lecture slides cds

8/13/2019 Lecture Slides Cds

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Defaultable bonds

Characterized by

Coupon c

Face value F

Recovery value R =random fractionof face value recovered on default

As before, will work directly with the risk neutral probability Q

Will model the term structure of default using a 1-step default probability

h(t) =Q(bond defaults in [t, t+ 1)| Ft).

Will calibrate h(t) to market prices.

Will modify the binomial lattice to include defaults.

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Binomial lattice for short rates

Binomial lattice for the short rate

nodes (i,j): date i= 0, . . . ,nand state j= 0, . . . , i

short rate rij

state transition probability

Q(i+1, s)| (i,j)= qu s=j+1qd s=j

0 otherwise

Split node (i,j) by introducing a variable that encodes whether or not defaulthas occurred before date i

(i,j, 0) = state jon date iwith default time >i

(i,j, 1) = state jon date iwith default time i

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Binomial lattice with defaults

(i+1,j,1)

(i,j,1)

(i+1,j+1,0)

(i,j,0)

(i+1,j,0)

(i+1,j+1,1)

Transitions from no-default state (i,j, 0)

Q

(i+ 1, s, )| (i,j, 0)

=

quhij s=j+ 1, = 1 (default)qu(1 hij) s=j+ 1, = 0 (no default)qdhij s=j, = 1 (default)qd(1 hij) s=j, = 0 (no default)0 otherwise

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5/28

Binomial lattice with defaults

(i+1,j,1)

(i,j,1)

(i+1,j+1,0)

(i,j,0)

(i+1,j,0)

(i+1,j+1,1)

Transitions from default state (i,j, 1): default state is an absorbing state

Q

(i+1, s, )| (i,j, 1)

=

qu s=j+ 1, = 1 (default)qd s=j, = 1 (default)0 otherwise

Conditional default probability hij is state dependent.5


6/28

Default-free zero-coupon bonds (ZCBs)

Default-free zero-coupon bonds with expiration T

pay $1 in every state at the expiration date T

no default possible

Pricing

ZTi,j, = price of a bond maturing on date T in node (i,j, )

Default events do not affect default-free bonds

ZTi,j,1=ZTi,j,0:=Z

Ti,j

Risk-neutral pricing:

ZTi,j= 1

1+rij

quZ

Ti+1,j+1+qdZ

Ti+1,j

Calibrate the short-rate lattice using the prices of the default-free ZCBs and other

default-free instruments.6


7/28

Defaultable ZCBs with no recovery

Defaultable zero-coupon bonds with expiration Tand no-recovery on default

pay $ 1 in every state at the expiration date Tprovided default has not

occurred at any date tT.If default occurs at any date tT, the bond pays 0, i.e. there isnorecovery.

PricingZ

T

i,j, = price of a bond maturing on date T in node (i,j, )No recovery implies that ZTi,j,1 0 in all default nodes (i,j, 1)

Risk-neutral pricing:

ZTi,j,0 = 1

1+rijqu(1 hij)Z

Ti+1,j+1,0+qd(1 hij)Z

Ti+1,j,0

+ 1

1+rij

quhijZ

Ti+1,j+1,1 0

+qdhijZTi+1,j,1 0

Calibratehijusing the prices of the defaultable ZCBs.

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8/28

Defaultable ZCBs with no-recovery (contd)

Risk-neutral prices

ZTi,j,0 = 1

hij

1+rijquZTi+1,j+1,0+qdZTi+1,j,0

e(rij+hij)EQi[ZTi+1,,]

where Q is the default-free risk-neutral probability.

Price of a defaultable ZCB is set by discounting the expected value by (rij+hij)

hij is the 1-period credit spread

conditional probability of default hijalso called the hazard rate.

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9/28

ZCBs with recovery

Assumption: Random recovery R is independent of the default and interestdynamics. Let R= E[R]

ZTi,j, = price of a bond maturing on date T in node (i,j, ) after recovery

ZTi,j,1 0 in all default nodes (i,j, 1)

Risk-neutral pricing

ZTi,j,0 = 1

1+rij

qu(1 hij)Z

Ti+1,j+1,0+ qd(1 hij)Z

Ti+1,j,0

+

1

1+rijquhijR+qdhijR

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10/28

Financial Engineering and Risk ManagementPricing defaultable bonds

Martin Haugh Garud Iyengar

Columbia UniversityIndustrial Engineering and Operations Research


11/28

State-independent hazard rates

Assumption: hazard rates hijare state-independent, i.e. hij=hi

Ensures that the default event is independent of interest rate dynamics

Let q(t) = risk-neutral probability that the bond survives until date t

Then q(t+1) = (1 ht)q(t) =t

k=0(1 hk)

Let I(t) denote the indicator variable that bond survives up to time t, i.e.

I(t) =

1 Bond is not in default at time t0 Otherwise

Then the indicator variable for default at time t is I(t 1) I(t).

From the definition ofI(t) is follows that

EQ0

[I(t)] =q(t)

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12/28

Pricing bonds with recovery

Assumption: Random recovery rate R is independent of interest rate dynamicsunder Q. Let R= EQ

0[R]

R is the fraction of the face value Fpaid on default

Pricing details:

t0= 0 is the current date

{t1, . . . , tn} are the future dates at which coupons are paid out.

The coupon is paid on date tkonly ifI(tk) = 1. Therefore the random cashflow associated with the coupon payment on date tk is cI(tk).

The face value Fis paid on date tnonly ifI(tn) = 1. Therefore, the random

cash flow associated with the face value payment on date tn is FI(tn).

The recovery R(tk)F is paid on date tk is the bond defaults on date tk.Therefore, the random cash flow associated with recovery on date tk isR(tk)F

I(tk1) I(tk)

.

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Pricing formula

Let B(t) denote the value of the cash account at time t.

Then the price P(0) of defaultable fixed coupon bond is given by

P(0) = EQ0

nk=1

cI(tk)

B(tk) +FI(tn)

B(t)

B(tn)+

nk=1

R(tk)F

B(tk) (I(tk1) I(tk))

=

nk=1

cEQ0 [I(tk)] E

Q0 1B(tk)+FEQ0 [I(tn)] EQ0 1B(tn)

+RF

nk=1

EQ0

[I(tk1)] EQ0

[I(tk)]

EQ0

1

B(tk)

=

nk=1

cq(tk)Ztk0 +Fq(tn)Ztn0 +RF

nk=1

q(tk1) q(tk)Ztk0=

nk=1

cq(tk)d(0, tk) +Fq(tn)d(0, tn) +RF

nk=1

q(tk1) q(tk)

d(0, tk)

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Calibrating hazard rates

Assume interest rate r is deterministic and known.

Model price P(h) of defaultable bond is a function h= (h0, . . . , hn1).

Observe market prices for mbonds

Pmkti = market price for i-th bond with expected recovery Ri, i= 1, . . . ,m

Assumption: default of all bonds induced by the same credit event.

Model calibration

Model price of i-th bond: Pi(h)

Pricing error: f(h) =mi=1(Pmkti Pi(h))2Calibration problem: minh0f(h)

Numerical example in spreadsheet bonds_cds.xlsx

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15/28

Financial Engineering and Risk ManagementCredit Default Swaps




16/28

Credit Default Swaps

The seller of acredit default swap(CDS) agrees to compensate the buyer in theevent of a loan default or some other credit event on a reference entity, e.g. a

corporation, or sovereign, in return for periodic premium payments.

N= notional principal amount of credit protectionAccrued interest = interest accumulated from last coupon to defaultS= coupon orspread

CDS seller has to compensate the buyer (1 R)Non default.2


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Credit Default Swaps: Numerical Example

Consider a (hypothetical) 2-year CDS

Notional principal N = $1 million

Spread S= 160bpsQuarterly premium payments

Suppose a default event occurs in month 16 of the 24 month protection periodand the recovery rate R= 45%.

Payments of protection buyer

Fixedpremiums in months 3, 6, 9, 12, 15= SN4

= $4000

Accrued interest in month 18 = SN12

= $1333.33

Payment of protection seller

Defaultcontingentpayment in month 18= (1 R)N = $550, 000

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B d l f h CDS h fl


18/28

Basic model for the CDS cash flows

Let {tk=k :k= 1, . . . , tn} denote times of coupon payments. Typically= 1

4, i.e. quarterly payments, and the dates of the payments are Mar. 20,

Jun. 20, Sept. 20, Dec. 20.If reference entity is not in default at time tk, the buyer pays the premiumSN where S is called theCDS spreadorcoupon.

If the reference entity defaultsat time (tk1, tk], the contractterminates at time tk. At time tk

the buyer pays the accrued interest (tk )SN, and

the buyer receives (1R)N where R denotes the recovery rate of theunderlying.

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CDS d il


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CDS contract details

Standardized by the International Swaps and Derivative Association in 1999.

Changes were made in 2003, and then again in 2009, may happen yet again ifCDSs become exchange traded.

Many difficult issues

How does one decide a credit event has occurred?

How does one determine the recovery rate?

Many many details . . .

How is the spread set? For junk bonds vs investment grade bonds? Forsovereigns?

When is coupon paid? In advance or in arrears?

How is the spread quoted?

Will focus on the basic model to illustrate the details of pricing and sensitivity to

hazard rates.5

CDS d f d f lt i k


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CDS spreads measure of default risk

Show later that CDS spread S(1 R)hwhere h is the hazard rate, orconditional probability of default.

For fixed R, the CDS spreads are directly proportional to the hazard rate h.

Plot of the 5yr CDS spreads for Ford, GM and AIG in the first 9 months of 2008.

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D l t d li ti f CDS


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Development and applications of CDS

Development credited to Blythe Masters from J. P. Morgan in 1994

J. P. Morgan extended a $4.8 billion credit line to Exxon to cover the

possible punitive damage from Exxon Valdez spillBought protection from the European Bank for Reconstruction andDevelopment using a CDS.

CDS market has grown tremendously

By the end of 2007, the CDS market had notional value of $62 trillion.

The Depository Trust and Clearing Corporation (DTCC) estimates grossnotional amount in 2012 as $25 trillion.

Initially developed for hedgingHedge concentrations of credit risk privately - maintain good client relations.

Hedge credit exposures where no publicly traded debt exists.

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Although a CDS can be used to protect against losses it is very different from aninsurance policy.

A CDS is a contract that can be written to cover anything.

Can buy protection even when one does not hold the debt.

CDS are easy to create and (until recently) completely unregulated.

Investing (speculation) quickly became the main application

Unfunded way to take a credit risk leverage possible.

Tailor credit exposure to match precise requirements.

Taking views on the credit quality of an reference credit.

Buying protection is easier than shorting the asset.

Arbitrage between the reference bond coupon and CDS spreads.

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CDS the financial crisis and the debt crisis


24/28

CDS, the financial crisis and the debt crisis

CDS positions are not transparent.

Riskiness of financial intermediaries cannot be accurately evaluated.

Threatened trust in all counterparties since no one knew who faced losses.

CDS trades are conducted on an OTC market.

Impossible for any dealer to know the previous deals of a customer.

Result: AIG was able to leverage its high credit rating to sell approximately$500billion worth of CDS.

Allowed a small number of CDS dealers to take a huge amount of risk.

The interconnected obligations of the dealers let to worries aboutcontagion.

CDS can be adversely affect the cost of borrowing of a firm or a country.

Speculators purchase CDS without holding underlying debt naked CDS.

This drives the spread higher, i.e. the firm starts to appear riskier.

The cost of borrowing of the firm increases, and can lead to its collapse.

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25/28

Financial Engineering and Risk ManagementPricing Credit Default Swaps



CDS pricing


26/28

CDS pricing

Value of CDS to buyer = Risk-neutral value of protection

Risk neutral value of premiums

Assumption: Default event uniformly distributed over the premium interval

Risk neutral value of single premium payment on date tk

SN EQ0

I(tk)

B(tk)

= SNq(tk)Z

tk0

=SNq(tk)d(0, tk)

Risk neutral value of all premium payments

nk=1

SNq(tk)d(0, tk)

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CDS pricing (contd)


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CDS pricing (contd)

Risk-neutral value of accrued interest if default (tk1, tk]

2SN EQ0 I(tk1) I(tk)B(tk) = SN2 q(tk1) q(tk)Ztk0

= SN

2

q(tk1) q(tk)

d(0, tk)

Risk neutral value of premium and accrued interest can approximated by

SN

nk=1

q(tk)d(0, tk) +SN

2

nk=1

q(tk1) q(tk)

d(0, tk)

= SN2

nk=1

q(tk1) +q(tk)

d(0, tk)

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CDS pricing (contd)


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CDS pricing (contd)

Risk-neutral present value of the protection (contigent payment)

(1 R)N

nk=1

EQ0 I(tk1) I(tk)B(tk)

= (1 R)Nn

k=1

q(tk1) q(tk)

d(0, tk)

par spread Spar = spread that makes the value of the contract equal to zero.

Spar = (1 R)

nk=1

q(tk1) q(tk)

d(0, tk)

2

n

k=1

q(tk1) +q(tk)

d(0, tk)

Suppose q(tk) (1 h)q(tk1). Then

Spar (1 R)h

(1 h/2)

Increasing in the hazard rate hand decreasing in recovery rate R

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