Implementation Of Intensity Model Approach To Constant .

Implementation of Intensity Model Approach toConstant Maturity Credit Default Swap PricingOhoe KimDepartment of Mathematics, Towson UniversityTowson, MD 212521

Abstract: Constant maturity credit default swaps (CMCDS) are useful as hedging tools. Inintensity model approach, the default time is defined as the first arrival time of the Poissonprocess. From the market quotes of CDS forward rates and bonds, we are able to numericallycompute the default probabilities. Approximating CMCDS price depends largely on CDSforward rates’ volatilities and their correlations. We implement the price algorithm based onBrigo’s work (2006). Starting with current market data such as CDS forward rates and nondefaultable bond prices, we describe steps involved to obtain the price CMCDS. We demonstratethe impact of convexity on the CMCDS price structure.2

IntroductionA Constant Maturity Credit Default Swap (CMCDS) is a combination of a ConstantMaturity Swap and Credit Default Swap. The valuation of a CMCDS is implemented in ExcelVBA and derives important quantities relating to CMCDS valuation based on the work of D.Brigo[1]1. Credit Default Swap (CDS)A CDS contract insures protection against default. If a third company C (Referencecredit) defaults at the time τ c with Ta τ c Tb , then B (Protection Seller) pays to A (Protectionbuyer) a certain cash amount LGD. In turn, A pays to B a rate R at time Ta 1 ,LTb or untildefault τ c .2. Constant Maturity Credit Default Swap (CMCDS)Consider a contract protecting in [ Ta , Tb ] against default of a reference credit C.If default occurs in [ Ta , Tb ], a protection payment LGD is made from the protection seller B to theprotection buyer A at the first T j following the default time, called protection leg. A pays to B ateach T j before default a C 1 – long CMCDS rate R i-1,i c (Ti 1 )B protection LGD at default τ c if Ta τ c Tb A R i-1,i c (Ti 1 ) at Ti Ta 1 , K , Tb or until τ c The value of the CMCDS to “B” is the value of the premium leg minus the value of theprotection leg.3

3. Constant Maturity Swap (CMS)A swap contract is when, on specified payment dates, party 1 agrees to pay the floatingLIBOR rate of a notional amount to party 2 and in return party 2 agrees to pay a fixed swap rateto party 1 of the same notional amount. A constant maturity swap differs in that neither groupwill be paying a fixed rate. In a constant maturity swap scenario, party 1 agrees to pay thefloating LIBOR rate on a notional amount to party 2 and party 2 will agree to pay a floating cperiod swap-rate (CMS rate) on the same notional amount to party 1. The CMS rate is notconstant and is calculated at the reset dates stated on the contract. The period of time ‘c’ used tocalculate the CMS rate will be stated in the contract. In essence both parties will be paying afluctuating rate. A CMS can be used to hedge against short term changes in interest rates.4. CDS PayoffInstead of considering the exact default time τ , the protection payments LGD is postponed tothe first time Ti following default and the premium payment R is paid at Ti as long as thedefault occurs after Ti . Consequently the CDS payoff as seen from B can be expressed asΠ PRCDSa ,b (t ) : b i a 1D (t , Ti )α i R1{τ Ti } b 1i a 1{Ti 1 τ Ti }D (t , Ti ) LGDwhere α i is the year fraction between Ti and Ti 1 and D (t , Ti ) is a stochastic discount factor attime t for maturity Ti .5. CDS Value at time tThe CDS price with respect to the risk neutral valuation can be written as4

PRCDSa ,b (t , R, LGD ) E{Π PRCDSa ,b (t ) Gt } 1{τ t }Pr(τ t Ft )E[Π PRCDSa ,b (t ) Ft ]where the filtration Gt includes default information and default free market information Ft .That is, Gt Ft σ ({τ u}, u t )Intensity ModelsWe describe the default by means of Poisson’s first arrival process. We assume thedefault is independent of all the default free market information. We denote default time by τ .We consider time in-homogeneous Poisson process and let λ (t ) be intensity (hazard) ratefunction. Having not defaulted before t, the risk neutral probability of defaulting in the nextdt instants isPr(τ [t , t dt ) τ t , Market information up to t ) λ (t )dtWe define the cumulative intensity function bytΓ (t ) λ (u )du0It is called hazard function. Assume λ (t ) is a deterministic function. We defineτξ : Γ(τ ) λ (u )du .0It is known that ξ is a standard exponential random variable. That is, Pr(ξ x) 1 e x . Wecan show thattλ ( u ) duPr(τ t ) Pr(Γ(τ ) Γ(t )) Pr(ξ Γ(t )) e 0 If λ (u ) is a stochastic process, then5

tλ ( u ) duPr(τ t ) Pr(Γ(τ ) Γ(t )) Pr(ξ Γ(t )) E[e 0] Pr ( s τ t ) et Γ ( s ) e Γ ( t )λ ( u ) du e sThe defaultable bond price P (t , T ) is defined as:T 1{τ t } P (t , T ) E[ D (t , T )1{τ T } Gt ] E[e t( r ( u ) λ ( u )) du]Thus, the survival probability looks like the price of a zero coupon bond in an interest rate modelwith short rate r replaced by λ (t ) and λ (t ) is interpreted as instantaneous credit spread. Inparticular,P (0, T ) E[ D (0, T )1{τ T }T( r ( u ) λ ( u )) duGt ] E[e 0] The Filtration Switching FormulaIt can be shown that the following switching formula is valid:E[1{τ T } Payoff Gt ] 1{τ t }Q{τ t Ft }E[1{τ T } Payoff Ft ]where Q represents probability.The proof of this result is in the reference [1]. Switching from Gt to Ft is useful because mosttimes it is easy to compute the Ft conditional expectation.CDS Forward RatesThe CDS forward rates can be defined as that rate R that makes the CDS value equal tozero at time t. That is, CDSa ,b (t , R, LGD ) E{Π PRCDSa ,b (t ) Gt } 0 . where Π represents CDSpayoff at time t .6

A defaultable zero coupon bond (DZCB), P (t , T ) is defined as1{τ t } P (t , T ) : E[ D(t , T )1{τ T } Gt ]Using the filtration switching formula, the following expression can be obtainedE[ D(t , T )1{τ T } Ft ] Q(τ t Ft ) P (t , T )CDS a ,b (t , R, LGD ) E[Π PRCDSa ,b (t ) Gt ] 1{τ t }Pr(τ t Ft )E[Π PRCDSa ,b (t ) Ft ] b b αEDtTRFL(,)1 i {τ Ti }it GD E D (t , Ti )1{Ti 1 τ Ti } Ft Q(τ t Ft ) i a 1i a 1 b b αRQτtFPtTL(,)() itiGD E D (t , Ti )1{Ti 1 τ Ti } Ft Q (τ t Ft ) i a 1i a 1 1{τ t }1{τ t }Hence, we have this expression,bR (t ) PRa ,bLGD bi a 1bE[ D(t , Ti )1{Ti 1 τ Ti } Ft ] α E[ D(t , T )1 τi a 1ii{ Ti }Ft ] LGD E[ D(t , T )1ii a 1b αi a 1i{Ti 1 τ Ti } Ft ]Pr(τ t Ft ) P (t , Ti )CDS OptionsConsider the option for a protection buyer to enter a CDS at a future time Ta 0 , Ta Tb ,paying a fixed premium rate K at times Ta 1 ,., Tb or in exchange for a protection payment LGDuntil default happens in [Ta , Tb ] . This option expires at Ta . The discounted CDS option payoff attime t is:Π callPRCDSa ,b (t ; K ) D(t , Ta ) CDS a ,b (Ta , Ra ,b (Ta ), LGD ) CDS a ,b (Ta , K , LGD ) D(t , Ta ) CDS a ,b (Ta , K , LGD ) 7

which can be also written:Π callPRCDSa ,b (t ; K ) b D(t , Ta ) α i Q(τ Ta FTa ) P (Ta , Ti ) ( Ra ,b (Ta ) K ) Q(τ Ta FTa ) i a 1 1{τ Ta }Thus, we obtain the market formula for CDS option:CallCDSa ,b (t , K , LGD ) E Π CallCDSa ,b (t , K ; LGD ) Gt { E 1{τ Ta } D(t , Ta )Ca ,b (Ta )( Ra ,b (Ta ) K ) Gt} 1{τ t } Ca ,b (t ) Ra ,b (t ) N (d1 (t )) KN (d 2 (t )) where d1,2ln( Ra ,b (t ) (Ta t )σ a2,b() )K2 σ a ,b Ta tOne period CDS forward rate R j (t )Ra ,b (t ) can be expressed as a linear combination of one period CDS forward rates R j (t )as the swap rate, S a ,b (t ) is expressed as a linear combination of forward ratesF j (t ) F (t , T j 1 , T j ) . The one period CDS rates is defined asR j (t ) R j 1, j (t ) : LGD E D(t , T j ) 1{T j 1 τ T j } Ft α j Q(τ t Ft ) P (t , T j )Let p(t , Ti ) be a zero-coupon bond, and Fi (t ) F (t , Ti 1 , Ti ) be a forward rate maturing at Ti . Theswap rate is defined as:bS a ,b (t ) α P(t , T ) F (t )ii a 1iib α P(t , T )i a 1iiLikewise, we define the credit default swap rate as:8

bRa ,b (t ) α R (t ) P (t , T )ii a 1ib α P (t , T )i a 1 iiibb Wi (t ) Ri (t ) W (0) R (t )i a 1i a 1iiOne period CDS rate R j (t ) is approximated by R% j (t ) . R% j (t ) is defined as:LGD [Ε[ D (t , Tj 1)1{τ T j 1 }P (t , T j )P (t , T j 1 )Ft ] Ε[ D (t , Tj )1{τ T j } Ft ]α jQ (τ t Ft ) P (t , T j )P (t , T j 1 ) LGDP (t , T j )P (t , T j 1 ) P (t , T j )α j P (t , T j ) P (t , T j 1 )LGD 1 α j (1 α j F j (t )) P (t , T j ) P (t , T j 1 )L GD 1 α j (1 α j F j (0)) P (t , T j ) Let Cˆ j 1, j (t ) α j Q(τ t Ft ) P (t , T j ) . Then, R% j (t ) is a martingale under the probability measure))Q j 1, j associated with numeraire C j 1, j .In other words, we have this expression P (t , Tj 1) α j %R j 1 (1 α j Fj (0) ) 1 P (t , Tj ) LGD This implies P (t , T j ) is completely determined by R%i and P (t , T j 1 ) . The dynamics of R%i isneeded to compute CMCDS values.9

One period CDS rate approximation and dynamicsFirst, R%i is a Martingale under Qˆ i measure as long as R%i remains positive.dR%i σ i R%i dZ ii under Qˆ i measureSecond, we need to change a probability measure from Qˆ i to Qˆ j for all i j . The followingresult is obtained and the proof is in the reference [1].dR%i σ i R%i dZ ii under Qˆ i measure j %σ h Rh σ i R%i ρi ,hdt dZ i j LGD h i 1 R% h αh R%i [ μi j ( R% )dt σ i Z i j ] under Qˆ j measurewhere ρi ,h is a correlation of R%i and R% hHence, Monte Carlo simulation is possible given R% (0) , the volatilities and correlations.Furthermore, the expected value of R%i (T j 1 ) under Qˆ j measure is computed.We assume the volatility σ i is piecewise constant. iσ h R% h (0)Let μ% i j ρi , hL h j 1R% h (0) GD αh Eˆ j 1, j σ i . Then we have i%T j 1σ h Rh (0) R%i (T j 1 ) R%i (0) exp μ% i j ( R% (0))du R%i (0)exp T j 1σ i ρih0LGD h j 1R% h (0) α h {} 10

Calibration TechniqueWe use the intensity models to obtain implied default probabilities from market quotes.We assume the intensity rate function γ (t ) to be deterministic and piecewise function.tΓ(t ) γ ( s )ds 0β ( t ) 1 (Ti 1i 1 Ti )γ i (t Tβ ( t ) 1 )γ β (t )jΓ j : γ ( s )ds (Ti Ti 1 )γ iTj0i 1We have the following expression for the Protection leg at time 0:bLGD E[ D(0, Ti )1{Ti 1 τ Ti } F0 ]i a 1 LGD 0 LGD LGD LGDb E[ D(0, T )1ii a 1bTii a 1Ti 1 bTii a 1Ti 1 b]Pr(τ [u, u du ))E[ D(0, Ti )]Pr(τ [u, u du ))P (0, Ti )γ (u ) exp( γ ( s )ds )duu γ i a 1{Ti 1 τ Ti }0Tiexp( Γi 1 γ i (u Ti 1 ))P(0, Ti )dui Ti 1We also have the expression for the Premium leg at time 0b E[ D(0, T )α R1 τii a 1 { Ti }]b E[ D(0, T )]α RE[1 τii a 1 ii{ Ti }]b P(0, T )α R Pr(τ T )i a 1iiib R P(0, Ti )α i Pr(τ Ti )i a 1It can be shown that11

CDSa ,b (t , R, LGD ; Γ( )) b P(t , Ti ) Rα i eΓ (t ) Γ (Ti ) LGDi a 1bTii a 1Ti 1 P(t , Ti )du (e Γ (u ) Γ (t ) )]In particular, under the piecewise assumption on γ (t ) γ i , γ i [Ti 1 , Ti ] , we have,CDS a ,b (t , R, LGD ; Γ( )) b P(t , Ti ) Rα i e ( Γ (Ti ) Γ (t )) LGDi a 1b γ Tii Ti 1i a 1exp( Γi 1 γ i (u Ti 1 ))P(0, Ti )duHere we use the discrete form of the CDS value at time 0 to implement in computation scheme toestimate the piecewise constant intensity.CDS a ,b (0, R, LGD , Γ)b R P (0, Ti )α i exp( Γ(Ti ))i a 1 LGDb P(0, T )γ (T ) exp( Γi a 1iibi 1 γ i (Ti Ti 1 ))(Ti Ti 1 ) R P (0, Ti )α i exp( Γ(Ti )) LGDi a 1b γ P(0, T ) exp( Γ )αi a 1iiiiwhere α i Ti Ti 1 and γ (t ) γ i , γ i [Ti 1 , Ti ]In the market Ta 0 and we have R quotes for Tb 1, 2,3,.,10 years, by setting Ti quarterly, wecan solve theMCDS0,1 (0, R0,1, LGD ; γ 1 γ 2 γ 3 γ 4 ; γ 1 ) 0MCDS0,2 (0, R0,2, LGD ; γ 1 ; γ 5 γ 6 γ 7 γ 8 ; γ 2 ) 0by Matlab Software. See the code in the Appendix.Numerical ExampleAt this point, we present some numerical examples, based on IBM Company CDS dataon 28th, October, 2008.12

Recovery Rate 40%Maturity Tb(yr)0.512345710Maturity 5277.1677.47279.439Table 1 Maturity dates and corresponding CDS quotes in bps for T0 28th, October, al 85%91.268%87.430%Table 2 Calibration with piecewise linear intensity on 28th, October, 2008.13