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: Vt;
• 2. Moving debt barrier: K(t);
• 3. Log-leverage ratio: Xt = 1

σ log(Vt/K(t));

• 4. Default of firm occurs at t∗, the first time Xt ≤ 0;
• 5. Zero recovery defaultable bond price:

¯ B(x, T) = EQ

x [e−rT 1{t∗>T}] = e−rT (1 − P(x, t))

• 6. Here P(x, t) is risk-neutral probability of default before t,

starting with X0 = x.

Multifirm Default Dynamics: Black-Cox 1976

First Passage Problems

If Xt is a time homogeneous Markov jump-diffusion we’ll need to solve Xt = mins≤t Xs.

• 1. First passage t∗ is the stopping time t∗ = inf{t|Xt ≤ 0}.
• 2. We want to compute functions like the CDF of t∗

P(x, t) = P[t∗ ≤ t|X0 = x] = Ex[1{Xt≤0]}] . . .

• 3. and the joint PDF for (t∗, Xt∗+)

p(x, y, t)dydt = P[t∗ ∈ (t, t + dt), Xt∗+ ∈ (y, y + dy)|X0 = x]

Known Results on First Passage

• 1. Exact formula for drifting Brownian motion

Xt = x + Wt + βt, x, y > 0: P(x, t; β) = N −x − βt √ t

• + e−2βxN

−x + βt √ t

• 2. First passage for diffusions is a rich area of probability,

with spectral and PDE methods.

• 3. Exact formulas exist for Kou-Wang jump-diffusion model,

and certain other “phase-type” jump-diffusions;

• 4. Various identities (eg Wiener-Hopf factorization) can be

used to compute other special cases, such as processes with

• ne-sided jumps.

TCBM: Time Changed Brownian Motion

We consider the special case when Xt = x + WTt + βTt, where the time change Tt is a non-decreasing process, independent of the drifting Brownian motion Wt + βt.

• 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 Tt 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 Tt ∼ CIR.

A sample path of L´ evy time change Tt

0.5 1 1.5 2 2.5 3 3.5 4 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|Xt ≤ 0}

• 2. First passage time of the second kind:

t(2) = inf{t|Tt ≥ τ ∗} where τ ∗ = inf{t|x + Wt + βt ≤ 0}

• 3. t(2) is relatively easy to compute:

P (2)(x, t) = Ex[E[1{τ ∗≤g}|Tt = s]] = ∞ P(x, s)ρt(ds)

Schematic

Figure: Three sample Brownian paths with first passage at t0.

Schematic

Figure: Three sample Brownian paths with first passage at t0, and a sample time change path that first crosses t0 at time s0.

Computing t(2)

When Tt is a L´ evy subordinator with known PDF ρt(z) and Laplace exponent ψ(u) := −1 t log E[e−uTt].

• 1. Probability of first passage before time t is:

P (2)(x, t) = E[P[τ ∗ ≤ g|Tt = g]] = ∞ P(x, z)ρt(dg) = 1 − e−βx π ∞

−∞

w sin(xw) w2 + β2 e−ψ((w2+β2)/2,t)dw.

• 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 s1(x) = t(2)(x)

sn(x) = s1(x)1{Xs1(x)+≤0}+sn−1(Xs1(x)+)1{Xs1(x)+>0}, n ≥ 2 converges a.s to t∗.

• 2. The joint PDF for (t∗, Xt∗) solves

p(x, y, t) = ˜ p(x, y, t)1y<0+

• [0,∞)2 dzds ˜

p(x, z, s)p(z, y, t−s) and the solution by iteration converges in the L1-norm.

Computing PDF of t∗

• 1. Joint PDF ˜

p(x0, x1, s) of (t(2), Xt(2)) is given by 2eβ(x1−x0) π2 PV

• (R+)2 dk1dk2

k2 cos |x1|k1 sin x0k2 k2

1 − k2 2

e−sψ((k2

1+β2)/2)ψ((k2

2 + β2)/2)

• 2. This is a 2-dim Fourier Transform.
• 3. We solve for PDF of t∗, p(x, t) by iterating

pn(x0, t) =

−∞

˜ p(x0, y, t)dy+

• [0,∞)2 dzds ˜

p(x0, z, s)pn−1(z, t−s)

• 4. Faster than PIDE methods and much faster than Monte

Carlo.

PDF of t∗ for the VG TCBM

1 2 3 4 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 2 3 4 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7

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 St = ex+rt+σWTt−σ2Tt/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 efficient, 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 Xt = x + WTt − Tt/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 Tt:

ψ(u, t) := − log EQ[e−uTt] = tc log(1 + u/a).

• 4. Time of default is τ = t(2).

Another Simple One-Firm Model: SEA Model

Self Exciting Affine (SEA) Model for Xt with stochastic volatility and contagion effects:

• 1. Log-leverage ratio Xt = x + WTt − Tt/2 where

dTt = λtdt + mdJt dλt = −bλtdt + dJt

• 2. Here Jt is increasing pure jump process with jump measure

ν(z) = ace−az, c, a > 0 on (0, ∞).

• 3. Laplace exponent of Tt is affine in λ:

ψ(u, t, λ) := − log EQ

0,λ[e−uTt] = 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 St and implied log-leverage ˆ Xt 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 St and implied log-leverage ˆ Xt 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 Xi

t = xi + σiW i T i

t − σ2

i T i t /2

• 2. Take a “one-factor” form for the time change (c.f.

Moosbrucker 2006 and Baxter 2006) T i

t = αiTt + (1 − αi) ˜

T i

t , αi ∈ [0, 1]

• 3. Here W 1, . . . , W I, ˜

T 1, . . . ˜ T I, T are mutually independent.

• 4. Then take

τ i = inf{t|T i

t ≤ ˜

ti}, ˜ ti = inf{t|xi + σiW i

t − σ2 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: W = −EQ[ T e−rtd[(b − Lt)+ − (a − Lt)+)]] V = EQ[ T e−rt[(b − Lt)+ − (a − Lt)+)]dt] where total portfolio loss at time t is Lt =

I

• i=1

ℓi1{τ i≤t} Fractional loss at default of firm i is ℓi = (1 − R)Ni/

• j

Nj

Computing CDOs

Perhaps the easiest to implement method is the Hockey stick expansion of Iscoe-Jackson-Kreinin-Ma 2007: (1 − x)+ ∼

100

• m=1

ωme−umx, for a set of values {ωm, um} which implies that (b − Lt)+ ∼ b

100

• m=1

ωme−umLt/b

Computing CDOs

1{τ i≤t} = 1{σ2

i Tt≥τ ∗ i } are independent RVs when conditioned on

Tt:

Proposition

Laplace transform of portfolio loss when αi = 1 is given by: Φ(u, t) = EQ[e−uLt] = ∞

I

• i=1
• 1 + (e−uℓi − 1)P(xi, σ2

i T)

• ρt(dT)

where P(x, t) = N −x + t/2 √ t

• + exN

−x − t/2 √ t

Back of the Envelope Flop Count

Consider a nonhomogeneous CDO tranche where the initial CDS curve of each firm i is fit with choice of (xi, σi). Maximal correlation case αi = 1 can be computed in O(1) × I × Nt × Nm ∼ O(1) × 106 flops Here:

• 1. Nt ∼ 100: number of t-gridpoints;
• 2. Nm ∼ 100: number of terms in “hockey stick expansion”.
• 3. I ∼ 100: number of firms;

Conclusions

• 1. TCBMs are a flexible class of processes, natural for

modeling credit and equities.

• 2. t(2) (but not t∗!) is relatively easy to compute for TCBMs.
• 3. We can compute true first passage t∗ for TCBMs in terms
• f t(2).
• 4. Using t(2) in one-firm TCBM structural credit models leads

to flexible and efficient calibration to CDS data.

• 5. Using time change leads to tractable and self-consistent

portfolio credit models with dynamic default correlation.

• 6. Some papers are available:

http://www.math.mcmaster.ca/tom/HurdTCBMRevised.pdf http://www.math.mcmaster.ca/tom/HurdKuznetsovFirstPassage.p