Unified Credit-Equity Modeling

Transcription

1 Unified Credit-Equity Modeling Rafael Mendoza-Arriaga Based on joint research with: Vadim Linetsky and Peter Carr The University of Texas at Austin McCombs School of Business (IROM) Recent Advancements in the Theory and Practice of Credit Derivatives Nice, France September 28-30, 2009 Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

2 Research Projects Single Firm Multi Firm Calendar Time Time Changes Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

3 Research Projects Single Firm Multi Firm The Constant Elasticity of Variance Model Calendar Time Time Changes Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

4 Research Projects Multi Firm The Constant Elasticity of Variance Model Single Firm Equity Default Swaps under the JDCEV process Calendar Time Time Changes Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

5 Research Projects Multi Firm Single Firm The Constant Elasticity of Variance Model Equity Default Swaps under the JDCEV process Time Changed Markov Processes in Unified Credit-Equity Modeling Calendar Time Time Changes Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

6 Research Projects Multi Firm Modeling Correlated Defaults by Multiple Firms (Future Research) Single Firm The Constant Elasticity of Variance Model Equity Default Swaps under the JDCEV process Time Changed Markov Processes in Unified Credit-Equity Modeling Calendar Time Time Changes Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

7 Literature Review Stock Option Pricing Literature Black-Scholes (geometric Brownian motion) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

8 Literature Review Stock Option Pricing Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

9 Literature Review Stock Option Pricing Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process No possibility of Bankruptcy! Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

10 Literature Review Stock Option Pricing Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process No possibility of Bankruptcy! Constant volatility and no jumps Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

11 Literature Review Stock Option Pricing Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process No possibility of Bankruptcy! Constant volatility and no jumps No volatility smiles! Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

12 Literature Review Stock Option Pricing Literature Black-Scholes (geometric Brownian motion) Subsequent Generations of Models (modeling the volatility smile) Infinite lifetime process No possibility of Bankruptcy! Constant volatility and no jumps No volatility smiles! Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

13 Literature Review Stock Option Pricing Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process Subsequent Generations of Models (modeling the volatility smile) Local Volatility (CEV, Dupier, etc) No possibility of Bankruptcy! Constant volatility and no jumps No volatility smiles! Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

14 Literature Review Stock Option Pricing Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process No possibility of Bankruptcy! Subsequent Generations of Models (modeling the volatility smile) Local Volatility (CEV, Dupier, etc) Stochastic Volatility (Heston, SABR, etc) Constant volatility and no jumps No volatility smiles! Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

15 Literature Review Stock Option Pricing Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process No possibility of Bankruptcy! Subsequent Generations of Models (modeling the volatility smile) Local Volatility (CEV, Dupier, etc) Stochastic Volatility (Heston, SABR, etc) Constant volatility and no jumps Jump Diffusion (Merton, Kou, etc) No volatility smiles! Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

16 Literature Review Stock Option Pricing Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process No possibility of Bankruptcy! Subsequent Generations of Models (modeling the volatility smile) Local Volatility (CEV, Dupier, etc) Stochastic Volatility (Heston, SABR, etc) Constant volatility and no jumps No volatility smiles! Jump Diffusion (Merton, Kou, etc) Pure Jump Models Based on Levy processes (VG, NIG, CGMY, etc) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

17 Literature Review Stock Option Pricing Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process No possibility of Bankruptcy! Constant volatility and no jumps No volatility smiles! Problem: Subsequent Generations of Models (modeling the volatility smile) Local Volatility (CEV, Dupier, etc) Stochastic Volatility (Heston, SABR, etc) Jump Diffusion (Merton, Kou, etc) These models ignore the possibilityof bankruptcy of the underlying firm In real world, firms have a positive probability of defaultin finite time Pure Jump Models Based on Levy processes (VG, NIG, CGMY, etc) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

18 Literature Review Stock Option Pricing Literature Credit Risk Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process No possibility of Bankruptcy! Subsequent Generations of Models (modeling the volatility smile) Local Volatility (CEV, Dupier, etc) Stochastic Volatility (Heston, SABR, etc) Reduced Form Framework Constant volatility and no jumps No volatility smiles! Jump Diffusion (Merton, Kou, etc) Pure Jump Models Based on Levy processes (VG, NIG, CGMY, etc) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

19 Literature Review Stock Option Pricing Literature Credit Risk Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process No possibility of Bankruptcy! Constant volatility and no jumps No volatility smiles! Subsequent Generations of Models (modeling the volatility smile) Local Volatility (CEV, Dupier, etc) Stochastic Volatility (Heston, SABR, etc) Jump Diffusion (Merton, Kou, etc) Pure Jump Models Based on Levy processes (VG, NIG, CGMY, etc) Reduced Form Framework Default Intensity Models Since Duffie & Singleton, Jarrow, Lando & Turnbull: A vast amount of research has been developed Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

20 Literature Review Stock Option Pricing Literature Credit Risk Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process No possibility of Bankruptcy! Constant volatility and no jumps No volatility smiles! Subsequent Generations of Models (modeling the volatility smile) Local Volatility (CEV, Dupier, etc) Stochastic Volatility (Heston, SABR, etc) Jump Diffusion (Merton, Kou, etc) Pure Jump Models Based on Levy processes (VG, NIG, CGMY, etc) Reduced Form Framework Default Intensity Models Since Duffie & Singleton, Jarrow, Lando & Turnbull: A vast amount of research has been developed Modeling Focus: Credit Default Events, Credit Spreads, Credit Derivatives, etc Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

21 Literature Review Stock Option Pricing Literature Credit Risk Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process No possibility of Bankruptcy! Constant volatility and no jumps No volatility smiles! Subsequent Generations of Models (modeling the volatility smile) Local Volatility (CEV, Dupier, etc) Stochastic Volatility (Heston, SABR, etc) Jump Diffusion (Merton, Kou, etc) Pure Jump Models Based on Levy processes (VG, NIG, CGMY, etc) Reduced Form Framework Default Intensity Models Since Duffie & Singleton, Jarrow, Lando & Turnbull: Problem: These credit models ignore the information of stock option markets A vast amount of research has been developed Disconnection between Equity Models andcredit Models Modeling Focus: Credit Default Events, Credit Spreads, Credit Derivatives, etc Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

22 Literature Review Stock Option Pricing Literature Credit Risk Literature Black-Scholes (geometric Brownian motion) Infinite lifetime process No possibility of Bankruptcy! Constant volatility and no jumps No volatility smiles! Subsequent Generations of Models (modeling the volatility smile) Local Volatility (CEV, Dupier, etc) Stochastic Volatility (Heston, SABR, etc) Jump Diffusion (Merton, Kou, etc) Pure Jump Models Based on Levy processes (VG, NIG, CGMY, etc) Unified Credit-Equity Modeling Reduced Form Framework Default Intensity Models Since Duffie & Singleton, Jarrow, Lando & Turnbull: A vast amount of research has been developed Modeling Focus: Credit Default Events, Credit Spreads, Credit Derivatives, etc Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

23 Motivating Example 2 weeks before bankruptcy (9/02/2008) Lehman Brothers (LEH) stock price price was $1613 Implied Volatility 200% 180% 160% 140% 120% 100% 18 days (9/20/2008) 46 days (10/18/2008) 137 days (1/17/2009) 228 days (4/18/2009) 501 days (1/16/2010) 80% 60% Moneyness (K/S) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

24 Motivating Example 2 weeks before bankruptcy (9/02/2008) Lehman Brothers (LEH) stock price price was $1613 Implied Volatility 200% 180% 160% 140% 120% 100% 18 days (9/20/2008) 46 days (10/18/2008) 137 days (1/17/2009) 228 days (4/18/2009) 501 days (1/16/2010) 80% 60% Moneyness (K/S) The stock price drop of 72% from the high $6219 to $1613! Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

25 Motivating Example 2 weeks before bankruptcy (9/02/2008) Lehman Brothers (LEH) stock price price was $1613 Implied Volatility 200% 180% 160% 140% 120% 100% 18 days (9/20/2008) 46 days (10/18/2008) 137 days (1/17/2009) 228 days (4/18/2009) 501 days (1/16/2010) 80% 60% Moneyness (K/S) The stock price drop of 72% from the high $6219 to $1613! Open Interest on Put contracts with strike prices K = 25 USD Maturing on 4/18/2009 (228 days) were 1529 contracts Maturing on 1/16/2010 (501 days) were 2791 contracts Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

26 The Case for the Next Generation of Unified Credit-Equity Models Put options provide default protection Deep out-of-the-money puts are essentially credit derivatives which close the link between equity and credit products Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

27 The Case for the Next Generation of Unified Credit-Equity Models Put options provide default protection Deep out-of-the-money puts are essentially credit derivatives which close the link between equity and credit products Pricing of equity derivatives should take into account the possibility of bankruptcy of the underlying firm Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

28 The Case for the Next Generation of Unified Credit-Equity Models Put options provide default protection Deep out-of-the-money puts are essentially credit derivatives which close the link between equity and credit products Pricing of equity derivatives should take into account the possibility of bankruptcy of the underlying firm Possibility of default contributes to the implied volatility skew in stock options Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

29 Research Goals Unified Credit Equity Framework Credit and equity derivatives on the same firm should be modeled within a unified framework Consistent pricing across Credit and Equity assets Consistent risk management and hedging Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

30 Research Goals Unified Credit Equity Framework Credit and equity derivatives on the same firm should be modeled within a unified framework Consistent pricing across Credit and Equity assets Consistent risk management and hedging Our Goal is to develop analytically tractable unified credit-equity models to improve pricing, calibration, and hedging Analytical tractability is desirable for fast computation of prices and Greeks, and calibration Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

31 Our Contributions We introduce a new analytically tractable class of credit-equity models Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

32 Our Contributions We introduce a new analytically tractable class of credit-equity models Our model architecture is based on applying random time changes to Markov diffusion processes to create new processes with desired properties Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

33 Our Contributions We introduce a new analytically tractable class of credit-equity models Our model architecture is based on applying random time changes to Markov diffusion processes to create new processes with desired properties We model the stock price as a time changed Markov process with state-dependent jumps, stochastic volatility, and default intensity (stock drops to zero in default) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

34 Our Contributions We introduce a new analytically tractable class of credit-equity models Our model architecture is based on applying random time changes to Markov diffusion processes to create new processes with desired properties We model the stock price as a time changed Markov process with state-dependent jumps, stochastic volatility, and default intensity (stock drops to zero in default) For the first time in the literature, we present state-dependent jumps that exhibit the leverage effect: Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

35 Our Contributions We introduce a new analytically tractable class of credit-equity models Our model architecture is based on applying random time changes to Markov diffusion processes to create new processes with desired properties We model the stock price as a time changed Markov process with state-dependent jumps, stochastic volatility, and default intensity (stock drops to zero in default) For the first time in the literature, we present state-dependent jumps that exhibit the leverage effect: As stock price falls arrival rates of large jumps increase Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

36 Our Contributions We introduce a new analytically tractable class of credit-equity models Our model architecture is based on applying random time changes to Markov diffusion processes to create new processes with desired properties We model the stock price as a time changed Markov process with state-dependent jumps, stochastic volatility, and default intensity (stock drops to zero in default) For the first time in the literature, we present state-dependent jumps that exhibit the leverage effect: As stock price falls arrival rates of large jumps increase As stock price rises arrival rate of large jumps decrease Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

37 Our Contributions (cont) In our model architecture, time changes of diffusions have the following effects: Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

38 Our Contributions (cont) In our model architecture, time changes of diffusions have the following effects: Lévy subordinator time change induces jumps with state-dependent Levy measure, including the possibility of a jump-to-default (stock drops to zero) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

39 Our Contributions (cont) In our model architecture, time changes of diffusions have the following effects: Lévy subordinator time change induces jumps with state-dependent Levy measure, including the possibility of a jump-to-default (stock drops to zero) Time integral of an activity rate process induces stochastic volatility in the diffusion dynamics, the Levy measure, and default intensity Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

40 Unifying Credit-Equity Models The Jump to Default Extended Diffusions (JDED) Before moving on to use time changes to construct models with jumps and stochastic volatility, we review the Jump-to-Default Extended Diffusion framework (JDED) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

41 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t We assume absolute priority: the stock holders do not receive any recovery in the event of default (ζ default time) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

42 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t Stock Price S(t) (ζ default time) Time (yrs) Model the pre-default stock dynamics under an EMM Q as: d S t = [ µ + h( S }{{} t ) ] S t dt + σ( S t ) S t db t Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

43 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t Stock Price S(t) (ζ default time) Time (yrs) Model the pre-default stock dynamics under an EMM Q as: d S t = [ µ + h( S }{{} t ) ] S t dt + σ( S t ) S t db t µ = r q Drift: short rate r minus the dividend yield q Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

44 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t Stock Price S(t) (ζ default time) Time (yrs) Model the pre-default stock dynamics under an EMM Q as: d S t = [ µ + h( S }{{} t ) ] S t dt + σ( S t ) }{{} S t db t σ(s) State dependent volatility Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

45 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t Stock Price S(t) (ζ default time) Time (yrs) Model the pre-default stock dynamics under an EMM Q as: d S t = [ µ + h( S }{{} t ) }{{} ] S t dt + σ( S t ) S t db t h(s) State dependent default intensity Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

46 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t Stock Price S(t) (ζ default time) Time (yrs) Model the pre-default stock dynamics under an EMM Q as: d S t = [ µ + h( S }{{} t ) }{{} ] S t dt + σ( S t ) S t db t h(s) State dependent default intensity Compensates for the jump-to-default and ensures the discounted martingale property Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

47 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t Stock Price S(t) (ζ default time) If the diffusion S t can hit zero: Time (yrs) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

48 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t Stock Price S(t) (ζ default time) 20 τ Time (yrs) If the diffusion S t can hit zero: Bankruptcy at the first hitting time of zero, τ 0 = inf { t : S } t = 0 Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

49 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t Stock Price S(t) (ζ default time) 20 ζ τ Time (yrs) Prior to τ 0 default could also arrive by a jump-to-default ζ with default intensity h( S), ζ = inf { t [0, τ 0 ] : } t 0 h( S u ) e, e Exp(1) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

50 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t Stock Price S(t) (ζ default time) 20 ζ τ Time (yrs) Prior to τ 0 default could also arrive by a jump-to-default ζ with default intensity h( S), { ζ = inf t [0, τ 0 ] : } t 0 h( S u ) e, e Exp(1) At time ζ the stock price S t jumps to zero and the firm defaults on its debt Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

51 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t Stock Price S(t) (ζ default time) The default time ζ is the earliest of: 20 0 ζ τ Time (yrs) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

52 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t Stock Price S(t) (ζ default time) The default time ζ is the earliest of: 1 The stock hits level zero by diffusion: τ ζ τ Time (yrs) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

53 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t Stock Price S(t) (ζ default time) The default time ζ is the earliest of: 1 The stock hits level zero by diffusion: τ 0 2 The stock jumps to zero from a positive value: ζ 20 0 ζ τ Time (yrs) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

54 Jump to Default Extended Diffusions (JDED) Defaultable Stock Price { St, ζ > t S t = 0, ζ t Stock Price S(t) Default Time ) ζ: ζ = min ( ζ, τ0 (ζ default time) The default time ζ is the earliest of: 1 The stock hits level zero by diffusion: τ 0 2 The stock jumps to zero from a positive value: ζ 20 ) ζ = min ( ζ, τ0 0 ζ τ Time (yrs) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

55 Contingent Claims Risk Neutral Survival Probability (no default by time T) Q (S, t; T ) = E [ ] 1 {ζ>t } [ = E e R ] T t h(s u)du }{{} 1 {τ 0 >T } }{{} ) Recall: Default time ζ = min ( ζ, τ0 Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

56 Contingent Claims Risk Neutral Survival Probability (no default by time T) Q (S, t; T ) = E [ ] 1 {ζ>t } [ = E e R ] T t h(s u)du }{{} 1 {τ 0 >T } }{{} ) Recall: Default time ζ = min ( ζ, τ0 1 No jump-to-default before maturity T, Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

57 Contingent Claims Risk Neutral Survival Probability (no default by time T) Q (S, t; T ) = E [ ] 1 {ζ>t } [ = E e R ] T t h(s u)du }{{} 1 {τ 0 >T } }{{} ) Recall: Default time ζ = min ( ζ, τ0 1 No jump-to-default before maturity T, 2 Diffusion does not hit zero before maturity T Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

58 Contingent Claims Defaultable Zero Coupon Bond (at time t) B (S, t; T ) = e r(t t) Q (S, t; T ) }{{} Disc Dollar if No Default occurs prior to maturity Recall:Q (S, t; T ) is the risk neutral survival probability R is a fraction of a dollar paid at maturity + e r(t t) R [1 Q (S, t; T )] }{{} Disc recovery R [0, 1] if Default occurs before maturity T Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

59 Contingent Claims Defaultable Zero Coupon Bond (at time t) B (S, t; T ) = e r(t t) Q (S, t; T ) }{{} Disc Dollar if No Default occurs prior to maturity Recall:Q (S, t; T ) is the risk neutral survival probability R is a fraction of a dollar paid at maturity + e r(t t) R [1 Q (S, t; T )] }{{} Disc recovery R [0, 1] if Default occurs before maturity Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

60 Contingent Claims Defaultable Zero Coupon Bond (at time t) B (S, t; T ) = e r(t t) Q (S, t; T ) }{{} Disc Dollar if No Default occurs prior to maturity Recall:Q (S, t; T ) is the risk neutral survival probability R is a fraction of a dollar paid at maturity + e r(t t) R [1 Q (S, t; T )] }{{} Disc recovery R [0, 1] if Default occurs before maturity Defaultable bonds with coupons are valued as portfolios of zero-coupon bonds Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

61 Contingent Claims Defaultable Zero Coupon Bond (at time t) B (S, t; T ) = e r(t t) Q (S, t; T ) }{{} Disc Dollar if No Default occurs prior to maturity Recall:Q (S, t; T ) is the risk neutral survival probability R is a fraction of a dollar paid at maturity + e r(t t) R [1 Q (S, t; T )] }{{} Disc recovery R [0, 1] if Default occurs before maturity Defaultable bonds with coupons are valued as portfolios of zero-coupon bonds Call Option C (S, t; K, T ) = e r(t t) E [e R ] T t h(s u )du (S T K) + 1 {τ0 >T } Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

62 Contingent Claims Put Payoff (Strike Price K > 0) (K S T ) + 1 {ζ>t } }{{} Put Payoff given no default by time T + K1 {ζ T } }{{} Recovery amount K if default occurs before maturity T Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

63 Contingent Claims Put Payoff (Strike Price K > 0) (K S T ) + 1 {ζ>t } }{{} Put Payoff given no default by time T + K1 {ζ T } }{{} Recovery amount K if default occurs before maturity T Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

64 Contingent Claims Put Payoff (Strike Price K > 0) (K S T ) + 1 {ζ>t } }{{} Put Payoff given no default by time T + K1 {ζ T } }{{} Recovery amount K if default occurs before maturity T Put Option Price P (S, t; K, T ) = e r(t t) E [e R ] T t h(s u)du (K S T ) + 1 {τ0 >T } + Ke r(t t) [1 Q (S, t; T )] NOTE A default claim is embedded in the Put Option Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

65 Jump-to-Default Extended Constant Elasticity of Variance (JDCEV) Model The JDCEV process (Carr and Linetsky (2006)) ds t = [µ + h(s t )]S t dt + σ(s t )S t db t, S 0 = S > 0 σ(s) = as β h(s) = b + c σ 2 (S) CEV Volatility (Power function of S) a > 0 β < 0 b 0 c 0 Default Intensity (Affine function of Variance) volatility scale parameter (fixing ATM volatility) volatility elasticity parameter constant default intensity sensitivity of the default intensity to variance For c = 0 and b = 0 the JDCEV reduces to the standard CEV process Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

66 Jump-to-Default Extended Constant Elasticity of Variance (JDCEV) Model The JDCEV process (Carr and Linetsky (2006)) ds t = [µ + h(s t )]S t dt + σ(s t )S t db t, S 0 = S > 0 σ(s) = as β h(s) = b + c σ 2 (S) CEV Volatility (Power function of S) Default Intensity (Affine function of Variance) The model is consistent with: leverage effect S σ(s) stock volatility credit spreads linkage σ(s) h(s) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

67 An Application of Jump to Default Extended Diffusions (JDED) Equity Default Swaps under the JDCEV Model Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

68 Equity Default Swaps (EDS) Credit-Type Instrument to bring protection in case of a Credit Event Credit Events: Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

69 Equity Default Swaps (EDS) Credit-Type Instrument to bring protection in case of a Credit Event Credit Events: 1 Reference Entity Defaults Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

70 Equity Default Swaps (EDS) Credit-Type Instrument to bring protection in case of a Credit Event Credit Events: 1 Reference Entity Defaults 2 Reference Stock Price drops significantly (L = 30%S 0 ) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

71 Equity Default Swaps (EDS) Credit-Type Instrument to bring protection in case of a Credit Event Credit Events: 1 Reference Entity Defaults 2 Reference Stock Price drops significantly (L = 30%S 0 ) Similar to CDS Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

72 Equity Default Swaps (EDS) Credit-Type Instrument to bring protection in case of a Credit Event Credit Events: 1 Reference Entity Defaults 2 Reference Stock Price drops significantly (L = 30%S 0 ) Similar to CDS Protection Buyer makes periodic Premium Payments on exchange of protection in case of a Credit Event Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

73 Equity Default Swaps (EDS) Credit-Type Instrument to bring protection in case of a Credit Event Credit Events: 1 Reference Entity Defaults 2 Reference Stock Price drops significantly (L = 30%S 0 ) Similar to CDS Protection Buyer makes periodic Premium Payments on exchange of protection in case of a Credit Event Protection Seller pays a recovery amount (1 r) for each dollar of principal at credit event time, if the event occurs prior to Maturity Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

74 Equity Default Swaps (EDS) Equity Default Swap (EDS) Protection Seller Protection Payment Hitting Level (L) T t Hits Level (L) S(t) Protection Buyer 20 L Time (yrs) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

75 Equity Default Swaps (EDS) Equity Default Swap (EDS) Protection Seller Protection Payment Default Event (or) Hitting Level (L) T t Hits Level (L) Or Default Occurs S(t) Protection Buyer 20 L Time (yrs) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

76 Equity Default Swaps (EDS) Equity Default Swap (EDS) Hits Level (L) Or Protection Seller Default Occurs Premium Payments + Accrued Interest Premium Payment Protection Payment S(t) t Default Event (or) Hitting Level (L) T t Protection Buyer 20 L Time (yrs) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

77 Equity Default Swaps (EDS) Equity Default Swap (EDS) Hits Level (L) Or Protection Seller Default Occurs Premium Payments + Accrued Interest Premium Payment Protection Payment S(t) t Acc Interest Default Event (or) Hitting Level (L) T t Protection Buyer 20 L Time (yrs) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

78 Equity Default Swaps (EDS): Balance Equation We want to obtain the EDS rate ϱ that balances out: ϱ = {ϱ PV(Protection Payment)=PV(Premium Payments + Accrued Interest)} Define: Credit Event Time TL PV(Protection Payment) PV(Premium Payments) PV(Accrued Interests) = min{first hitting time to L, Default Time} ϱ E [ (1 r) E ϱ t N [ e r T L e r T L 1{T L T } i=1 e r t i E ( T L t [ ] 1 {T L t i } ] [ T L t ] ) 1 {T L T } ] t r T TL ϱ r Time Interval Recovery Maturity Credit Event Time EDS rate Risk Free Rate Details Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

79 Equity Default Swaps (EDS) Advantages of EDS over CDS Transparency on which an EDS payoff is triggered It is easy to know whether a firm stock price has crossed a lower threshold (L) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

80 Equity Default Swaps (EDS) Advantages of EDS over CDS Transparency on which an EDS payoff is triggered It is easy to know whether a firm stock price has crossed a lower threshold (L) Using the Stock Price as the state variable to determine a credit event allows investors to have a Exposure to Firms for which CDS are not usually traded (as in the case of firms with high yield debt) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

81 Equity Default Swaps (EDS) Advantages of EDS over CDS Transparency on which an EDS payoff is triggered It is easy to know whether a firm stock price has crossed a lower threshold (L) Using the Stock Price as the state variable to determine a credit event allows investors to have a Exposure to Firms for which CDS are not usually traded (as in the case of firms with high yield debt) EDS closes the gap between equity and credit instruments since it is structurally similar to the credit default swap Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

82 Time-Changing the Jump to Default Extended Diffusions (JDED) Under the jump-to-default extended diffusion framework (including JDCEV), the pre-default stock process evolves continuously and may experience a single jump to default Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

83 Time-Changing the Jump to Default Extended Diffusions (JDED) Under the jump-to-default extended diffusion framework (including JDCEV), the pre-default stock process evolves continuously and may experience a single jump to default Our contribution is to construct far-reaching extensions by introducing jumps and stochastic volatility by means of time-changes Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

84 Time-Changing the Jump to Default Extended Diffusions (JDED) Time Changes of Markov Processes in Credit-Equity Modeling Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

85 General Panorama Continuous Markov Process w/ Default Intensity Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

86 General Panorama Time Changes Continuous Markov Process w/ Default Intensity Bochner Levy Subordination Absolute Continuous Time Changes Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

87 General Panorama Continuous Markov Process w/ Default Intensity Bochner Levy Subordination Absolute Continuous Time Changes Time Changes Levy Subordination & Absolute Continuous Time Changes Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

88 General Panorama Continuous Markov Process w/ Default Intensity Jump-Diffusion Process w/ Stochastic Volatility Default Intensity Bochner Levy Subordination Absolute Continuous Time Changes Time Changes Levy Subordination & Absolute Continuous Time Changes Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

89 General Panorama Continuous Markov Process w/ Default Intensity Jump-Diffusion Process w/ Stochastic Volatility Default Intensity Bochner Levy Subordination Absolute Continuous Time Changes Time Changes Levy Subordination & Absolute Continuous Time Changes Analytical Unified Credit and Equity Option Pricing Formulas Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

90 General Panorama Continuous Markov Process w/ Default Intensity Jump-Diffusion Process w/ Stochastic Volatility Default Intensity Bochner Levy Subordination Absolute Continuous Time Changes Time Changes Levy Subordination & Absolute Continuous Time Changes Analytical Unified Credit and Equity Option Pricing Formulas f(x) L 2 Laplace Transform Approach Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

91 General Panorama Continuous Markov Process w/ Default Intensity Jump-Diffusion Process w/ Stochastic Volatility Default Intensity Bochner Levy Subordination Absolute Continuous Time Changes Time Changes Levy Subordination & Absolute Continuous Time Changes Analytical Unified Credit and Equity Option Pricing Formulas f(x) L 2 f(x) L 2 Laplace Transform Approach Spectral Expansion Approach Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

92 Time-Changed Process Y t = X Tt Time Changed Process Construction Y t = X Tt X t is a background process (eg JDCEV) T t is a random clock process independent of X t Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

93 Time-Changed Process Y t = X Tt Time Changed Process Construction Y t = X Tt X t is a background process (eg JDCEV) T t is a random clock process independent of X t Random Clock {T t, t 0} Non-decreasing RCLL process starting at T 0 = 0 and E [T t ] < We are interested in TC with analytically tractable Laplace Transform (LT): L(t, λ) = E [ e λt ] t < Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

94 Time-Changed Process Y t = X Tt Time Changed Process Construction Y t = X Tt X t is a background process (eg JDCEV) T t is a random clock process independent of X t Random Clock {T t, t 0} Non-decreasing RCLL process starting at T 0 = 0 and E [T t ] < We are interested in TC with analytically tractable Laplace Transform (LT): L(t, λ) = E [ e λt ] t < 1 Lévy Subordinators with LT L(t, λ) = e ϕ(λ)t induce jumps Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

95 Time-Changed Process Y t = X Tt Time Changed Process Construction Y t = X Tt X t is a background process (eg JDCEV) T t is a random clock process independent of X t Random Clock {T t, t 0} Non-decreasing RCLL process starting at T 0 = 0 and E [T t ] < We are interested in TC with analytically tractable Laplace Transform (LT): L(t, λ) = E [ e λt ] t < 1 Lévy Subordinators with LT L(t, λ) = e ϕ(λ)t induce jumps 2 Absolutely Continuous (AC) time changes induce stochastic volatility Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

96 Time-Changed Process Y t = X Tt Time Changed Process Construction Y t = X Tt X t is a background process (eg JDCEV) T t is a random clock process independent of X t Random Clock {T t, t 0} Non-decreasing RCLL process starting at T 0 = 0 and E [T t ] < We are interested in TC with analytically tractable Laplace Transform (LT): L(t, λ) = E [ e λt ] t < 1 Lévy Subordinators with LT L(t, λ) = e ϕ(λ)t induce jumps 2 Absolutely Continuous (AC) time changes induce stochastic volatility 3 Composite Time Changes induce jumps & stochastic volatility Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

97 Illustration of Lévy Subordinators Y = XTt where X t = B t and T t = t+ Compound Poisson Process with Exponential Jumps 06 Background Process X(t) Time Process X(t) Time T(t) Time t Y(t)=X (T(t)) Time T(t) Time Changed Process Y(t)=X(T(t)) Time (t) When jump in T (t) arrives, the clock skips ahead, and time-changed process is generated by cutting out the corresponding piece of the diffusion sample path in which T (t) skips ahead Jumps arriving at (expected) time intervals 1/α = 1/4 yrs of (expected) jump size 1/η = 01 yrs Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

98 Illustration of Lévy Subordinators Y = XTt where X t = B t and T t = t+ Compound Poisson Process with Exponential Jumps 06 Background Process X(t) Time Process X(t) Time T(t) Time t Y(t)=X (T(t)) Time T(t) Time Changed Process Y(t)=X(T(t)) Time (t) When jump in T (t) arrives, the clock skips ahead, and time-changed process is generated by cutting out the corresponding piece of the diffusion sample path in which T (t) skips ahead Jumps arriving at (expected) time intervals 1/α = 1/4 yrs of (expected) jump size 1/η = 01 yrs Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

99 Illustration of Lévy Subordinators Y = XTt where X t = B t and T t = t+ Compound Poisson Process with Exponential Jumps 06 Background Process X(t) Time Process X(t) Time T(t) Time t Y(t)=X (T(t)) Time T(t) Time Changed Process Y(t)=X(T(t)) Time (t) When jump in T (t) arrives, the clock skips ahead, and time-changed process is generated by cutting out the corresponding piece of the diffusion sample path in which T (t) skips ahead Jumps arriving at (expected) time intervals 1/α = 1/4 yrs of (expected) jump size 1/η = 01 yrs Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

100 Illustration of Lévy Subordinators Y = XTt where X t = B t and T t = t+ Compound Poisson Process with Exponential Jumps 06 Background Process X(t) Time Process X(t) Time T(t) Time t Y(t)=X (T(t)) Time T(t) Time Changed Process Y(t)=X(T(t)) Time (t) When jump in T (t) arrives, the clock skips ahead, and time-changed process is generated by cutting out the corresponding piece of the diffusion sample path in which T (t) skips ahead Jumps arriving at (expected) time intervals 1/α = 1/4 yrs of (expected) jump size 1/η = 01 yrs Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

101 Illustration of Lévy Subordinators Y = XTt where X t = B t and T t = t+ Compound Poisson Process with Exponential Jumps 06 Background Process X(t) Time Process X(t) Time T(t) Time t Y(t)=X (T(t)) Time T(t) Time Changed Process Y(t)=X(T(t)) Time (t) When jump in T (t) arrives, the clock skips ahead, and time-changed process is generated by cutting out the corresponding piece of the diffusion sample path in which T (t) skips ahead Jumps arriving at (expected) time intervals 1/α = 1/4 yrs of (expected) jump size 1/η = 01 yrs Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

102 Illustration of Lévy Subordinators Y = XTt where X t = B t and T t = t+ Compound Poisson Process with Exponential Jumps 06 Background Process X(t) Time Process X(t) Time T(t) Time t Y(t)=X (T(t)) Time T(t) Time Changed Process Y(t)=X(T(t)) Time (t) When jump in T (t) arrives, the clock skips ahead, and time-changed process is generated by cutting out the corresponding piece of the diffusion sample path in which T (t) skips ahead Jumps arriving at (expected) time intervals 1/α = 1/4 yrs of (expected) jump size 1/η = 01 yrs Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

103 Illustration of Lévy Subordinators Y = XTt where X t = B t and T t = t+ Compound Poisson Process with Exponential Jumps 06 Background Process X(t) 2 Time Process Time T(t) Time t X(t) Y(t)=X (T(t)) Time T(t) Time Changed Process Y(t)=X(T(t)) Time (t) When jump in T (t) arrives, the clock skips ahead, and time-changed process is generated by cutting out the corresponding piece of the diffusion sample path in which T (t) skips ahead Jumps arriving at (expected) time intervals 1/α = 1/4 yrs of (expected) jump size 1/η = 01 yrs Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

104 Examples of Lévy Subordinators Three Parameter Lévy measure: ν(ds) = Cs Y 1 e ηs ds where C > 0, η > 0, Y < 1 Details C changes the time scale of the process (simultaneously modifies the intensity of jumps of all sizes) Y controls the small size jumps η defines the decay rate of big jumps Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

105 Examples of Lévy Subordinators Three Parameter Lévy measure: ν(ds) = Cs Y 1 e ηs ds where C > 0, η > 0, Y < 1 Details C changes the time scale of the process (simultaneously modifies the intensity of jumps of all sizes) Y controls the small size jumps η defines the decay rate of big jumps Lévy-Khintchine formula L(t, λ) = e ϕ(λ)t γλ CΓ( Y )[(λ + η) Y η Y ], Y 0 where ϕ(λ) = γλ + C ln(1 + λ/η), Y = 0 Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

106 Absolutely Continuous Time Changes Absolutely Continuous Time Changes (AC) An AC Time change is the time integral of some positive function V (z) of a Markov process {Z t, t 0}, T t = t 0 V (Z u)du We are interested in cases with Laplace Transform in closed form: [ L z (t, λ) = E z e λ R ] t 0 V (Z u)du Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

107 Absolutely Continuous Time Changes Absolutely Continuous Time Changes (AC) An AC Time change is the time integral of some positive function V (z) of a Markov process {Z t, t 0}, T t = t 0 V (Z u)du We are interested in cases with Laplace Transform in closed form: [ L z (t, λ) = E z e λ R ] t 0 V (Z u)du Example: The Cox-Ingersoll-Ross (CIR) process: dv t = κ(θ V t )dt + σ V Vt dw t with V 0 = v > 0, rate of mean reversion κ > 0, long-run level θ > 0, and volatility σ V > 0 Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

108 Absolutely Continuous Time Changes The Laplace Transform of the Integrated CIR process: [ L v (t, λ) = E v e λ R t Vudu] 0 = A(t, λ)e B(t,λ)v A =! 2ϖe (ϖ+κ)t/2 2κθ σ 2V 2λ(e ϖt 1) q (ϖ + κ)(e ϖt, B = 1) + 2ϖ (ϖ + κ)(e ϖt 1) + 2ϖ, ϖ = 2σV 2 λ + κ2 Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

109 Absolutely Continuous Time Changes The Laplace Transform of the Integrated CIR process: [ L v (t, λ) = E v e λ R t Vudu] 0 = A(t, λ)e B(t,λ)v A =! 2ϖe (ϖ+κ)t/2 2κθ σ 2V 2λ(e ϖt 1) q (ϖ + κ)(e ϖt, B = 1) + 2ϖ (ϖ + κ)(e ϖt 1) + 2ϖ, ϖ = 2σV 2 λ + κ2 This is the Zero Coupon Bond formula under the CIR interest rate r t = λv t Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

110 Illustration of Absolutely Continuous Time Changes CIR parameters κ = 7, θ = 2, V 0 = 05 and σ v = 2 CIR Process Time Process V(t) Time T(t) Time (t) Time t P ro c e s s e s X (t) v s Y (t)= X (T (t)) X (t) T im e (t) Time speeds up or slows down based on the amount of new information arriving and the amount trading (trading time) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

111 Illustration of Absolutely Continuous Time Changes CIR parameters κ = 7, θ = 2, V 0 = 05 and σ v = 2 CIR Process Time Process V(t) Time T(t) Time (t) Time t 0 5 P ro c e s s e s X (t) v s Y (t)= X (T (t)) X (t) X (T (t)) T im e (t) Time speeds up or slows down based on the amount of new information arriving and the amount trading (trading time) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

112 Illustration of Absolutely Continuous Time Changes CIR parameters κ = 7, θ = 2, V 0 = 05 and σ v = 2 CIR Process Time Process 3 25 V(t) Time T(t) Slow Fast Time (t) Time t 0 5 P ro c e s s e s X (t) v s Y (t)= X (T (t)) X (t) X (T (t)) Slow Fast T im e (t) Time speeds up or slows down based on the amount of new information arriving and the amount trading (trading time) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

113 Composite Time Changes Composite Time Changes A Composite Time Change induces both jumps and stochastic volatility T t = T 1 T 2 t T 1 t T 2 2 is a Lévy Subordinator is and AC time change Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

114 Composite Time Changes Composite Time Changes A Composite Time Change induces both jumps and stochastic volatility T t = T 1 T 2 t T 1 t T 2 2 is a Lévy Subordinator is and AC time change Laplace Transform of the Composite Time Change It is obtained by first conditioning wrt the AC time change E[e λt t ] = E[e T 2 t ϕ(λ) ] = L z (t, ϕ(λ)) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

115 Quick Summary We have: Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

116 Quick Summary We have: 1 A Jump-to-Default Extended Diffusion process: E [ ] [ f (X t ) 1 {ζ>t} = E e R ] t 0 h(xu)du f (X t ) 1 {τ0 >t} Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

117 Quick Summary We have: 1 A Jump-to-Default Extended Diffusion process: E [ ] [ f (X t ) 1 {ζ>t} = E e R ] t 0 h(xu)du f (X t ) 1 {τ0 >t} 2 A time-changed process Y t = X Tt with the Laplace transform for the time change T t given in closed form, E [ e λtt ] = L (t, λ) Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1

118 Quick Summary We have: 1 A Jump-to-Default Extended Diffusion process: E [ ] [ f (X t ) 1 {ζ>t} = E e R ] t 0 h(xu)du f (X t ) 1 {τ0 >t} 2 A time-changed process Y t = X Tt with the Laplace transform for the time change T t given in closed form, E [ e λtt ] = L (t, λ) How do we evaluate contingent claims written on the time-changed process Y t? E [ ] f (Y t ) 1 {ζ>tt } Rafael Mendoza (McCombs) Unified Credit-Equity Modeling Credit Risk / 1