Credit Default Swap Pricing based on ISDA Standard Upfront ...rmi-quant.nus.edu.sg/assets/documentations... · Credit Curve (Survival Probability Curve) If we assume default is a

Credit Default Swap Pricingbased on ISDA Standard Upfront Model

Summarized by Wu Chen

Risk Management Institute, National University of Singapore

[email protected]

March 8, 2017

Summarized by Wu Chen (RMI) CDS Valuation March 8, 2017 1 / 47

Outline

1 OverviewChinese Credit Derivative MarketThe Standard CDS Contract

2 CDS Pricing based on ISDA ModelIntroductionNotations and TerminologyCDS PricingISDA ModelCalibration of the Credit Curve

3 Appendix


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3 Appendix


Chinese Credit Derivatives Market

• In late 2010, China unveiled its onshore credit derivatives market withthe launch of two Credit Risk Mitigation (CRM) products: the CRMAgreement (CRMA) and the CRM Warrant (CRMW).

• However, the market have remained languid since the launch more thanfive years ago.

• In the wake of the disappointing performance of first generation of creditderivative instruments, in mid 2016, the National Association ofFinancial Markets Institutional Investors (NAFMII) ratified a majorrevision of its credit derivatives guidelines with the introduction of twonew products: CDS and CLN, which return to an international CDSstandard feature.

• In 31st Oct 2016, Some Banks (Bank Of China, China ConstructionBank, etc.) start their first trade in CDS with total notional 300MMCNY.

• This Presentation focus on: Standard CDS Contract Pricing based onISDA Standard Upfront Model


CDS Contract

For i-th period,

• Premium Payment = N ×DCC(si, ei)× C (no default)= N ×DCC(si, τ)× C (default at time τ)

• Default Payment = N × (1−RR)

where:

si: accural start date

ei: accural end date

τ : default date

C: fixed coupon amount

N : Notional Amount

DCC(t1, t2): Day Count Convention between date t1 and t2RR: Recovery Rate


Differences before and after ‘Big Bang’

• ‘Big Bang’: in 2009 ISDA (International Swaps and DerivativesAssociation) issued the ‘Big Bang’ protocol in an attempt to restart themarket by standardising CDS contracts.

• Before ‘Big Bang’: CDSs were traded at par (have zero cost of entry).CDSs were quoted by par spread, which makes the clean PV of theCDS zero. For all protection buyers, the cost of entry is zero and theyonly needs to pay regular coupon payments.

• After ‘Big Bang’: All CDS contracts now have a standard coupon andup-front charge, quoted as a percentage of the notional - PointsUp-Front (PUF) or par spread. This amount is quoted as if it is paidby the protection buyer. In other words, protection buyer will pay orreceive an upfront charge when they enter into a CDS contract.


Differences before and after ‘Big Bang’

However, Upfront Amount can also be negative, in which case it is paidto the protection buyer.


The Standard CDS Contract

• Legal Effective Date. This was T + 1 (i.e. the same as the step-in date),which could cause risk to offsetting trades (as there may be a delay in creditevent becoming known). It is now T − 60 (credit events) or T − 90 (successionevents).

• Accrued Interest. For legacy trades only accrued interest from the step-indate (T + 1). If the trade date was more than 30 days before the first coupondate, the first coupon is reduced to just be the accrued interest over theshortened period (short stub); if there are less than 30 days, nothing is paid onthe (nominal) first coupon and on the second coupon (effectively the first) thefull accrued over the extended period (long stub) is paid - i.e. the normalpremium for this period plus the portion not paid on the first coupon date.Following this front stub period, normal coupons are paid. For standard CDS,interest is accrued from the previous IMM date (the prior coupon date), so allcoupons are paid in full. This accrued interest (from the prior coupon date tothe step-in date) is rebated to the protection buyer, on the cash settlementdate.

• coupons. Previously CDSs were issued with zero cost of entry, so the couponwas such that the fair value of the CDS was zero (par spread). Now CDSs areissued with standard coupons, and and upfront free is paid.

• ...Summarized by Wu Chen (RMI) CDS Valuation March 8, 2017 8 / 47

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3 Appendix


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3 Appendix


Introduction to ISDA Model

The ISDA Standard CDS Model is maintained by Markit, It is writtenin the C language and is the evolution of the JP Morgan CDS pricingroutines. The source code is written by JP Morgan Engineer.



Their documentations are mainly focused on standard contract specification and wecan only get clues of mathematical formula form its original source code. However,the source code have large amount of codes and is hard to read.

Luckily, OpenGamma has summarized these source code based on their

understanding. Thus, the following summary of ISDA model are mainly based on

OpenGamma’s quant research paper and some textbooks.




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3 Appendix


Day Count Convention

We will denote the year-fraction, 4t, between two dates, t1 and t2 as:

4t = DCC(t1, t2)

ISDA Model:

• The DCC for premium payments: ACT/360

• The DCC for curves: ACT/365F


Interest Rate Curve

The Price of zero-coupon bond at some time t, for expiry at T ≥ t, is defined asP (t, T ).If the instantaneous spot rate r(t) is deterministic, the price of a zero-coupon bondis given by

P (t, T ) = e−∫ Tt r(s)ds (1)

If r(t) follows some stochastic process, the price of a zero-coupon bond is given by

P (t, T ) = EQ[e−∫ Tt r(s)ds|Ft] (2)

Also, we can define the instantaneous forward rates, f(t, s), s ≥ t andf(t, t) = r(t), such that

P (t, T ) = e−∫ Tt f(t,s)ds ⇐⇒ f(t, s) = −∂ln[P (t, s)]

∂s= − 1

P (t, s)

∂P (t, s)

∂s(3)

Finally, we define the continuously compounded yield on the zero coupon bondR(t, T ), as

P (t, T ) = e−R(t,T )(T−t) ⇐⇒ R(t, T ) = − 1

T − t ln[P (t, T )] (4)

When t = 0 we may drop the first argument and simply write P (T ), f(T ) and R(T )

for the discount factor, forward rate and zero rate to time T .


Credit Curve (Survival Probability Curve)

If we assume default is a Poisson process, with an deterministic intensity λ(t).If the default time is τ , then the probability of default over an infinitesimaltime period dt, given no default to time t is

P(t < τ ≤ t+ dt|τ > t) = λ(t)dt (5)

The probability of surviving to at least time, T > t, (assuming no default hasoccurred up to time t) is given by

Q(t, T ) = P(τ > T |τ > t) = E(Iτ>T |Ft) = e−∫ Ttλ(s)ds (6)

If we assume intensity follows a stochastic process, then the survivalprobability is given by

Q(t, T ) = EQ[e−∫ Ttλ(s)ds|Ft] (7)

Also, we can extend this analogy and define the forward hazard rate, h(t, T ) as

Q(t, T ) = e−∫ Tth(t,s)ds ⇐⇒ h(t, s) = −∂ln[Q(t, s)]

∂s= − 1

Q(t, s)

∂Q(t, s)

∂s(8)


Credit Curve (Survival Probability Curve)

and the zero hazard rate, Λ(t, T ) as

Q(t, T ) = e−Λ(t,T )(T−t) ⇐⇒ Λ(t, T ) = − 1

T − tln[Q(t, T )] (9)

The forward hazard rate represents the (infinitesimal) probability of a defaultbetween times T and T + dT , conditional on survival to time T , as seen fromtime t < T . The unconditional probability of default between times T andT + dt is given by

P(T < τ ≤ T + dt|τ > t) = Q(t, T )h(t, T ) = −∂Q(t, T )

∂T(10)

When t = 0 we may drop the first argument and simply write Q(T ), h(T ) andΛ(T ) for the survival probability, forward hazard rate and zero hazard rate totime T .

Note that when t = 0, equation is just the probability density function for

default time τ .


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3 Appendix


CDS Pricing

We assume that a continuous time interest rate and credit curve isextraneously given, and are defined from the trade date (t = 0) to atleast the expiry of the CDS.We consider the prices at cash-settlement date rather than the tradedate. Also, we take the trade date as t = 0 and the maturity as T . Thestep-in(effective protection date) is te and the valuation(cash-settledate) is tv.


The Protection Leg

The protection leg of a CDS only consists of a random payment ofN(1−RR(τ)) at default time τ , if default time τ is before the expiry of theCDS (time T ). Thus, the present value of this leg is given by:

PVProtection Leg = NEQ[e−∫ τtvr(s)ds(1−RR(τ))Iτ<T ]

= N(1−RR)EQ[e−∫ τtvr(s)dsIτ<T ]

= − N(1−RR)

P (tv)

∫ T

0

P (s)dQ(s)

dsds

= − N(1−RR)

P (tv)

∫ T

0

P (s)dQ(s) (11)

Where RR = EQ[RR(τ)] is the expected recovery rate; second equality is

given under assumptions that recovery rates are independent of interest,

hazard rates and the default time; third equality is given under assumptions

that interest rates and hazard rates are independet; EQ[·] is the expectation

taken under risk-neutral measure.


The Premium Leg

The premium leg consists of two parts: Coupon payments up to the expiry of theCDS, which cease if a default occurs; and a single payment of accrued premium inthe event of a default (this is not included in all CDS contracts but is for SNAC).The Chinamoney’s summary of Chinese CDS specifications indicates that thechinese version of CDS has accured interest.Assume there are M remaining payments with

• payment times t1, t2, ..., tM

• period end times e1, e2, ..., eM

• year fraction 41,42, ...,4MThe present value of the premiums only is

PVpremium only =NCEQ[M∑i=1

4ie−∫ titvr(s)dsIei<τ

]

=NC

P (tv)

M∑i=1

4iP (ti)Q(ei) (12)

The quantity P (T )Q(T ) ≡ B(T ) is known as the risky discount factor - the PV of

the premium payments is just the risky discounted value of the cash flows.


The Premium Leg

The second part of the premium leg is the accrued interest paid on default. Ifthe accrual start and end times are (s1, e1), ..., (si, ei), ..., (sM , eM ), its PV isgiven by

PVaccured interest =NCEQ[M∑i=1

DCC(si, τ)e−∫ τtvr(s)dsIsi<τ<ei

]

=− NC

P (tv)

M∑i=1

[∫ ei

si

DCC(si, t)P (t)dQ(t)

dtdt

]

=− NC

P (tv)

M∑i=1

[ηi

∫ ei

si

(t− si)P (t)dQ(t)

dtdt

](13)

Where the third equality is due to different day count convention applied for

curves and accured interest. We usually use ACT/365 for measure year

fractions for the curves and ACT/360 for accured interest. In this case, we

have ηi = DCCaccured(si, ei)/DCCcurve(si, ei).


The Premium Leg

Thus, the full PV of the premium leg is given by:

PVpremium =PVpremium only + PVaccured interest

=NC

P (tv)

M∑i=1

[4iP (ti)Q(ei)− ηi

∫ ei

si

(t− si)P (t)dQ(t)

dtdt

](14)

This scales linearly with coupon C. The Risky PV01 (RPV01) isdefined as the value of the premium leg per unit of coupon, so:

PVpremium = C ×RPV 01dirty (15)


Clean PV v.s. Dirty PV

The valuation we disscussed above is the dirty PV, since it considersthe accured interest. For clarity we label the Risky PV01 given above,RPV 01dirty and RPV01 without a subscript as clean RPV01. Thus,the cash settlement for the buyer of protection is given by:

PVdirty = PVProtection Leg − C ×RPV 01dirty (16)

As with bonds, this value has a sawtooth pattern against time, drivenby the discrete premium payments. To smooth out this pattern, theaccrued interest between the accrued start date (immediately beforethe step-in date) and the step-in date (protection effective date T+1business day adjusted) is subtracted from the premium payments. So

RPV 01 = RPV 01dirty −N ×DCC(s1, te) (17)

Thus, we have

PVclean =PVProtection Leg − C ×RPV 01 (18)

PVdirty =PVProtection Leg − C ×RPV 01−NC ×DCC(s1, te) (19)


Points Up-front (PUF) and Par Spread

• Points Up-front (PUF)

PUF =PVcleanN

(20)

• Par Spread: the coupon that makes the clean PV of the CDSzero is known as the par spread, Sp, which is given by

Sp(T ) =PVProtection Leg

RPV 01

=−(1−RR)

∫ T0P (s) dQ(s)

dsds∑M

i=1


∫ eisi

(t− si)P (t) dQ(t)dt

dt]− P (tv)×DCC(si, te)

(21)


Points Up-front (PUF) and Par Spread


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3 Appendix


ISDA Model

The ISDA Model makes the assumption that both yield curve andcredit curve are piecewise constant in their respective(instantaneous) forward rates, which will simplify the calculationfor the integrals in protection leg and premium leg.

• Yield Curve: Constructed from money market rates (spot (L)iborrates) with maturities out to 1Y, and swap rates with maturitiesout to 30Y [Single Curve Discounting for OTC market, OISDiscounting for trades on exchange]

• Credit Curve: Calibrated from market price


ISDA Model Curve

We assume that the yield curve has ny nodes at time T y = {ty1, ty2, ...., t

yny}

and the credit curve has nc nodes at T c = {tc1, tc2, ...., tcnc}. At the ith node ofthe yield curve the discount factor is given by Pi = exp(−tyiRi) and at ith

node of the credit curve the survival probability is given by Qi = exp(−tciΛi).Then we have:

Q(t) = exp(−[tciΛi(t

ci+1 − t) + tci+1Λi+1(t− tci )

4tci

]) for ti < t < ti+1 (22)

where 4tci = tci+1 − tci . The interpolation is linear in the log of the survivalprobability. Thus we have the forward hazard rate:

Q(t) =−tciΛi + tci+1Λi+1

4tci≡ hi+1 for ti < t < ti+1 (23)

which justifies our previous statement that the forward rates are piecewiseconstant. This allows us to write

Q(t) = Qiehi+1(t−tci ) (24)

Λ(t) =Λit

ci + hi+1(t− tci )

tfor ti < t < ti+1 (25)


ISDA Model Curve

For times before the first node we have simply

Q(t) = e−h1t for 0 < t < t1 (26)

and after the last node

Q(t) = Qnce−hnc (t−tcnc ) for t > tcnc (27)

This property also holds for yield curve constructed from the money marketrates and swap rates. In summary, the both forward interest rate curve andforward hazard rare curve are piecewise constant curves.

We let the chronologically ordered combined set of unique nodes be

T = T y ∪ T c = {t1, t2, ..., tn}

where n ≤ ny + nc. Obviously, we are able to guarantee that both the forward

interest rate and forward hazard rate are constant.


ISDA Model Curve

Summary: Piecewise constant and flat extrapolation for (instantaneous)forward rate / forward hazard rate.


ISDA Model Curve

We use following notation for later derivation of formulas.

• Qi ≡ Q(ti), P i ≡ P (ti)

• 4ti = ti+1 − ti• fi: the constant forward interest rate when ti−1 < t < ti

• hi: the constant forward hazard rate when ti−1 < t < ti

• fi ≡ fi4ti−1 = ln(P i−1)− ln(P i)

• hi ≡ hi4ti−1 = ln(Qi−1)− ln(Qi)

• risky discount factor Bi = P iQi


The Protection Leg

From previous part, for protection leg we have:

PVProtection Leg = −N(1−RR)

P (tv)

∫ T

0

P (s)dQ(s)

=N(1−RR)

P (tv)

n∑i=1

Ii where Ii = −∫ ti

ti−1

P (s)dQ(s)

(28)

The individual integral elements are given by:

Ii = −∫ ti

ti−1

P (t)dQ(t) = P i−1Qi−1hi

∫ ti

ti−1

e−(fi+hi)(t−ti−1)dt

=hi

fi + hi(Bi−1 −Bi) =

hi

fi + hi(Bi−1 −Bi) (29)


The Premium Leg

We have same simplification for the premium leg:

PVpremium =NC

P (tv)

M∑i=1


∫ ei

si

(t− si)P (t)dQ(t)

dtdt

]

=NC

P (tv)

M∑i=1

[4iP (ti)Q(ei) + ηiIi] where Ii = −∫ ei

si

(t− si)P (t)dQ(t)

dtdt

(30)

Since the time interval (si, ei) could possibly contains at least one or morenodes (in yield curve and credit curve). If we now truncate our set ofcombined nodes, so it only contains the nk nodes between sk and ekexclusively, then add sk and ek as the first and last node (i.e. t0 = sk andtnk = ek) we have

Ik =−nk∑i=1

∫ ti

ti−1

(t− sk)P (t)dQ(t)

dtdt

=

nk∑i=1

[hi

fi + hi

(4ti−1

(Bi−1 −Bifi + hi

−Bi)

+ (ti−1 − sk)(Bi−1 −Bi))](31)


Formula given in ISDA Standard CDS Model C Code

The lastest ISDA Source Code’s (version 1.8.2) implementation for Ik gives theformula as:

IkISDA =

nk∑i=1

[hi

fi + hi

(∆ti−1

(Bi−1 − Bifi + hi

− Bi)

+ (ti−1 − sk)(Bi−1 − Bi

))](32)

where sk = sk − 1/730, the accrual start is reduced by half a day (on a 365-basedday count), the reason for this is unclear. This is simply:

IkISDA = IkOpenGamma +1

730

nk∑i=1

hi

fi + hi

(Bi−1 − Bi

)(33)

In a note in December 2012, Markit states that the formula (32) is incorrect. Thecorrected equation is (in our notation)

IkMarkit flx =

nk∑i=1

[∆ti−1hi

fi + hi

(Bi−1 − Bifi + hi

− Bi)]

(34)

The above equation can be rewrite as:

IkMarkit flx = IkOpenGamma −nk∑i=1

[hi

fi + hi

((ti−1 − sk)

(Bi−1 − Bi

))](35)





Both Markit and Bloomberg still use the code with equation (32) and are yetto switch to equation (34). (Source: OpenGamma Research Paper in Oct 3rd,2014)

Nevertheless some groups have started using the Markit fix. (Source:

OpenGamma Research Paper in Oct 3rd, 2014)


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3 Appendix


Calibration of the Credit Curve

For a particular reference entity, we consider following two cases:

• There is only one CDS tenor available with reliable pricinginformation (i.e is liquid)

• CDS quotes are available for several tenors on the same name, andone is interested in building a credit curve that will exactly repriceall of the quoted instruments.

For market observed clean price, PVclean, define the price residualfunction:

G(λ) = (PVProtection Leg(λ)− C ×RPV 01(λ)− PVclean)/N (36)


The Single CDS Case

Figure 1: The normalised price (i.e. unit notional) for a standard 5Y CDS against hazardrate for a range of recovery rates. The CDS premium is 500bps.

From the above figure, G(λ) is monotonic in λ and is guaranteed to have a rootG(λ∗) = 0 provided that

−CNM∑i=1

4iP (ti) ≤ P (tv)PVclean < (1−RR)N (37)

The above inequality is also the no arbitrage condition for PVclean.


Multiple CDSs

If CDS quotes are available for several tenors on the same name, theproblem is then the curve construction problem. Based on theinterpolation scheme and a set of nodes, we work with the zero hazardrate curve Λ(t). Let Λ = (Λ1, ...,Λn)T be the vector of curve nodes andV = (V1, ..., Vn)T be the vector of market prices, then we must solvethe vector equation:

G(Λ) = V (38)

where G(·) is the function that prices all the CDSs given the creditcurve that results from the nodes Λ. We need in effect to invert thisequation, and have

Λ = G−1(V) (39)


Multiple CDSs

This can be handled by a multi-dimensional root finding technique.Since the interpolator used in the ISDA model is linear in the log ofthe survival probability, the survival probability (or equivalently thezero hazard rate) at a particular time only depends on the value of thetwo adjacent nodes. Thus the vector equation may be broken down as:

G1(Λ1) = V1

G2(Λ1,Λ2) = V2...

Gn(Λ1,Λ2, ...,Λn) = Vn (40)

We can iteratively solve the equation from the first one to the last one.


Bloomberg Multiple CDSs’ Credit Curve Input


Thank you!


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3 Appendix


Appendix


Documents

Credit Default Swap Pricing based on ISDA Standard Upfront ...rmi-quant.nus.edu.sg/assets/documentations... · Credit Curve (Survival Probability Curve) If we assume default is a