A Simple Model of Credit Spreads with Incomplete Information

Transcription

1 A Simple Model of Credit Spreads with Incomplete Information Chuang Yi McMaster University April, 2007 Joint work with Alexander Tchernitser from Bank of Montreal (BMO). The opinions expressed here are those of the authors, and do not necessarily reflect the views of BMO. 1

2 Outline Introduction Merton s Model and Related Literature Randomized Merton with Drifted Brownian Motion Randomized Merton with Ornstein-Uhlenbeck Conclusions and Future Research 2

3 Introduction Yield Spread: CS(T ) = 1 T log B(0,T ) B(0,T ) CDS Credit Spread Term Structure of Credit Spread (TSCS) Varying Shapes: upward, downward, hump-shaped Non Zero Short Spread Non Zero Long Term Spread Real Motivation 3

4 Merton s Model Set Up Asset: V t dvt = rvtdt + σvtdwt, V0 > 0 Debt: K < V 0 with Time to Maturity: T Default Time: τ τ = { T VT < K + V T K. 4

5 Analysis Probability of Default: P D(T ) P D(T ) := P [V T < K] = Φ log V 0 K + (r 1 2 σ2 )T σ T Expected Recovery Rate: RR(T ) RR(T ) := E[ V T K V T < K] = V Φ 0 K ert ( log V 0 K +(r+ 1 2 σ2 )T σ T P D(T ) ) Expected Loss Given Default: LGD := 1 RR 5

6 Default Free Bond: B(0, T ) = Ke rt Defaultable Bond: B(0, T ) = Ke rt [(1 P D) + RR P D] Credit Spread: CS(T ) where CS(T ) = 1 log[1 P D LGD] T = 1 ( T log Φ(d ) + V 0 K ert Φ( d + ) ) d = log V 0 K + (r 1 2 σ2 )T σ T d + = d + σt 6

7 Credit Spreads in Merton s Model 7

8 Structural Models: Black-Cox (1976), Longstaff-Schwartz (1995) Leland (1994), Leland-Toft (1996) Dufresne-Goldstein (2001) Fouque-Sircar-Solna (2005) Hybrid Models: Zhou (2001), Chen-Kou (2004) Duffie-Lando (2001), Giesecke (2004) Coculescu-Geman-Jeanblanc (2006) Linetsky (2006), Carr-Linetsky (2006) Intensity-based Models: Jarrow-Turnbull (1995) Lando (1998), Duffie-Singleton (1999) 8

9 Randomized Merton Set Up Solvency Ratio: X t := log V t K t X t = X 0 + µt + σw t. Noisy Observation: y 0 = X 0 + w, where w N(0, σ 2 0 ) Information Available: X 0 > 0 and y 0 > 0 Independence Assumption: X 0 B t, conditional on y 0 Default Time: τ = T, if X T < 0. 9

10 Conditional on y 0 X 0 N(y 0, σ 2 0 ), denote φ x 0 as its pdf X t N(µ x, σ 2 x) where µ x (t) = y 0 + µt, σ 2 x (t) = σ2 0 + σ2 t. Z t := µt + σw t N(µ z, σz 2 ) where µ z (t) = µt, σz 2 (t) = σ2 t. denote φ z (t) as its pdf 10

11 Conditional Default Probability P D(T ) = P (X T < 0 X 0 > 0, y 0 ), = P (X 0 > 0, µt + σb T < X 0 y 0 ) P (X 0 > 0 y 0 ) = 1 Φ(y 0 /σ 0 ) + 0 x0 φ x 0 φ z dzdx 0 = 1 Φ(y 0 /σ 0 ) πσ0 2 exp( (x 0 y 0 ) 2 2σ0 2 )Φ( x 0 + µt σ T )dx 0 11

12 Asymptotics of Default Probability As T +0 : lim T +0 P D(T ) T = σ 2πσ 0 Φ(y 0 /σ 0 ) exp( y2 0 2σ0 2 ) AS T + : lim P D(T ) = T + 0 µ > µ = 0 1 µ < 0. As σ 0 +0 : lim P D(T ) = Φ( y 0 + µt σ 0 +0 σ T ) which is Merton s case with: y 0 = log V 0 K, µ = r 1 2 σ2 12

13 Relationship with Merton We do not assume the form of V t or K t Merton assumes V t to be GBM and K t to be constant K We assume randomized initial X 0 Merton assumes exact starting point V 0 Merton is a sub-model of ours, Ito lemma implies d[log V t K ] = (r 1 2 σ2 )dt + σdw t 13

14 Candidate Approximation of Default Probability P D(T ) = P (X 0 > 0, X T < 0 y 0 ) P (X 0 > 0 y 0 ) = P (X T < 0 y 0 ) P (X 0 0, X T < 0 y 0 ) P (X 0 > 0 y 0 ) P (X T < 0 y 0 ) P (X 0 0 y 0 ) P (X 0 > 0 y 0 ) = Φ( y 0+µT σ0 2+σ2 T Φ( y 0 σ 0 ) ) Φ( y 0 σ 0 ). 14

15 Warnings of the Approximation Theoretically speaking, it is always less than real P D(T ) It may go negative, when σ 0 is huge comparatively to y 0 It works fairly well, when σ 0 is small comparatively to y 0 15

16 Conditional Expected Recovery Rate RR(T ) = E[e X T X T < 0, X 0 > 0, y 0 ] = E[e X 0+Z T X 0 + Z T < 0, X 0 > 0, y 0 ] = 1 P (X T < 0, X 0 > 0 y 0 ) + 0 x0 ex 0+z φ x0 φ z dzdx 0. = exp(µt σ2 T ) P D(T )Φ(y 0 /σ 0 ) + 0 Φ( x 0 + µt + σ 2 T σ T )e x 0φ x0 dx 0. 16

17 Asymptotics of Recovery Rate As σ 0 0: lim RR(T ) = exp(µ x + σ2 µ x z σ )Φ( σ z σ z ) Φ( µ. x σ z ) = exp(y 0 + µt + 1 Φ 2 σ2 T ) ( y 0+µT +σ 2 ) T σ T ( Φ y ) 0+µT σ T Recall Merton s Recovery: RR(T ) = V Φ 0 K ert Φ ( log V ) 0 K +(r+ 2 1 σ2 )T σ T ( log V ) 0 K +(r 2 1 σ2 )T σ T 17

18 Candidate Approximation of Recovery Rate Approximation: RR(T ) = E[e X T X T < 0, X 0 > 0, y 0 ] E[e X T X T < 0, y 0 ] = exp(µ x + σ2 µ x x 2 )Φ( σ x σ x ) Φ( µ. x σ x ) Warning: The same warning should be announced as in proxy of default probability 18

19 Term Structure of Credit Spreads Under assumption of constant interest rate: CS(y 0, T ) = 1 T log[1 P D(T ) LGD(T )] = 1 T log Φ(y 0/σ 0 ) 1 T log [Φ(y 0 /σ 0 ) φ x0 g(x 0, T )dx 0 ] g(x, T ) = exp(µt σ2 T )Φ( x + µt + σ2 T σ T )e x Φ( x + µt σ T ) 19

20 Asymptotics of Credit Spread lim σ0 +0 CS(y 0, T ) = 1 T log ( Φ( y 0 + µt σ T ) + ey 0 exp(µt σ2 T )Φ( y 0 + µt + σ 2 T σ T ) ) Recall Merton s Spread: 1 T log Φ( log V 0 K + (r 1 2 σ2 )T σ ) + V V0 0 log T K ert Φ( K + (r σ2 )T σ T ) 20

21 Properties of Credit Spreads Non Zero Short Spread: where CS(y 0, +0) = 1 2 σ2 φ 0 Φ(y 0 /σ 0 ) φ 0 = 1 2πσ0 2 exp( y2 0 2σ0 2 ) Long Term Spread: could be positive or zero, depending on µ Varying Shapes of Term Structure of Credit Spreads 21

22 Numerical Results Credit Spreads of Randomized Merton µ = 0.01,σ = 0.12, y 0 = 0.25, σ 0 =

23 Credit Spreads of Randomized Merton µ = 0.01,σ = 0.1, y 0 = 0.35, σ 0 =

24 Term Structure of Credit Spreads, varying noise σ 0 : µ = 0.01: σ = 0.12, y 0 =

25 Credit Spreads of Randomized Merton µ = 0.01,σ = 0.12, y 0 = 0.35, varying σ 0 25

26 Approximation of Credit Spreads of Randomized Merton: µ = 0.01,σ = 0.12, y 0 = 0.35, σ 0 =

27 Approximation of Credit Spreads of Randomized Merton: µ = 0.01,σ = 0.12, y 0 = 0.35, σ 0 =

28 Randomized Merton Model and Merton Model fit to Financial Sector BBB CDS data on May 24, The fitted Generalized Merton parameters are µ = , σ = , y 0 = and σ 0 = The fitted Merton parameters are µ = , σ = and y 0 =

29 Randomized Merton with Ornstein-Uhlenbeck Set Up Solvency Ratio: X t := log V t K t X t = X 0 e κt + θ(1 e κt ) + σ t 0 eκ(s t) dw s. Noisy Observation: y 0 = X 0 + w, where w N(0, σ 2 0 ) Information Available: X 0 > 0 and y 0 > 0 Independence Assumption: X 0 B t, conditional on y 0 Default Time: τ τ = { T XT < 0 + X T 0. 29

30 Conditional Distributions X t = M t + Z t N(µ x (t), σ x (t)) M t = X 0 e κt N(µ m (t), σ m (t)) Z t = θ(1 e κt ) + σ µ m (t) = y 0 e κt σ 2 m (t) = σ2 0 e 2κt µ z (t) = θ(1 e κt ) σ 2 z (t) = σ2 2κ (1 e 2κt ) µ x (t) = µ m (t) + µ z (t) σ 2 x (t) = σ2 m (t) + σ2 z (t) t 0 eκ(s t) db s N(µ z (t), σ z (t)) denote φ m (t) and φ z (t) as the pdf of M t and Z t respectively 30

31 Results Conditional Default Probability: P D(T ) P D(T ) = 1 Φ(y 0 /σ 0 ) + 0 φ m (T )Φ ( m + µ z(t ) σ z (T ) ) dm Conditional Expected Recovery Rate: RR(T ) RR(T ) = exp(µ z σ2 z ) P D(T )Φ(y 0 /σ 0 ) + 0 Φ( m + µ z + σ 2 z σ z )e m φ m (T )dm where φ m (T ) = 1 y0e κt )2 exp{ (m 2πσ0 2 e 2κT 2σ0 2 } e 2κT 31

32 Under assumption of constant interest rate: Credit Spreads: CS(y 0, T ) = 1 T log Φ(y 0/σ 0 ) 1 T log where [ Φ(y 0 /σ 0 ) φ m (T )g(m, T )dm ] g(m, T ) = exp(µ z σ2 z )Φ( m + µ z + σ 2 z σ z )e m Φ( m + µ z σ z ) 32

33 Properties of Credit Spreads Non Zero Short Spread: CS(y 0, +0) = 1 2 σ2 φ 0 Φ(y 0 /σ 0 ). Zero Long Term Spread: CS(y 0, + ) = 0 Theoretical Humped/Downward Term Structure of Credit Spreads 33

34 Term Structure of Credit Spreads, varying σ 0 : κ = 0.01, θ = 0.4 σ = 0.12, y 0 =

35 Conclusions and Future Research simple, easy to implement incomplete information is considered positive short spread varying shapes of term structure of credit spread generalize Merton s model 35

36 Future Research develope optimal calibration scheme redefine time to default incorporate stochastic volatility effects add jumps to solvency ratio

37 Acknowledgement Thanks to all my colleagues in BMO Market Risk Group, especially Paul Kim, Xiaofang Ma, Arsene Moukoukou, Jerry Shen, Raphael Yan, Bill Sajko, Tom Merrall. Chuang appreciates the continuing support from his Ph.D supervisor Dr. Tom Hurd and his Ph.D committee members Dr. Matheus Grasselli and Dr. Peter Miu. All Errors are Mine. 36

38 THANK YOU! Chuang Yi yichuang 37