“Time Changed Markov Processes
in United Credit-Equity Modeling”
Rafael Mendoza-Arriaga∗
Northwestern University
Department of Industrial Engineering and
Management Sciences
Rutgers University
Feb 2008
∗ Joint
work with Peter Carr and Vadim Linetsky
Motivation. 1.- Local Volatility Models
• In the Black-Scholes-Merton model, the stock price
follows GBM, which is a process with infinite lifetime (i.e. No possibility of bankruptcy) and with
constant volatility. Not very realistic assumptions!!!
• Newer Stock options pricing models focus on modeling the implied volatility smile/skew by various
means, such as local volatility, stochastic volatility, and jumps, but ignore the possibility of default
of the firm underlying the option contract.
• Real-World firms have positive probability of bankruptcy
in finite time.
1
Motivation. 2- Credit Models
• Credit risk literature focuses on modeling default,
credit spreads, and the pricing of corporate bonds
and credit derivatives,
• This literature ignores the information in the equity
options market
• Clear disconnection between equity and credit derivatives, even though evidence shows the usage of equity (put) options as protection vehicles in case of
default.
2
Motivation. 3-GM Example
• In 2006 financial analysts considered GM to have high default
risk.
• In particular, GM equity Put Options with Low Strikes had
the highest Open Interest:
1. Jan 07 $10 Put @ $1.50 (ImpVol ≈100%) with OI 493,051
contracts.
2. Jan 07 $5 Put @ $0.50 (ImpVol ≈120%) with OI 205,703
contracts.
3. Jan 07 $7.50, @ $1.00 (ImpVol ≈110%) and $2.50, @
$0.15 (ImpVol ≈133%) Puts also had substantial OI
• Total outstanding notional for Jan 07 and Jan 08 Puts with
strikes $2.50-$10 ≈130 million shares at $21 USD.
3
Motivation. 4- Far from Continuous
The GM stock price was $21.19 in Feb. 22, 2006
4
Motivation. 5- Far from Constant Vol.
140
130
Implied Volatility, %
120
110
100
90
80
70
60
50
40
0
5
10
15
20
25
30
Strike Price
Implied Volatility Skew for GM Jan 2007 Puts on Feb
22, 2006.
Historical Volatility of GM stock price over the
previous 12 months ≈ 46%
5
Observations
• Put options on the stock provide default protection and can
be used to manage default risk.
• Deep out-of-the-money put options are essentially credit derivatives.
• The possibility of default contributes to the implied volatility
skew in stock options: there is a linkage between implied
volatilities skew in options markets and credit spreads in credit
markets.
(Default Prob ⇑↔ S ⇓→ σ ⇑)
• Corporate liabilities, credit derivatives, and equity derivatives
should be modeled within a unified framework.
6
Our Approach
• We develop an analytically tractable unified modeling framework for corporate debt, credit derivatives, and equity derivatives.
• In the reduced-form, intensity-based framework, we model Defaultable Stock as the fundamental state variable.
• We view corporate debt, equity derivatives, and credit derivatives as contingent claims written on the defaultable stock.
• We introduce default into major classes of equity derivatives
models, incorporating jumps and stochastic volatility based on
time changes of Markov processes with killing.
7
GENERAL PANORAMA
We start off with…
… which we modify under a time change…
Continuous
Continuous Markov
Markov
Process
Process w/killing
w/killing
Jump
Jump Diffusion
Diffusion
Process
Process w/Stochastic
w/Stochastic
Volatility
Volatility &
& Killing
Killing
Time Changes
Bochner’s
Bochner’sLevy
Levy
Subordination
Subordination
Bochner’s
Bochner’sLevy
Levy
Subordination
Subordination
Abs.
Abs.Cts.
Cts.Time
Time
Changes
Changes
Abs.
Abs.Cts.
Cts.Time
Time
Change
Change
…to move on from here….
…and get a model for options and credit
Analytical Option and Credit Pricing
f ( x) ∈ L2 ((0, ∞), m)
Spectral
SpectralExp.
Exp.
Approach
Approach
f ( x) ∉ L2 ((0, ∞), m)
Laplace
LaplaceTransform
Transform
Approach
8
Approach
-1-
Jump-to-Default Extended Diffusion (JDD)
• Model the pre-default stock dynamics under an EMM Q as a
diffusion:
dSt = [µ + λ (St )] Stdt + σ (St ) St dBt,
S0 = S > 0
where µ = r − q, r, q, σ and λ are the short rate, dividend
yield, volatility, and default intensity.
• If the diffusion can hit zero, we kill it at the first hitting time
of zero, T0 , and send it to a cemetery (bankruptcy) state ∆,
where it remains forever.
• Prior to T0, a jump-to-default arrives at the first jump time of
a doubly stochastic Poisson process with intensity λ(St). The
time of default
¯Z t
¾
½ is:
¯
λ (Su ) du ≥ e ,
e ≈ Exp(1)
ζ = inf t ∈ [0, T0 ] ¯¯
0
• Assume stock holders do not receive any recovery in the event
of default. Addition of λ in the drift r − q + λ compensates
for default to insure that the discounted gain process to the
stock holders is a martingale under the EMM.
9
Corporate Bonds and European Options
under JDD
• The time-t price of a defaultable zero coupon bond with face
value of $1 and recovery R ∈ [0, 1] at maturity:
B (S, t; T ) = e−r(T −t) Q (S, t; T ) + e−r(T −t) R [1 − Q (S, t; T )]
where Q (S, t; T ) is the risk neutral
· R T survival probability:
¸
λ(Su )du
−
Q (S, t; T ) = E e t
1{T0>T }
• Likewise, the time-t price of a Call Option with strike price
K > 0 is given by:
· RT
¸
−
λ(Su )du
C (S, t; K, T ) = e−r(T −t) E e t
(ST − K)+ 1{T0 >T }
10
Corporate Bonds and European Options
under JDD
• The payoff Put Option with strike price K > 0 can be decomposed as:
(K − ST )+ 1{ζ>T } + K 1{ζ≤T }
Where the first part is the payoff given no default by time T
and a recovery amount K at time T in case of default occurs
prior maturity.
• Therefore, we price the European Put option as:
· RT
¸
−
λ(Su )du
P (S, t; K, T ) = e−r(T −t) E e t
(K − ST )+ 1{T0>T }
+Ke−r(T −t) [1 − Q (S, t; T )]
NOTICE. A Credit Derivative is embedded in the Put Option!!!
11
A Jump-to-Default Extended Constant Elasticity
of Variance (JDCEV) Model
• The JDCEV process is a 1D diffusion introduced by Carr and
Linetsky (2006):
dSt = [µ + λ(St)]St dt + σ(St )St dBt , S0 = S > 0,
where the local volatility is CEV and default intensity is linear
in instantaneous variance:
σ(S) = aS β , λ(S) = b + c σ 2 (S) = b + c a2S 2β ,
where a is the volatility scale parameter (fixing ATM volatility), β < 0 is the volatility elasticity parameter, b is the constant part of the default intensity, and c controls the contribution of the stock return variance in default intensity.
• This specification is consistent with the leverage effect St ⇓→
σ ⇑ and the linkage between stock volatility and credit spreads
σ ⇑↔ λ ⇑.
12
Inducing Jumps and Stochastic Volatility Via
Time Changes
• JDD models and in particular the JDCEV process assume that
the prior to default process evolves continuously, with a single
jump to default.
• Even though models like the JDCEV process provide enough
structure to capture the leverage effect and volatility skews,
these can be enhanced by allowing jumps and stochastic volatility that better represent reality.
• This current work allows a more general class of models via
Time Changes: pre-default stock price is a process with jumps
and stochastic volatility + jump to default
13
Introduction to Time Changes Tt
• {Tt, t ≥ 0} is an increasing RCLL process starting at T0 = 0
and E [Tt ] < ∞.
• We are interested in time changes with analytically tractable
Laplace Transform (LT):
£ −λTt ¤
L(t, λ) = E e
<∞
– Lévy Subordinators: increasing Lévy processes (stationary
and independent increments) with positive jumps and nonnegative drift.
– Absolutely Continuous (A.C.) time changes:
Z t
Tt =
Vu du
0
with such activity rate process Vu that the LT. is known
in closed form.
– Composed Time Changes, e.g., subordinate first, then do
an A.C. time change. These time changes induce both
Jumps + Stoch. Volatility
14
The Expectation Operator
• To compute expectations, we can condition on the time change
(recall that X and T are independent):
¯
£
¤¤
¤
£ £
¯
Ex 1{ζ>Tt}f (XTt ) = Ex Ex 1{ζ>s}f (Xs) Tt = s
• We need to calculate the expectation operator of the original
diffusion process Xt :
£
¤
Ex 1{ζ>t}f (Xt)
• To gain tractability we choose two important approaches:
1. Resolvent Operator, which is the LT of the expectation
operator. Generalized methodology.
2. Spectral Repressentation, which is particular case when
the infinitesimal generator is symmetric and the payoff is
in the space of functions f ∈ L2 (D, m)
15
The Resolvent Operator Approach
• For any Markov process X, the Resolvent Operator Rs is defined as the Laplace Transform of the Expectation Operator:
Z ∞
£
¤
−st
(Rs f )(x) :=
e Ex 1{ζ>t} f (Xt ) dt
0
• Conversely, the Bromwich Laplace inversion formula reads:
Z ²+i∞
£
¤
1
Ex 1{ζ>t}f (Xt) =
est (Rs f )(x)ds
2πi ²−i∞
• Therefore, if we know the Resolvent we can solve for the
Expectation and vice versa.
• NOTICE. The time t enters in this expression only through
the exponential es t .This is crucial for time changes!
16
The Spectral Approach
• When X is a 1D diffusion, the generator G is a self-adjoint
2
operator (i.e., (Gu,
R v) = (u, Gv)) in L (D, m) with the inner
product (f, g) = D f (x)g(x)m(x)dx, where m(x) is the speed
density of the 1D diffusion (σ is the diffusion coefficient and
µ is the drift):
R 2µ(x)
2
dx
m(x) = 2
e σ2(x) .
σ (x)
• If f ∈ L2 (D, m), we can apply the Spectral Theorem to write
down the spectral representation for the expectation operator:
∞
X
¤
£
Ex 1{ζ>t}f (Xt) =
e−λn tcn ϕn (x),
n=1
where λn and ϕn are the eigenvalues and eigenfunctions of −G:
−Gϕ(x) = λn ϕ(x),
and the expansion coefficients are: cn = (f, ϕn ) . If the spectrum is continuous, the integral takes place of the sum.
17
Time Changing in the Resolvent Approach
• The expectation operator for the time changed process can
now be expressed in terms of the resolvent of the original process X and the Laplace transform of the time change T :
¯
¤¤
£
¤
£ £
Ex 1{ζ>Tt}f (XTt ) = Ex Ex 1{ζ>s}f (Xs)¯ Tt = s
·Z ²+i∞
¸
1
=
E
esTt (Rs f )(x)ds
2πi
²−i∞
Z ²+i∞
£ ¤
1
=
E esTt (Rsf )(x)ds
2πi ²−i∞
Z ²+i∞
1
=
L (t, −s) (Rs f )(x)ds.
2πi ²−i∞
18
Time Changing in the Spectral Approach
• If f ∈ L2 (D, m) and the spectrum of G is discrete, we can
write:
Ex
£
∞
X
¤
£
¤
1{ζ>Tt}f (XTt ) =
E e−λnTt cnϕn(x)
n=1
=
∞
X
L (t, λn ) cn ϕn (x),
n=1
where L (t, λn ) is the Laplace Transform of Tt. If the spectrum
is continuous, the integral takes place of the sum.
• For f ∈
/ L2 (D, m) one needs to use the Laplace transform
approach.
19
Lévy Subordinators
• A Lévy subordinator is a non-decreasing Lévy process {Tt , t ≥
0} with positive jumps and non-negative drift with Laplace
Transform (LT):
L (t, λ) = E[e−λTt ] = e−tφ(λ)
• with the Laplce exponent given by the Lévy-Khintchine formula:
Z
¡
¢
−λs
φ (λ) = γλ +
1−e
ν (ds)
(0,∞)
• Where
the drift γ ≥ 0, the Lévy measure ν (ds) satisfies
R
(s ∧ 1) ν (ds) < ∞, and the transition probability πt (ds)
(0,∞)
is:
Z
e−λs π (ds) = e−tφ(λ)
[0,∞)
20
Lévy Subordinators (Cont.)
• T starts at T0 , drifts at rate γ ≥ 0, and experiences positive
jumps controlled by ν (ds).
• ν (ds) describes the arrival rates of jumps so that jumps of sizes
in some Borel set A bounded away from zero
R occur according
to a Poisson process with intensity ν (A) = A ν (ds).
R
• If + ν (ds) < ∞, the subordinator is a compound Poisson proR
R
cess with intensity α = + ν (ds) and the jump size distribution
R
−1
is given by α ν.
• Otherwise, it is an infinite activity jump process (infinite number of jumps in any finite time interval).
21
Examples of Subordinators
• Lévy measure (3 parameter class):
ν(ds) = Cs−Y −1 e−ηs ds
C > 0, η > 0, Y < 1
Where Y controls the short term jumps, η the long term jumps
and C is a general scaling factor.
• For Y ∈ (0, 1): Tempered Stable subordinators
– Special case Y = 1/2: Inverse Gaussian process
– Limiting case Y = 0: Gamma process
22
Examples of Subordinators (Cont.)
• For Y 6= 0 the Laplace exponent is given by:
φ(λ) = γλ − CΓ(−Y )[(λ + η)Y − η Y ]
• For Y = 0:
φ(λ) = γλ + C ln(1 + λ/η)
• The processes with Y < 0 are compound Poisson processes
with gamma distributed jump sizes
– Special case with Y = −1: compound Poisson process
with exponential jumps:
αλ
ν(ds) = αηe−ηs ds, φ(λ) = γλ +
λ+η
• The processes with Y ∈ [0, 1) are infinite activity.
23
Inducing Jumps by Lévy Subordination
• If the original process Xt is a diffusion and Tt is a subordinator with Lévy measure ν, the time changed process Yt = XTt
acquires jumps with the state-dependent Lévy density:
Z ∞
π Y (x, y) =
p(t; x, y)ν(dt)
0
where p(t; x, y) is the transition probability density of the original process X from x to y in time t.
• The default intensity for the process Y is:
Z ∞
λY (x) = γλ(x) +
Pd (x, t)ν(dt)
0
where Pd (x, t) is the probability of default by time t if started
at x for the original process X.
24
Lévy Subordination Yt = XTt (Xt = Bt and
Tt = t + CP P )
Background Process X(t)
0.6
Time Process
2
0.5
1.8
X(t)
0.4
0.3
1.6
0.2
0.1
1.4
-0.1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Time T(t)
Time Changed Process Y(t)=X(T(t))
Time T(t)
0
1.2
1
0.8
0.6
0.6
Y(t)=X (T(t))
0.5
0.4
0.4
0.3
0.2
0.2
0.1
0
0
-0.1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
0
1.7
0.2 0.4 0.6 0.8
Time t
Time (t)
CPP parameters α = 4,1/η = 0.1
25
1
Absolutely Continuous Time Changes
• Let {Zt , t ≥ 0} be a n-dimensional Markov process independent
Rt
of X. Define Tt = 0 V (Zu )du, where V (z) is some positive
function. We are interested in analytically tractable LT:
Rt
−λ
V (Zu )du
0
Lz (t, λ) = Ez [e
]
• Example: The Cox-Ingersoll-Ross (CIR) process:
√
dVt = κ(θ − Vt)dt + σV VtdWt
with V0 = v > 0, rate of mean reversion κ > 0, long-run level
θ > 0, and volatility σV > 0. The CIR Laplace Transform (CIR
zero-coupon bond):
Rt
−λ
V (Zu )du
0
Lv (t, λ) = Ez [e
] = A(t, λ)e−B(t,λ)v
µ
A(t, λ) =
2$e($+κ)t/2
¶ 2κθ
σ2
V
,
($ + κ)(e$t − 1) + 2$
q
2λ(e$t − 1)
2
2
B(t, λ) =
2σ
λ
+
κ
,
$
=
V
($ + κ)(e$t − 1) + 2$
26
Absolutely Continuous Time Changes
√
κ = 4, θ = V0 = 2 and σv = 2
CIR Process
3
Time Process
V(t)
2.5
2
1
1.5
0.9
1
0.8
0.5
0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Time (t)
Processes X(t) vs Y(t)=X(T(t))
0.5
Time T(t)
0
0.6
0.5
0.4
X(t)
0.3
X(T(t))
0.2
0.4
0.1
0.3
0
0
0.2
0.1
0.2 0.3
0.4
0.5
Time t
0.1
0
-0.1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Time (t)
27
0.6 0.7
0.8
0.9
1
Composite Time Changes
• Consider a composite time change process:
Tt = TT1t2
where Tt1 is a subordinator with Laplace exponent φ and Tt2
is an integral of some positive function of a Markov process
with LT Lz (t, λ).
• T is obtained by time changing a Lévy subordinator T 1 with
an AC time change T 2 .
• Conditioning on Tt2 , we have:
E[e−λTt ] = E[e−Tt φ(λ)] = Lz (t, φ(λ))
2
• Composing Levy and A.C. time changes induces both jumps
and stochastic volatility!
28
A Credit-Equity Model
• Model defaultable stock:
St = 1{t<τd } eρt XTt .
• X is a JDCEV:
dXt = [µ + λ(Xt)]Xt dt + σ(Xt )Xt dBt, X0 = x > 0,
σ(x) = axβ , λ(x) = b + c σ 2 (x) = b + c a2 x2β .
¯R
o
¯ t
• ζ is the lifetime of X. ζ = inf t ∈ [0, T0 ] ¯ 0 λ (Xu ) du ≥ e ,
e ≈ Exp(1)
n
• Tt is either a subordinator, an A.C., or a composite time
change.
• The default time τd is:
τd = inf{t ≥ 0 : ζ ≤ Tt}.
At τd the stock jumps to zero and stays there (default).
29
A Credit-Equity Model (Cont.)
• Intensity λ has to be added in the drift of X to compensate
for jump to zero, and ρ and µ are parameters to be selected
to make the discounted process into a martingale:
E[St2 |Ft1 ] = e(r−q)(t2−t1)St1 , t1 ≤ t2,
where r and q are the risk-free rate and dividend yield.
• If Tt is a subordinator, then µ can be arbitrary and
ρ = r − q + φ(−µ).
• If Tt is an A.C. time change, then
µ = 0, ρ = r − q.
30
Pricing Corporate Bonds
• Survival probability:
1
Q(τd > t) = Q(ζ > Tt) =
2πi
Z
ε+i∞
L(t, −s)(Rs 1)(x)ds,
ε−i∞
where Q(ζ > t) is the survival probability of X with lifetime ζ.
• A defaultable zero-coupon bond with unit face value, maturity
t > 0, and recovery R ∈ [0, 1]:
BR (x, t) = e−rt Q(τd > t) + Re−rt [1 − Q(τd > t)]
= e−rt R + e−rt (1 − R)Q(τd > t).
• In this case the function f (x) = 1 is NOT in L2 (D, m) thus
we need to use the Resolvent Approach!
31
Resolvent
• In this model the resolvent is available in closed form:
Z ∞
Rs f (x) =
Gs (x, y)f (y)dy
0
with the resolvent kernel (for µ + b > 0):
Γ(ν/2 + 1/2 − k(s))
Gs (x, y) =
(µ + b)Γ(1 + ν)y
µ ¶c+1/2−β
x
−2β
−2β
eA(y −x )
y
×Mk(s), 2ν (A(x ∧ y)−2β )Wk(s), 2ν (A(x ∨ y)−2β )
where
1 + 2c
ν−1
s
µ+b
ν=
, k(s) =
−
, A= 2 ,
2|β|
2
2|β|(µ + b)
a |β|
and Mk,m (z) and Wk,m (z) are the first and second Whittaker
functions (related to the Kummer and Tricomi confluent hypergeometric functions).
• Analytical tractability is due to a connection of the JDCEV
process to Bessel processes.
32
Computing the Survival Probability
• In this case the Laplace transform can be inverted analytically using the Cauchy Residue Theorem, yielding the survival
probability for the process X:
Q(ζ > t) =
∞
X
e−(b+ωn)t
n=0
1
×A 2|β| xe−Ax
−2β
1 F1 (1
Γ(1 + c/|β|)Γ(n + 1/(2|β|))
Γ(ν + 1)Γ(1/(2|β|))n!
− n + c/|β|, ν + 1, Ax−2β ),
where 1 F1 (a; b; x) is the Kummer confluent hypergeometric
function and ω = 2|β|(µ + b).
• To obtain the survival probability for the time changed process
XTt , just replace e−(b+ωn)t with L(t, b + ωn)!
33
Pricing Stock Options
• A call option with payoff (St − K)+ at t has no recovery if the
firm defaults:
C(x; K, t) = e−rt Ex[(eρt XTt − K)+ 1{τd >t} ].
• A put option with payoff (K − St)+ can be decomposed into
two parts: the put payoff (K − St)+1{τd >t} , given no default by
t, and a recovery payment equal to the strike K in the event
of default τd ≤ t:
P (x; K, t) = P0 (x; K, t) + PD (x; K, t),
P0 (x; K, t) = e−rt Ex[(K − eρt XTt )+1{τd >t} ],
PD (x; K, t) = Ke−rt [1 − Q(τd > t)].
• The recovery part of the put is a credit derivative that pays a
fixed cash amount K at maturity t if and only if the underlying
firm has defaulted by t. The put option contains an embedded
credit derivative ⇒ UNIFIED CREDIT-EQUITY!
34
Pricing Put Options
The put payoff is in L2 , and the spectral expansion is (k := e−ρt K):
∞
X
P0 (x; K, t) = e(ρ−r)tEx [(k − XTt )+ 1{τd >t} ] =
L(t, λn )cn ϕn (x),
n=1
with the eigenvalues, eigenfunctions, and expansion coefficients
(for µ + b > 0):
λn = ωn + 2c(µ + b) + b, ω = 2|β|(µ + b),
s
(n − 1)!(µ + b)(2c + 1) −Ax−2β (ν)
xe
Ln−1 (Ax−2β ),
Γ(ν + n)
p
ν/2+1
2c+1−2β
A
k
Γ(ν + n)
p
cn =
Γ(ν + 1) (µ + b)(2c + 1)(n − 1)!
¶
¾
½
µ
c
¢
¡
1 − n, |β|
+1
Γ(ν
+
1)(n
−
1)!
|β|
,
×
; Ak−2β −
Ak−2β
Lν+1
2 F2
c
n−1
ν + 1, |β| + 2
c + |β|
Γ(ν + n + 1)
ν
ϕn (x) = A 2
where 2F2 is the generalized hypergeometric function and Ln(ν) are
generalized Laguerre polynomials.
35
Numerical Results
• Assume a background process {Xt, t > 0} following a JDCEV,
and a composite time change Inverse Gaussian Process + CIR
with the following parameters:
JDCEV
S
a
β
c
b
r
q
50
10
−1
0.5
0.01
0.05
0
CIR
IG
V
θ
σV
κ
γ
η
C
1
1
1
4
0
p8
2 2/π
Note that γ = 0, thus the time changed process is a pure
jump process!.
36
Implied Volatility
Implied Volatility
61%
1/4
Implied Volatility
1/2
1
49%
2
3
37%
25%
13%
30
35
40
45
50
55
45
26.19
26.39
27.53
29.61
31.34
50
21.41
22.64
24.30
26.72
28.64
60
65
Strike
Time/Strike
1/4
1/2
1
2
3
30
62.04
51.94
45.74
43.03
42.80
35
47.94
41.47
38.24
37.68
38.34
40
35.52
32.72
32.14
33.23
34.55
55
20.09
20.72
22.12
24.45
26.39
60
20.28
19.84
20.65
22.66
24.52
65
20.88
19.46
19.64
21.25
22.96
Implied volatility smile/skew curves as functions of the strike price. Current
stock price level is 50, local volatility σ = aS β = 20%
37
Credit Spreads and Default Probability
Credit Spreads
Default Probability
80.0%
8.5%
70.0%
30
7.5%
Probability of default
Credit spreads
40
6.5%
50
60
5.5%
70
4.5%
3.5%
60.0%
50.0%
40.0%
30
40
30.0%
50
20.0%
2.5%
60
70
10.0%
1.5%
0.0%
0
5
10
15
20
25
30
Time to maturity (years)
35
40
45
50
0
5
10
15
20
25
30
Time to maturity (years)
35
40
45
Credit spreads and default probabilities as functions of time to
maturity for current stock price levels S = 30, 40, 50, 60, 70.
38
50
Killing Rate
Killing Rate (k)
Killing Rate (k)
1.2
28
1
25
22
0.8
JDCEV
16
k(S,V)
k(S,V)
19
CEV
JDCEV
0.6
CEV
13
0.4
10
7
0.2
4
1
0
1
1.5
2
2.5
3
3.5
4
Stock Price (S)
4.5
5
5.5
6
6.5
7
7
11
15
19
23
27
31
35
39
43
Stock Price (S)
Killing rate (default intensity) k(S,V) as a function of the stock
price S when the activity rate is fixed at V = 1. The solid line is
the default intensity in the model obtained by time changing the
JDCEV process. The dashed line is the default intensity obtained
by time changing the standard CEV process with b = c = 0 without
the jump to default
39
47
Conclusions
• Starting from an analytically tractable diffusion Xt and an
increasing process Tt with known Laplace transform, the time
changed process XTt is also analytically tractable.
• Time changing diffusions with Lévy subordinators induces jumps,
while time changing with A.C. time changes induces stochastic volatility and composite time changes induce both.
• By pairing analytically tractable diffusions with killing with analytically tractable time changes we are able to generate analytically tractable Credit-Equity models with jumps, stochastic
volatility, and default.
• These Hybrid credit-equity models feature linkages between
corporate credit spreads in the credit markets and implied
volatility skews in the stock options markets.
40
THANK YOU!
41