Structural Models of Credit Risk and Some Applications

Structural Models of Credit Risk and Some Applications Albert Cohen Actuarial Science Program Department of Mathematics Department of Statistics and Probability albert@math.msu.edu August 29, 2018

Outline 1 Review of Some Structural Models Fundamentals of Structural Models Merton Model Black-Cox Model Moody s KMV Model Correlated Asset-Recovery Model

Single-Name Default Models Single-name default models typically fall into one of three main categories:

Single-Name Default Models Single-name default models typically fall into one of three main categories: Structural Models. Attempts to explain default in terms of fundamental properties, such as the firms balance sheet and economic conditions (Merton 1974, Black-Cox 1976, Leland 1994 etc).

Single-Name Default Models Single-name default models typically fall into one of three main categories: Structural Models. Attempts to explain default in terms of fundamental properties, such as the firms balance sheet and economic conditions (Merton 1974, Black-Cox 1976, Leland 1994 etc). Reduced Form (Intensity) Models. Directly postulates a model for the instantaneous probability of default via an exogenous process λ t (Jarrow-Turnbul 1995, Duffie-Singleton 1999 etc) via:

Single-Name Default Models Single-name default models typically fall into one of three main categories: Structural Models. Attempts to explain default in terms of fundamental properties, such as the firms balance sheet and economic conditions (Merton 1974, Black-Cox 1976, Leland 1994 etc). Reduced Form (Intensity) Models. Directly postulates a model for the instantaneous probability of default via an exogenous process λ t (Jarrow-Turnbul 1995, Duffie-Singleton 1999 etc) via: P[τ [t + dt) F t ] = λ t dt

Single-Name Default Models Single-name default models typically fall into one of three main categories: Structural Models. Attempts to explain default in terms of fundamental properties, such as the firms balance sheet and economic conditions (Merton 1974, Black-Cox 1976, Leland 1994 etc). Reduced Form (Intensity) Models. Directly postulates a model for the instantaneous probability of default via an exogenous process λ t (Jarrow-Turnbul 1995, Duffie-Singleton 1999 etc) via: P[τ [t + dt) F t ] = λ t dt Hybrid Models. Incorporates features from structural and reduced-form models by postulating that the default intensity is a function of the stock or of firm value (Madan-Unal 2000, Atlan-Leblanc 2005, Carr-Linetsky 2006 etc).

What Exactly Are Structural Models Supposed To Do?

What Exactly Are Structural Models Supposed To Do? Price both corporate debt and equity securities (perhaps even equity derivatives and CDS!) Important for buyers/investors, sellers/firms, and advisors

What Exactly Are Structural Models Supposed To Do? Price both corporate debt and equity securities (perhaps even equity derivatives and CDS!) Important for buyers/investors, sellers/firms, and advisors Estimate default probabilities of firms. Useful to commercial banks, investment banks, rating agencies etc.

What Exactly Are Structural Models Supposed To Do? Price both corporate debt and equity securities (perhaps even equity derivatives and CDS!) Important for buyers/investors, sellers/firms, and advisors Estimate default probabilities of firms. Useful to commercial banks, investment banks, rating agencies etc. Determining optimal capital structure decisions. Essential part of economic capital estimation (Moody s Portfolio Manager and RiskFrontier)

What Exactly Are Structural Models Supposed To Do? Price both corporate debt and equity securities (perhaps even equity derivatives and CDS!) Important for buyers/investors, sellers/firms, and advisors Estimate default probabilities of firms. Useful to commercial banks, investment banks, rating agencies etc. Determining optimal capital structure decisions. Essential part of economic capital estimation (Moody s Portfolio Manager and RiskFrontier) Analyzing most corporate decisions that affects cash flows. Can aid in determining most value-maximizing decisions

How Do We Mathematically Define Default? Several possible triggers for default:

How Do We Mathematically Define Default? Several possible triggers for default: Zero Net Worth Trigger. Asset value falls below debt outstanding. But... firms often continue to operate even with negative net worth. Might issue more stock and pay coupon rather than default.

How Do We Mathematically Define Default? Several possible triggers for default: Zero Net Worth Trigger. Asset value falls below debt outstanding. But... firms often continue to operate even with negative net worth. Might issue more stock and pay coupon rather than default. Zero CashflowTrigger. Cashflows inadequate to cover operating costs. But... zero current net cash flow doesn t always imply zero equity value. Shares often have positive value in this scenario. Firm could again issue more stock rather than default.

How Do We Mathematically Define Default? Several possible triggers for default: Zero Net Worth Trigger. Asset value falls below debt outstanding. But... firms often continue to operate even with negative net worth. Might issue more stock and pay coupon rather than default. Zero CashflowTrigger. Cashflows inadequate to cover operating costs. But... zero current net cash flow doesn t always imply zero equity value. Shares often have positive value in this scenario. Firm could again issue more stock rather than default. Optimal (Endogenous) Default. Maximize equity value. The trigger is chosen so that E(A) A = 0 A=Trigger

Simplest Framework for Structural Models

Simplest Framework for Structural Models Assume that the firm s assets A t is the sum of equity E t and debt D t.

Simplest Framework for Structural Models Assume that the firm s assets A t is the sum of equity E t and debt D t. Assumptions on the assets process must be postulated.

Simplest Framework for Structural Models Assume that the firm s assets A t is the sum of equity E t and debt D t. Assumptions on the assets process must be postulated. Assumptions on the debt structure must be postulated.

Simplest Framework for Structural Models Assume that the firm s assets A t is the sum of equity E t and debt D t. Assumptions on the assets process must be postulated. Assumptions on the debt structure must be postulated. A default mechanism τ and payout structure f must be postulated.

Simplest Framework for Structural Models Assume that the firm s assets A t is the sum of equity E t and debt D t. Assumptions on the assets process must be postulated. Assumptions on the debt structure must be postulated. A default mechanism τ and payout structure f must be postulated. The main idea, pioneered by Merton, is that the firms equity E t is then modeled as a call option on the assets A t of the firm:

Simplest Framework for Structural Models Assume that the firm s assets A t is the sum of equity E t and debt D t. Assumptions on the assets process must be postulated. Assumptions on the debt structure must be postulated. A default mechanism τ and payout structure f must be postulated. The main idea, pioneered by Merton, is that the firms equity E t is then modeled as a call option on the assets A t of the firm: [ ] E(t) = E Q e r(t t) f

Simplest Framework for Structural Models Assume that the firm s assets A t is the sum of equity E t and debt D t. Assumptions on the assets process must be postulated. Assumptions on the debt structure must be postulated. A default mechanism τ and payout structure f must be postulated. The main idea, pioneered by Merton, is that the firms equity E t is then modeled as a call option on the assets A t of the firm: [ ] E(t) = E Q e r(t t) f This connects equity E t with debt value D t. In particular if D t has a tractable debt structure it provides a methodology to compute default probability, bond price, credit spreads, recovery rates, etc.

Merton Model Set-up

Merton Model Set-up Underlying asset is modeled as GBM under P: da t = µ A A t dt + σ A A t dw A t

Merton Model Set-up Underlying asset is modeled as GBM under P: da t = µ A A t dt + σ A A t dw A t Default is implicitly assumed to coincide with the event { AT < B } : τ Merton = T 1 {AT B} + 1 {A T > B}

Merton Model Set-up Underlying asset is modeled as GBM under P: da t = µ A A t dt + σ A A t dw A t Default is implicitly assumed to coincide with the event { AT < B } : τ Merton = T 1 {AT B} + 1 {A T > B} This results in a turnover of the company s assets to bondholders if assets are worth less than the total value of bond outstanding.

Merton Model Set-up Underlying asset is modeled as GBM under P: da t = µ A A t dt + σ A A t dw A t Default is implicitly assumed to coincide with the event { AT < B } : τ Merton = T 1 {AT B} + 1 {A T > B} This results in a turnover of the company s assets to bondholders if assets are worth less than the total value of bond outstanding. At maturity the bond value (payoff) is: payoff Merton = B T = A T 1 {τ=t } + B1 {τ= }

Merton Model Bond Price Consequently,

Merton Model Bond Price Consequently, B t = e r(t t) Ẽ t [B T ] = e r(t t) Ẽ t [A T 1 {τ=t } + B1 {τ= } [ B ] = e r(t t) A P t [A T da] + B P t [A T da] 0 = e r(t t) Ẽ t [A T A T < B] P t [A T < B] + e r(t t) B P t [A T B] = Be r(t t) e r(t t) Ẽ t [( B A T ) A T < B] P t [A T < B] := Be r(t t) Be r(t t) Ẽ t [Loss Default] P t [Default]. Here, Loss denotes the loss per unit of face in the event { AT < B }. B

Merton Model PD and LGD

Merton Model PD and LGD PD Q Merton = P[A T < B] = 1 N (d )

Merton Model PD and LGD PD Q Merton = P[A T < B] = 1 N (d ) LGD Q Merton = 1 B Ẽ[ B A t A T < B] = 1 e r(t t) Ā B N ( d + ) N ( d )

Merton Model PD and LGD PD Q Merton = P[A T < B] = 1 N (d ) LGD Q Merton = 1 B Ẽ[ B A t A T < B] = 1 e r(t t) Ā B N ( d + ) N ( d ) Real-world probabilities from risk-neutral probabilities via µ r (Wang Transfrom:)

Merton Model PD and LGD PD Q Merton = P[A T < B] = 1 N (d ) LGD Q Merton = 1 B Ẽ[ B A t A T < B] = 1 e r(t t) Ā B N ( d + ) N ( d ) Real-world probabilities from risk-neutral probabilities via µ r (Wang Transfrom:) ( PDMerton P = N N 1 (PDMerton Q ) µ ) A r T σ A

Merton Model Credit Spread

Merton Model Credit Spread The yield-to-maturity credit spread Y (t, T ) is defined as the spread over the risk free rate r which prices the bond:

Merton Model Credit Spread The yield-to-maturity credit spread Y (t, T ) is defined as the spread over the risk free rate r which prices the bond: (r+y (t,t ))(T t) B t = Be

Merton Model Credit Spread The yield-to-maturity credit spread Y (t, T ) is defined as the spread over the risk free rate r which prices the bond: Solving for Y yields (r+y (t,t ))(T t) B t = Be Y (t, T ) = 1 ( ) B T t ln r f B t

Merton Model Credit Spread The yield-to-maturity credit spread Y (t, T ) is defined as the spread over the risk free rate r which prices the bond: Solving for Y yields (r+y (t,t ))(T t) B t = Be Y (t, T ) = 1 ( ) B T t ln r f B t For the Merton Model: Y Merton (t, T ) = 1 [ T t ln N (d ) A ] t B er(t t) N ( d + ) where

Merton Model Credit Spread The yield-to-maturity credit spread Y (t, T ) is defined as the spread over the risk free rate r which prices the bond: Solving for Y yields (r+y (t,t ))(T t) B t = Be Y (t, T ) = 1 ( ) B T t ln r f B t For the Merton Model: Y Merton (t, T ) = 1 [ T t ln N (d ) A ] t B er(t t) N ( d + ) where d ± = ( ) ln At + (r ± B 1 2 σ2 A (T t)) σ A T t

Merton Model Parameter Estimation

Merton Model Parameter Estimation To compute the desired quantities, model asks for A and σ A as inputs.

Merton Model Parameter Estimation To compute the desired quantities, model asks for A and σ A as inputs. These are unobservable from the markets.

Merton Model Parameter Estimation To compute the desired quantities, model asks for A and σ A as inputs. These are unobservable from the markets. The solution is to imply them from the market.

Merton Model Parameter Estimation There are many ways to do this, but one is to solve the equations

Merton Model Parameter Estimation There are many ways to do this, but one is to solve the equations Et market = A t N (d 1 ) Be r(t t) N (d 2 ) σe market Et market = σ A A t N (d 1 )

Merton Model Parameter Estimation There are many ways to do this, but one is to solve the equations Et market = A t N (d 1 ) Be r(t t) N (d 2 ) σe market Et market = σ A A t N (d 1 ) Note that the second equation is retained from the general theorem relating an option s (V ) volatility to the underlying asset (S) volatility: σ option = σ stock Ω Ω(S, t) := S V S V (S, t).

Merton Model Criticisms

Merton Model Criticisms The assumption that the firm can only default at debt maturity is empirically violated.

Merton Model Criticisms The assumption that the firm can only default at debt maturity is empirically violated. Model tends to underestimate default probabilities and credit spreads, especially for short maturities. Indeed lim T 0 1 T P(A T B) = 0

Merton Model Criticisms The assumption that the firm can only default at debt maturity is empirically violated. Model tends to underestimate default probabilities and credit spreads, especially for short maturities. Indeed lim T 0 1 T P(A T B) = 0 Does not take into account empirical evidence that recovery is correlated to probability of default.

Merton Model Criticisms The assumption that the firm can only default at debt maturity is empirically violated. Model tends to underestimate default probabilities and credit spreads, especially for short maturities. Indeed lim T 0 1 T P(A T B) = 0 Does not take into account empirical evidence that recovery is correlated to probability of default. Does not take into account liquidity issues.

Black-Cox Model Setup

Black-Cox Model Setup Same asset dynamics as in Merton model:

Black-Cox Model Setup Same asset dynamics as in Merton model: da t = µ A A t dt + σ A A t dw A t

Black-Cox Model Setup Same asset dynamics as in Merton model: da t = µ A A t dt + σ A A t dw A t Default can happen at times other than maturity, in particular when the assets fall below a prescribed default point DP:

Black-Cox Model Setup Same asset dynamics as in Merton model: da t = µ A A t dt + σ A A t dw A t Default can happen at times other than maturity, in particular when the assets fall below a prescribed default point DP: τ Black Cox = min{inf{t 0 : A t DP}, τ Merton }

Black-Cox Model Setup Same asset dynamics as in Merton model: da t = µ A A t dt + σ A A t dw A t Default can happen at times other than maturity, in particular when the assets fall below a prescribed default point DP: τ Black Cox = min{inf{t 0 : A t DP}, τ Merton } In the original Black-Cox paper, DP was taken to be DP = Ce γ(t t).

Black-Cox Model Setup Same asset dynamics as in Merton model: da t = µ A A t dt + σ A A t dw A t Default can happen at times other than maturity, in particular when the assets fall below a prescribed default point DP: τ Black Cox = min{inf{t 0 : A t DP}, τ Merton } In the original Black-Cox paper, DP was taken to be DP = Ce γ(t t). The payoff to the bondholder is payoff Black Cox = A τ 1 {τ T } + B1 {τ>t }

Black-Cox Model PD and LGD PD Q Black Cox = 1 p d p c p d = N (d ) ( A p c = Ce γ(t t) ) 1 2(r γ)/σ 2 A (2 ln Ce γ(t t) N σ A T t where d is the same formula as for Merton but with B = Ce γ(t t). d ) (1) Default can happen at times other than maturity, in particular when the assets fall below a prescribed default point DP:

Black-Cox Model PD and LGD PD Q Black Cox = 1 p d p c p d = N (d ) ( A p c = Ce γ(t t) ) 1 2(r γ)/σ 2 A (2 ln Ce γ(t t) N σ A T t where d is the same formula as for Merton but with B = Ce γ(t t). d ) (1) Default can happen at times other than maturity, in particular when the assets fall below a prescribed default point DP: τ Black Cox = min{inf{t 0 : A t DP}, τ Merton }

Black-Cox Model PD and LGD PD Q Black Cox = 1 p d p c p d = N (d ) ( A p c = Ce γ(t t) ) 1 2(r γ)/σ 2 A (2 ln Ce γ(t t) N σ A T t where d is the same formula as for Merton but with B = Ce γ(t t). d ) (1) Default can happen at times other than maturity, in particular when the assets fall below a prescribed default point DP: τ Black Cox = min{inf{t 0 : A t DP}, τ Merton } In the original Black-Cox paper, DP was taken to be DP = Ke γ(t t).

KMV Model Setup

KMV Model Setup Moody s starts by defining a distance-to-default DD: DD = ln A DP + (µ A σa 2 /2)(T t) σ A T t which is just d in the Merton model with B replaced with DP.

KMV Model Setup Moody s starts by defining a distance-to-default DD: DD = ln A DP + (µ A σa 2 /2)(T t) σ A T t which is just d in the Merton model with B replaced with DP. Default point DP chosen as the short term debt plus half of long-term debt: DP = ST + 1 2 LT

KMV Model Setup Moody s starts by defining a distance-to-default DD: DD = ln A DP + (µ A σa 2 /2)(T t) σ A T t which is just d in the Merton model with B replaced with DP. Default point DP chosen as the short term debt plus half of long-term debt: DP = ST + 1 2 LT Then calibrate DD to empirical distribution Ψ of realized defaults from proprietary database: PD Moody s = Ψ(DD)

KMV Model Setup Moody s starts by defining a distance-to-default DD: DD = ln A DP + (µ A σa 2 /2)(T t) σ A T t which is just d in the Merton model with B replaced with DP. Default point DP chosen as the short term debt plus half of long-term debt: DP = ST + 1 2 LT Then calibrate DD to empirical distribution Ψ of realized defaults from proprietary database: PD Moody s = Ψ(DD) Compared to Merton s estimate of PD (i.e. PD Merton = N (d )) prediction power is far superior.

Bond Price under Correlated A-R Model

Bond Price under Correlated A-R Model Define the quantities: A t, the asset value at time t > 0 R t, the recovery amount at time t > 0. We model recovery as a portion of asset, and thus follows the same dynamics as the asset.

Bond Price under Correlated A-R Model Define the quantities: A t, the asset value at time t > 0 R t, the recovery amount at time t > 0. We model recovery as a portion of asset, and thus follows the same dynamics as the asset. The dynamics for the asset and recovery are da t A t dr t R t = r A dt + σ A dw A t = r R dt + σ R dw R t (2) where the drivers are correlated via E[dW A t dw R t ] = ρ A,R dt. (3)

Bond Price under Correlated A-R Model Furthermore, assume Existence of a risk-neutral measure P with risk-free rate r f. Default τ is adapted to the filtration F t = σ(a t ) with maturity T. It follows that B t (ω) = Be r f (T t) P t [τ > T ] + Ẽ t [ e r f (T t) R τ 1 {τ=t } ] + Ẽ t [ e r f (τ t) R τ 1 {τ<t } ] (4)

Example-Basic Hazard Rate Model As an initial example to motivate bond pricing formula, assume that default is in fact independent of the Asset filtration, and is in fact memoryless: P[τ > u] = e λu for some λ > 0 and all u 0. Ẽ t [e r f s R s ] = e r f t R t. It follows that P[τ > u τ > t] = e λ(u t) and B t (ω) = Be r f (T t) e λ(t t) + R t (ω) (1 e λ(t t) ) and CreditSpread [ t,t (ω) = ] 1 T t ln e λ(t t) + Rt(ω) (1 e Be λ(t t) ) r f (T t) ( ) lim t T CreditSpread t,t (ω) = λ 1 R T (ω). B Note that the short term credit spread is positive a.e. as long as P [ R T < B ] = 1.

Two Factor Merton Model - Default at Maturity If we allow for a shadow recovery process as in equation (5), then we are left with an exchange option where the payoff at expiry is G(A T, R T ) = R T 1 {AT < B} + B1 {AT B}. (5) Define α := δ + 1 2 σ2 R (1 ρ2 A,R ) + γ (r f 1 2 σ2 A γ := ρ A,R σ R σ A δ := (r f 1 2 σ2 R ) γ(r f 1 2 σ2 A ). ) + γ2 σ 2 A 2 (6) Note that ρ A,R = γ = 1 implies that δ = 0 and α = r f.

Two Factor Merton Model - Default at Maturity Consequently, [ ] B t = e r f (T t) Ẽ t R T 1 {AT < B} + B1 {AT [ ] B} = Ẽ t e r f (T t) R T 1 {AT < B} + Be r f (T t) P t [A T B] ( = R t e (α r f )(T ln B t) A Φ t (r f 1 2 σ2 A )(T t) ) γσ2 A (T t) σ A T t ( ( )) + Be r f (T t) ln B A 1 Φ t (r f 1 2 σ2 A )(T t) σ A T t (7) Note that we retain the classical 1d Merton price if ρ A,R = γ = 1.

Two Factor Merton Model - Default at Maturity Recovery metrics: Ẽ t [Loss Default] = Ẽt[ B R T A T < B] ( B = 1 R t Be α(t t) Φ ln B At (r f 1 2 σ2 A )(T t) γσ2 A σ A T t Φ ( ln B At (r f 1 2 σ2 A )(T t) σ A T t (T t) ( ) ln B A P t [Default] = Φ t (r f 1 2 σ2 A )(T t) σ A T t ( ) (2D Merton) Credit Spread t,t = 1 T t ln 1 1 P t [Default]Ẽ t [Loss Default] (8) ) )

Complete Recovery Upon Default and No Default If it happens that the entire arbitrage-free value of the bond is recovered upon default, i.e. Ẽ t [R s ] = Be r f (T s) then the entire no-default value is retained: [ ] B t (ω) = Be r f (T t) P t [τ > T ] + Ẽ t e r f (T t) B1 {τ=t } + T 0 e r f (s t) Be r f (T s) P t [τ ds] = Be r f (T t) (9) (10) Of course, if P t [τ > T ] = 1, then P t [τ ds] = 0 for all s [t, T ] and so B t = Be r f (T t) once again.

Two Factor Black-Cox Model - Stopping Times Let DP define our default point written into the bondholder covenant on the asset A, and define τ DP as the first time A reaches this default point.

Two Factor Black-Cox Model - Stopping Times Let DP define our default point written into the bondholder covenant on the asset A, and define τ DP as the first time A reaches this default point. Furthermore, define our default time τ as τ DP : {τ DP < T τ = T : {τ DP T } { A T < B } : otherwise.

Two Factor Black-Cox Model - Stopping Times Let DP define our default point written into the bondholder covenant on the asset A, and define τ DP as the first time A reaches this default point. Furthermore, define our default time τ as τ DP : {τ DP < T τ = T : {τ DP T } { A T < B } : otherwise. And, finally, define f DP,t (s) = P t [τ DP ds] g(x, x 1, t, α 1 ) = 1 ( x α1 t ) φ σ 1 t σ 1 t ( x1 α 1 t ) G(x 1, t, α 1 ) = Φ e σ 1 t ( [ 1 e ( 2α 1 x 1 σ 2 1 ) 4x 2 1 4x 1 x 2σ 2 1 t ) ] ( x1 α 1 t ) Φ σ 1 t (11)

Probability of Default To calculate f DP,0 (t), first consider for µ := r f 1 2 σ2 A the probability [ F (x, T ) := P inf 0 t T ( ) ] σ A Z A (t) + µt > x. (12) By also defining τ µ x := inf {t Z A (t) + µt x}, we can see that F (x, T ) = P[τ µ x > T ]. (13) The following quantities are commonly known: [ F (x, T ) = P inf 0 t T ( ) ] σ A Z A (t) + µt > x = G( x, T, µ) ( µt x ) ( 2µx = Φ exp σ A T σa 2 ) f DP,0 (t) = t F ( ln DP A 0, t ) ( µt + x ) Φ σ A T (14)

Weak Covenant Model Assume that DP < B. ] Define g DP,t,T (s, a) = P t [A T da, τ DP ds Assume Ẽ t [e r f s R s ] = e r f t R t. B t = Ẽ t [e r f (τ DP t) R τdp 1 {τdp T }] + Be r f (T t) P t [A T B, τ DP > T ] ( DP ) [ ] = R t γẽt e [(δ r f )+ 1 2 σ2 R (1 ρ2 A,R )](τdp t) 1 A {τdp T } t + Be r f (T t) P t [A T B, τ DP > T ] = R t ( DP A t ) γ T t s=0 e (δ r f )s e 1 2 σ2 R (1 ρ2 A,R )s f DP,0 (s)ds + Be r f (T t) P t [A T B, τ DP > T ]. (15)

Weak Covenant Model We can also complete this calculation using the fact that e r f t R t is a local martingale, and so for the bounded stopping time τ := min {τ DP, T }, e r f t R t = Ẽ t [e r f τ R τ ] = Ẽ t [e r f τ R τ 1 {τ T } ] + Ẽ t [e r f τ R τ 1 {τ>t } ] = Ẽ t [e r f τ DP R τdp 1 {τdp T }] + Ẽ t [e r f T R T 1 {τdp >T }] (16) It follows that B t = R t + Be r f (T t) P t [A T B, τ DP > T ] R t A γ e (r f δ 1 2 σ2 R (1 ρ2 A,R ))T t DP a γ P t [A T da, τ DP > T ] (17)

Weak Covenant Model We calculate the remaining integral using the identities from the Double Lookbacks paper. We begin with a standard Brownian motion W on a probability space and define X t = µt + σw t τ a = min {t : X t = a} X t = min 0 s t X s g(x, y, t, µ) := 1 ( x µt σ t φ σ t ) ( 1 exp ( 4y 2 4xy ) ) (18) 2σ 2 t

Weak Covenant Model The authors of the Double Lookback paper prove that P [X t dx, X t y] = g( x, y, t, µ)dx = 1 ( x + µt ) ( σ t φ σ 1 exp t = 1 σ t φ ( (x µt) ) ( ( σ 1 exp t ( 4y 2 4xy ) ) 2σ 2 t 4y(x y) ) ) 2σ 2 t (19)

Weak Covenant Model By setting it follows that and so for y = ln DP A 0, µ := r f 1 2 σ2 A σ := σ A (20) X t = ln A t A 0 X t = min ln A (21) s 0 s t A 0 A t = A 0 e Xt { ( )} DP {τ DP > T } = X T > ln = A 0 { τ X y } > T. (22)

Weak Covenant Model Using this notation, we combine the above to retain, DP a γ P t [A T da, τ DP > T [ ( = Ẽ 0 e γ = y = (A t ) γ ln (A t)+x T t ) 1 {τ X y >T t} ] = Ẽ t [ e γ(x+ln At) P 0 [X T t dx, τ X y > T t y (A t ) γ y ( 1 e 2πσ 2A (T t) ( 1 e 2πσ 2A (T t) ] A γ T 1 {τ DP >T } ] ] ) (x µ(t t))2 2σ A 2 γ(t t) 2σ A 2 (T t) dx ) (x µ(t t))2 +4y(x y) 2σ A 2 γ(t t) 2σ A 2 (T t) dx. (23)