numerical implementation with the premium leg. For simplicity, as in Section 4.1,let θ(·, ·) is a constant quantity θ. Apply (5.1) and (5.2) to (4.3), the values of D(t, ti) andS(t, ti) can be interpolated with piecewise constant forward interest rates and forward hazardrates. As for (4.5), assume there are ni nodes of T = {τ1, . . . , τn} residing within (si, ei). suchthat si = τ i0 < τ i1 < . . . < τ ini < τ ini+1 = ei. Let the forward interest rate and defaut intensitybe denoted as f ij and hij at (τ ij−1, τ

ij ]. The curve values at the node τ ij are Di

j = D(t, τ ij) and

Sij = S(t, τ ij). Then (4.5) is described as below,

Va(t; c) =− Pc

D(t, tc)θ

N∑i=1

∫ ei

si

(τ − si)D(t, τ)∂S(t, τ)

∂τdτ

=Pc

D(t, tc)

N∑i=1

ni+1∑j=1

hijD(t, τ ij−1)S(t, τ ij−1)θ

∫ τ ij

τ ij−1

(τ − si)e−(f ij+hij)(τ−τ ij−1)dτ

=Pc

D(t, tc)

N∑i=1

ni+1∑j=1

hijθ

f ij + hij

(Dij−1S

ij−1 −Di

jSij

f ij + hij−Di

jSij(τ

ij − si) +Di

j−1Sij−1(τ ij−1 − si)

)(5.3)

The value of the premium leg in (4.6) is thus computed as:

V Dprem(t; c) =

Pc

D(t, tc)

N∑i=1

(∆(ti−1, ti)D(t, ti)S(t, ti) +

ni+1∑j=1

hijθ

f ij + hij

(Dij−1S

ij−1 −Di

jSij

f ij + hij

−DijS

ij(τ

ij − si) +Di

j−1Sij−1(τ ij−1 − si)

)).

Remark. If ni = 0, then si = τ i0 < τ i1 = ei. Above equations can be simplifies as below:

Va(t; c) =Pc

D(t, tc)

N∑i=1

hi1θ

f i1 + hi1

(Di

0Si0 −Di

1Si1

f i1 + hi1−Di

1Si1(τ i1 − τ i0)

),

and

V Dprem(t; c) =

Pc

D(t, tc)

N∑i=1

(∆(ti−1, ti)D(t, ti)S(t, ti)

+hi1θ

f i1 + hi1

(Di

0Si0 −Di

1Si1

f i1 + hi1−Di

1Si1(τ i1 − τ i0)

)).

Adopting the similar approach, we can obtain the numerical expression of the other leg.Suppose T = T ∩ [tc, T ] = {τ0, . . . τm} with τ0 = tc and τm = T . Formulate Di = D(t, τi) andSi = S(t, τi). The numerical formulation of the protection leg in (4.7) is determined as

V Dprot(t) =−P (1−RR)

D(t, tc)

m∑i=1

∫ τi

τi−1

D(t, τ)∂S(t, τ)

∂τdτ

=P (1−RR)

D(t, tc)

m∑i=1

hi

fi + hi

(Di−1Si−1 − DjSi

).

With the results above one can obtain the dirty and clean values of the CDS contract.However, in practice the numerical stability is subject to the influence of the division termhij/(f

ij + hij) (or hi/(fi + hi)) as f ij + hij (or fi + hi) may be close to zero. To avoid this, one

choice is to add a small quantity, say 10−50 as in the ISDA model, and another is to applyTaylor expansion (see [2]).

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5.3 The curve calibration

In the short run, within one year for example, one can obtain the discount curve with the zerocash rate. Analogous to (3.3), it is

D(t, T ) =1

1 +R(t, T )(T − t)

where R(t, T ) is the simple zero cash rate.The discount curve to the longer run can be implied from the interest rate swap (IRS)

through the bootstrapping process (see [3]). Since the discount curve in CDS valuation isrisk-free, the single curve discount technique is relevant.

Suppose the IRS spread (swap rate) with maturity Tj observed at time t in the market issj . Define ∆Ti = Ti − Ti−1, and the discount factor over [t, Tj ] can be attained iteratively with

D(t, Tj) =1− sj

∑j−1i=1 ∆TiD(t, Ti)

1 + sj∆Tj.

The equations above provides the individual points of the discount curve. The inter-pointvalues are attainable via the log-linear interpolation to these calibrated value.

The S(t, ·) with different tenors is also bootstrapped from the market observation. Supposen CDS contracts with the same reference obligation have different time-to-maturities. Theirclean values are VC = {V1, V2, . . . , Vn} with maturities T = {T1, . . . , Tn}. Given the discountcurve, the forward hazard rates with the same maturity nodes follow the equations below:

V C(t; c, T1, h1) = V1,

V C(t; c, T2, h1, h2) = V2,...

V C(t; c, T2, h1, . . . , fn) = Vn.

The root finding process starts from the first equation solving h1 for h(t, τ) with 0 ≤ τ ≤ T1.With the solved h1, h2 can be solved for h(t, τ) with T1 ≤ τ ≤ T2. In this way the forwardhazard rates h(t, ·) at the remaining nodes, along with the corresponding term structures ofS(t, ·) and Λ(t, ·), can be recursively calibrated.

5.4 The RMI CDS Calculator

Before introducing the CDS Calculator on the RMI online dashboard, we further clarify oneissue on the expressions of si, ei and ti. It is ambiguous in the modeling and programming thatwhen the payment is made and when the protection takes effect in each period. In Section 4.1,we state ti ≥ ei = si+1 for i < N . One may equate ti with ei (and si+1), or make ei (and si+1)slightly previous to ti.

In ISDA Standard Model, si (i ≤ N) and ei (i < N) are rolled back one day except thematurity date5. The interpretation of this treatment, in our opinion, is that the protection maybe effective from the beginning of the date at ti (or the end of ti minus one day). In otherwords, the payment is made at the end of the (adjusted) IMM date, but the renewed accrualstarts and the protection becomes effective from the beginning of that date. In the given Ccode of ISDA, however, besides this one-day rollback, 1/730 (half a day) is further subtractedfrom the times si, ei and τi. The reason is unknown. We call the result with such adjustmentto (5.3) as the ISDA Model.

5This exception gives rise to an extra day of protection.

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According to [2], there is one correction to the half-day adjustment reevaluating (5.3) asbelow:

Va(t; c) =Pc

D(t, tc)

N∑i=1

ni+1∑j=1

hijθ

f ij + hij

(Dij−1S

ij−1 −Di

jSij

f ij + hij−Di

jSij(τ

ij − τ ij−1)

).

This valuation may not be identical to (5.3). It seems misusing the result for ni = 0. We referto the valuation applying the equation above as the Markit Fix Model.

In the RMI Model, (5.3) is adopted with one-day rollback but without the half-day furtheradjustment. In spite of the trivial numerical differences in the valuation practice, all threeapproaches are available in the RMI CDS Calculator.

The C program of NUS RMI CDS calculator for Chinese markets is modified from the opensource code provided by ISDA, whose interfaces are illustrated in Figure 5. Plot 5a showcasesthe calculator for the upfront models, selected by UPFRONT MODELS in Panel Option 1,with a single CDS par spread in the Input Section 6. This means the term structure is at a flatlevel. Plot 5b on the right exhibits the fair value model calculator, selected by FAIR VALUEMODELS in Panel Option 1, with Input Section 6 taking in a series of par spreads with differenttenors. The Input Sections 2 to 6, as well as Button 7 and the Output Section 8 are detailed asfollows:

(a) The CDS Calculator for the upfront model (b) The CDS Calculator for the fair value model

Figure 5: The RMI CDS Calculator interfaces

• Input Section 2 : the Trade Date is a customer input variable being the current dateby default. It fixes the Effective Date and Cash Settle Date to the next business day.The First Accrual Begin Date is changeable while defaulted to the last IMM date. TheMaturity Date and Maturity Year specify the expiry date of the CDS. The notional withthe default value 10, 000, 000 is amendable. There are two items under the Credit Ratingoption: “AA+ or higher” and “others”. The former entails the 100 bps Coupon whilethe latter 20 bps Coupon. It is stipulated in Chinese standardized contracts. Lastly, theRecovery Rate (%) is set to 25 but any value between 0 to 100 is acceptable.

• Input Section 3 : under Discounting Curve, FR007 and SHIBOR3M are two candidatesfor the curve construction. The table in this section displays the Type (Type M beingthe deposit rate and Type S the swap rate), Tenor (spanning from 1 day to 10 year) andRate (in percentage). The default values is given under Rate, but they are subject to thechanges from the users.

• Input Section 4 : the “Market” option can be switched between Exchange and Interbank.The selection determines which Chinese business day calendar, either that of the Chinesestock exchange or that of the Chinese interbank market, is used for the valuation.

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• Input Section 5 : Model has three options: 1) ISDA Standard Upfront/Fair Value Model,2) Markit Fixed Upfront/Fair Value Model, and 3) RMI Upfront/Fair Value Model. Theyare in agreement with the ISDA Model Markit Fix Model and RMI Model mentionedearlier.

• Input Section 6 : the Credit Curve Rate (bps) or CDS Spread (bps) is specified. If in PanelOption 1 UPFRONT MODEL is selected, a single CDS Spread value (basis points) shouldbe manually inputted. In this case, the term structure of the CDS par spreads is flat. Onthe other hand, if FAIR VALUE MODEL is otherwise selected, a table with Date andCredit Curve Rate (bps) is given. Each CDS par spread is related to a specified tenor. Nodefault values are given in this section, as the OTC transaction data is not available forRMI to calibrate the credit curve. The users should specify the CDS par spread(s) basedon their understanding of the specific contracts.

• Button 7 : the COMPUTE button will produce the results in Output Section 8, providedthe all the previous settings are properly determined.

• Output Section 8 : the resulting Price, Principal, Accrued are the clean price, the cleanvalue and the accrued interest of the CDS contract, respectively, along with the SpreadDV01, Interest Rate DV01, Recovery Risk and Default Exposure, all of which are definedin Section 4.

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References

[1] ISDA and Markit Group Limited, “ISDA Standard CDS Converter Specification”. 2009.

[2] Richard White, “The Pricing and Risk Management of Credit Default Swaps, with a Focuson the ISDA Model”. OpenGamma Quantitative Research. 2013.

[3] W. Cui, M. Dai, S.G. Kou, Y.Q. Zhang, C.X. Zhang and X.H. Zhu,, “Interest Rate SwapValuation in the Chinese Market”. Innovations in Insurance, Risk and Asset Management,Springer, forthcoming.

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c© 2018 NUS Risk Management Institute (RMI). All Rights Reserved.

The content in this document is for information purposes only. This information is, to thebest of our knowledge, accurate and reliable as per the date indicated in this technical reportand NUS Risk Management Institute (RMI) makes no warranty of any kind, either express orimplied, regarding its completeness or accuracy. The descriptions contained may reflect ouropinions and are subject to change without notice.

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