Credit Risk using Time Changed Brownian Motions Tom Hurd - PDF document

Credit Risk using Time Changed Brownian Motions Tom Hurd Mathematics and Statistics McMaster University Joint work with Alexey Kuznetsov (New Brunswick) and Zhuowei Zhou (Mac) 2nd Princeton Credit Conference 2008 Structural Credit Models Time Changed Brownian Motion t (2) Approximates t ∗ Models Using TCBMs Multifirm Modeling Conclusions

Structural Credit Model: Black-Cox 1976 A time-consistent modification of Merton’s 1974 model: 1. Value of a firm: V t ; 2. Moving debt barrier: K ( t ) ; 3. Log-leverage ratio: X t = 1 σ log( V t /K ( t )) ; 4. Default of firm occurs at t ∗ , the first time X t ≤ 0 ; 5. Zero recovery defaultable bond price: x [ e − rT 1 { t ∗ >T } ] = e − rT (1 − P ( x, t )) B ( x, T ) = E Q ¯ 6. Here P ( x, t ) is risk-neutral probability of default before t , starting with X 0 = x . Multifirm Default Dynamics: Black-Cox 1976

First Passage Problems If X t is a time homogeneous Markov jump-di ff usion we’ll need to solve X t = min s ≤ t X s . 1. First passage t ∗ is the stopping time t ∗ = inf { t | X t ≤ 0 } . 2. We want to compute functions like the CDF of t ∗ P ( x, t ) = P [ t ∗ ≤ t | X 0 = x ] = E x [ 1 { X t ≤ 0] } ] . . . 3. and the joint PDF for ( t ∗ , X t ∗ + ) p ( x, y, t ) dydt = P [ t ∗ ∈ ( t, t + dt ) , X t ∗ + ∈ ( y, y + dy ) | X 0 = x ] Known Results on First Passage 1. Exact formula for drifting Brownian motion X t = x + W t + β t , x, y > 0 : � − x − β t � � − x + β t � + e − 2 β x N P ( x, t ; β ) = N √ √ t t 2. First passage for di ff usions is a rich area of probability, with spectral and PDE methods. 3. Exact formulas exist for Kou-Wang jump-di ff usion model, and certain other “phase-type” jump-di ff usions; 4. Various identities (eg Wiener-Hopf factorization) can be used to compute other special cases, such as processes with one-sided jumps.

TCBM: Time Changed Brownian Motion We consider the special case when X t = x + W T t + β T t , where the time change T t is a non-decreasing process, independent of the drifting Brownian motion W t + β t . 1. Most of the favourite L´ evy models in finance, such as Kou-Wang, VG, NIG, hyperbolic, can be written as TCBMs. In these cases T is itself a L´ evy process. 2. For example, the Variance Gamma (VG) model arises by taking T t to be a gamma process. 3. Stochastic volatility models (with a zero correlation condition) can also be written as TCBM. For example, the zero correlation Heston model has T t ∼ CIR . A sample path of L´ evy time change T t 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4

Default Dynamics in “Brownian Time” spread Default Dynamics in Real Time spread

First passage times for TCBMs 1. The usual first passage time: t ∗ = inf { t | X t ≤ 0 } 2. First passage time of the second kind: t (2) = inf { t | T t ≥ τ ∗ } where τ ∗ = inf { t | x + W t + β t ≤ 0 } 3. t (2) is relatively easy to compute: � ∞ P (2) ( x, t ) = E x [ E [ 1 { τ ∗ ≤ g } | T t = s ]] = P ( x, s ) ρ t ( ds ) 0 Schematic Figure: Three sample Brownian paths with first passage at t 0 .

Schematic Figure: Three sample Brownian paths with first passage at t 0 , and a sample time change path that first crosses t 0 at time s 0 . Computing t (2) When T t is a L´ evy subordinator with known PDF ρ t ( z ) and Laplace exponent ψ ( u ) := − 1 t log E [ e − uT t ] . 1. Probability of first passage before time t is: � ∞ E [ P [ τ ∗ ≤ g | T t = g ]] = P (2) ( x, t ) = P ( x, z ) ρ t ( dg ) 0 � ∞ = 1 − e − β x w sin( xw ) w 2 + β 2 e − ψ (( w 2 + β 2 ) / 2 ,t ) dw. π −∞ 2. Robust and fast to compute using the Fast Fourier Transform.

t (2) Approximates t ∗ Theorem (A Kuznetsov and TH) For any β ≤ 0 and L´ evy subordinator G : 1. The sequence of stopping times defined by s 1 ( x ) = t (2) ( x ) s n ( x ) = s 1 ( x ) 1 { X s 1( x )+ ≤ 0 } + s n − 1 ( X s 1 ( x )+ ) 1 { X s 1( x )+ > 0 } , n ≥ 2 converges a.s to t ∗ . 2. The joint PDF for ( t ∗ , X t ∗ ) solves �� p ( x, y, t ) = ˜ p ( x, y, t ) 1 y< 0 + [0 , ∞ ) 2 dzds ˜ p ( x, z, s ) p ( z, y, t − s ) and the solution by iteration converges in the L 1 -norm. Computing PDF of t ∗ p ( x 0 , x 1 , s ) of ( t (2) , X t (2) ) is given by 1. Joint PDF ˜ 2 e β ( x 1 − x 0 ) �� PV ( R + ) 2 dk 1 dk 2 π 2 k 2 cos | x 1 | k 1 sin x 0 k 2 e − s ψ (( k 2 1 + β 2 ) / 2) ψ (( k 2 2 + β 2 ) / 2) k 2 1 − k 2 2 2. This is a 2-dim Fourier Transform. 3. We solve for PDF of t ∗ , p ( x, t ) by iterating � 0 �� p n ( x 0 , t ) = p ( x 0 , z, s ) p n − 1 ( z, t − s ) p ( x 0 , y, t ) dy + ˜ [0 , ∞ ) 2 dzds ˜ −∞ 4. Faster than PIDE methods and much faster than Monte Carlo.

PDF of t ∗ for the VG TCBM 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 1 2 3 4 5 0 1 2 3 4 5 Figure: The density of the VG first passage time for two sets of parameters. The circles show the “exact” result; the three solid lines show the first three approximations. Models Using TCBMs Numerous modeling avenues are open if we replace t ∗ by t (2) in first passage problems: 1. Barrier options in model S t = e x + rt + σ W Tt − σ 2 T t / 2 (note: β = − 1 / 2 ); 2. Defaultable bonds and credit derivatives in Black-Cox style models; 3. Hybrid credit-equity models where the stock price hits 0 at the time of default. 4. Multifirm credit models in which the time-changes are dynamically correlated are e ffi cient, consistent and flexible. 5. These are under active investigation with Zhuowei Zhou.

A Simple One-Firm Structural Model: VG Model Adapted from the Variance Gamma option pricing model (Madan-Seneta 1990): 1. Log-leverage ratio X t = x + W T t − T t / 2 (here β = − 1 / 2 ); 2. T gamma process, the increasing pure jump process with jump measure ν ( z ) = ce − az /z, c, a > 0 on (0 , ∞ ) ; 3. Laplace exponent of T t : ψ ( u, t ) := − log E Q [ e − uT t ] = tc log(1 + u/a ) . 4. Time of default is τ = t (2) . Another Simple One-Firm Model: SEA Model Self Exciting A ffi ne (SEA) Model for X t with stochastic volatility and contagion e ff ects: 1. Log-leverage ratio X t = x + W T t − T t / 2 where dT t = λ t dt + mdJ t d λ t = − b λ t dt + dJ t 2. Here J t is increasing pure jump process with jump measure ν ( z ) = ace − az , c, a > 0 on (0 , ∞ ) . 3. Laplace exponent of T t is a ffi ne in λ : ψ ( u, t, λ ) := − log E Q 0 , λ [ e − uT t ] = A ( u, t ) + B ( u, t ) λ . Here A, B are explicit. 4. J -shocks can cause instantaneous defaults and a simultaneous jump in credit spreads.

VG Model Results Figure: Best VG fit to one day Ford CDS curve, with variation in parameters VG Model Results (ctd): Ford spread

VG Model Results (ctd) Figure: Daily closing Ford stock price S t and implied log-leverage ˆ X t for June-Aug 2006 SEA Model Results Figure: Best fit parameters and Sensitivities

SEA Model Results (ctd) spread SEA Model Results (ctd) Figure: Daily closing Ford stock price S t and implied log-leverage ˆ X t for June-Aug 2006

A “Bottom-up” Structural Portfolio Credit Model We can embed the TCBM models into a multifirm setting: 1. For firm i, i = 1 , . . . , I , take log-leverage ratios t = x i + σ i W i X i t − σ 2 i T i t / 2 T i 2. Take a “one-factor” form for the time change (c.f. Moosbrucker 2006 and Baxter 2006) t , α i ∈ [0 , 1] T i t = α i T t + (1 − α i ) ˜ T i 3. Here W 1 , . . . , W I , ˜ T 1 , . . . ˜ T I , T are mutually independent. 4. Then take τ i = inf { t | T i t i = inf { t | x i + σ i W i t i } , t − σ 2 t ≤ ˜ ˜ i t/ 2 ≤ 0 } 5. This model has dynamic default correlation. Synthetic CDOs Idealized default and premium legs of a synthetic CDO [ a, b ] -tranche on firms i = 1 , . . . , I have price: � T e − rt d [( b − L t ) + − ( a − L t ) + )]] − E Q [ W = 0 � T e − rt [( b − L t ) + − ( a − L t ) + )] dt ] E Q [ V = 0 where total portfolio loss at time t is I � L t = ℓ i 1 { τ i ≤ t } i =1 Fractional loss at default of firm i is � ℓ i = (1 − R ) N i / N j j

Computing CDOs Perhaps the easiest to implement method is the Hockey stick expansion of Iscoe-Jackson-Kreinin-Ma 2007: 100 (1 − x ) + ∼ � ω m e − u m x , for a set of values { ω m , u m } m =1 which implies that 100 ( b − L t ) + ∼ b � ω m e − u m L t /b m =1 Computing CDOs 1 { τ i ≤ t } = 1 { σ 2 i } are independent RVs when conditioned on i T t ≥ τ ∗ T t : Proposition Laplace transform of portfolio loss when α i = 1 is given by: � ∞ I 1 + ( e − u ℓ i − 1) P ( x i , σ 2 � � Φ ( u, t ) = E Q [ e − uL t ] = � i T ) ρ t ( dT ) 0 i =1 where � − x + t/ 2 � � − x − t/ 2 � + e x N P ( x, t ) = N √ √ t t