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 2009
1/1
Single – Firm
Multi – Firm
Research Projects
Calendar Time
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Time Changes
Credit Risk 2009
2/1
Multi – Firm
Research Projects
Single – Firm
The Constant
Elasticity of Variance
Model
Calendar Time
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Time Changes
Credit Risk 2009
2/1
Multi – Firm
Research Projects
Single – Firm
The Constant
Elasticity of Variance
Model
Equity Default Swaps
under the JDCEV
process
Calendar Time
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Time Changes
Credit Risk 2009
2/1
Multi – Firm
Research Projects
Single – Firm
The Constant
Elasticity of Variance
Model
Equity Default Swaps
under the JDCEV
process
Calendar Time
Rafael Mendoza (McCombs)
Time Changed Markov
Processes in Unified
Credit-Equity Modeling
Unified Credit-Equity Modeling
Time Changes
Credit Risk 2009
2/1
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
Calendar Time
Rafael Mendoza (McCombs)
Time Changed Markov
Processes in Unified
Credit-Equity Modeling
Unified Credit-Equity Modeling
Time Changes
Credit Risk 2009
2/1
Literature Review
Stock Option Pricing Literature
Black-Scholes
(geometric Brownian motion)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
3/1
Literature Review
Stock Option Pricing Literature
Black-Scholes
(geometric Brownian motion)
Infinite lifetime process
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
3/1
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 2009
3/1
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 2009
3/1
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 2009
3/1
Literature Review
Stock Option Pricing Literature
Black-Scholes
Subsequent
Generations of Models
(geometric Brownian motion)
(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 2009
3/1
Literature Review
Stock Option Pricing Literature
Black-Scholes
Subsequent
Generations of Models
(geometric Brownian motion)
(modeling the
volatility smile)
Infinite lifetime process
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 2009
3/1
Literature Review
Stock Option Pricing Literature
Black-Scholes
Subsequent
Generations of Models
(geometric Brownian motion)
(modeling the
volatility smile)
Infinite lifetime process
Local Volatility
No possibility
of Bankruptcy!
Stochastic Volatility
(CEV, Dupier, etc.)
(Heston, SABR, etc.)
Constant volatility
and no jumps
No volatility smiles!
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
3/1
Literature Review
Stock Option Pricing Literature
Black-Scholes
Subsequent
Generations of Models
(geometric Brownian motion)
(modeling the
volatility smile)
Infinite lifetime process
Local Volatility
No possibility
of Bankruptcy!
Stochastic Volatility
(CEV, Dupier, etc.)
(Heston, SABR, etc.)
Jump Diffusion
Constant volatility
and no jumps
(Merton, Kou, etc.)
No volatility smiles!
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
3/1
Literature Review
Stock Option Pricing Literature
Black-Scholes
Subsequent
Generations of Models
(geometric Brownian motion)
(modeling the
volatility smile)
Infinite lifetime process
Local Volatility
No possibility
of Bankruptcy!
Stochastic Volatility
(CEV, Dupier, etc.)
(Heston, SABR, etc.)
Jump Diffusion
Constant volatility
and no jumps
No volatility smiles!
(Merton, Kou, etc.)
Pure Jump Models
Based on
Levy processes
(VG, NIG, CGMY, etc.)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
3/1
Literature Review
Stock Option Pricing Literature
Black-Scholes
Subsequent
Generations of Models
tcy
(geometric Brownian motion)
(modelingpthe
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k
volatility
an
fb
yo
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:
i
l
Infinite lifetime processem ibi Local Volatility
l
ob ss firm(CEV, Dupier, etc.)
e.
Pr he po ying
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rl
No possibility e
v
e
a
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or und
Stochastic
s h lt in Volatility
of Bankruptcy!
ign e
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(Merton, Kou, etc.)
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Constant volatility
ive
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i
s
and noojumps
Pure Jump Models
p
No volatility smiles!
Based on
Levy processes
(VG, NIG, CGMY, etc.)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
3/1
Literature Review
Stock Option Pricing Literature
Black-Scholes
Subsequent
Generations of Models
(geometric Brownian motion)
(modeling the
volatility smile)
Infinite lifetime process
Local Volatility
No possibility
of Bankruptcy!
Stochastic Volatility
Credit Risk Literature
Reduced Form
Framework
(CEV, Dupier, etc.)
(Heston, SABR, etc.)
Jump Diffusion
Constant volatility
and no jumps
No volatility smiles!
(Merton, Kou, etc.)
Pure Jump Models
Based on
Levy processes
(VG, NIG, CGMY, etc.)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
3/1
Literature Review
Stock Option Pricing Literature
Black-Scholes
Subsequent
Generations of Models
(geometric Brownian motion)
(modeling the
volatility smile)
Infinite lifetime process
Local Volatility
No possibility
of Bankruptcy!
Stochastic Volatility
Credit Risk Literature
Reduced Form
Framework
Default Intensity Models
(CEV, Dupier, etc.)
(Heston, SABR, etc.)
Jump Diffusion
Constant volatility
and no jumps
No volatility smiles!
Since Duffie & Singleton,
Jarrow, Lando &
Turnbull:
A vast amount
of research has been
developed
(Merton, Kou, etc.)
Pure Jump Models
Based on
Levy processes
(VG, NIG, CGMY, etc.)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
3/1
Literature Review
Stock Option Pricing Literature
Black-Scholes
Subsequent
Generations of Models
(geometric Brownian motion)
(modeling the
volatility smile)
Infinite lifetime process
Local Volatility
No possibility
of Bankruptcy!
Stochastic Volatility
Credit Risk Literature
Reduced Form
Framework
Default Intensity Models
(CEV, Dupier, etc.)
(Heston, SABR, etc.)
Jump Diffusion
Constant volatility
and no jumps
No volatility smiles!
A vast amount
of research has been
developed
(Merton, Kou, etc.)
Pure Jump Models
Based on
Levy processes
(VG, NIG, CGMY, etc.)
Rafael Mendoza (McCombs)
Since Duffie & Singleton,
Jarrow, Lando &
Turnbull:
Unified Credit-Equity Modeling
Modeling Focus:
Credit Default Events,
Credit Spreads,
Credit Derivatives, etc
Credit Risk 2009
3/1
Literature Review
Stock Option Pricing Literature
Black-Scholes
Subsequent
Generations of Models
(geometric Brownian motion)
(modeling the
volatility smile)
Infinite lifetime process
Local Volatility
No possibility
of Bankruptcy!
Stochastic Volatility
Credit Risk Literature
Reduced Form
Framework
Default Intensity Models
(CEV, Dupier, etc.)
: els f
od n&oSingleton,
lem mDuffie
Since
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o s
t
o
ati Lando
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rm ke
ar
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se
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(Merton, Kou, etc.)
ls
de
MoModeling Focus:
y
Pure Jump Models
t
ui
Based on
Eq
(Heston, SABR, etc.)
Constant volatility
and no jumps
No volatility smiles!
Levy processes
(VG, NIG, CGMY, etc.)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Default Events,
Credit Spreads,
Credit Derivatives, etc
Credit Risk 2009
3/1
Literature Review
Stock Option Pricing Literature
Credit Risk Literature
Black-Scholes
Subsequent
Generations of Models
(geometric Brownian motion)
(modeling the
volatility smile)
Infinite lifetime process
Local Volatility
No possibility
of Bankruptcy!
Stochastic Volatility
Reduced Form
Framework
Default Intensity Models
(CEV, Dupier, etc.)
(Heston, SABR, etc.)
Jump Diffusion
Constant volatility
and no jumps
No volatility smiles!
Since Duffie & Singleton,
Jarrow, Lando &
Turnbull:
A vast amount
of research has been
developed
(Merton, Kou, etc.)
Pure Jump Models
Based on
Levy processes
Modeling Focus:
Credit Default Events,
Credit Spreads,
Unified
Credit Derivatives, etc
Credit-Equity
Modeling
(VG, NIG, CGMY, etc.)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
3/1
Motivating Example
2 weeks before bankruptcy (9/02/2008) Lehman Brothers (LEH)
stock price price was $16.13
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)
200%
Implied Volatility
180%
160%
140%
120%
100%
80%
60%
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
Moneyness (K/S)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
4/1
Motivating Example
2 weeks before bankruptcy (9/02/2008) Lehman Brothers (LEH)
stock price price was $16.13
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)
200%
Implied Volatility
180%
160%
140%
120%
100%
80%
60%
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
Moneyness (K/S)
The stock price drop of 72% from the high $62.19 to $16.13!
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
4/1
Motivating Example
2 weeks before bankruptcy (9/02/2008) Lehman Brothers (LEH)
stock price price was $16.13
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)
200%
Implied Volatility
180%
160%
140%
120%
100%
80%
60%
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
Moneyness (K/S)
The stock price drop of 72% from the high $62.19 to $16.13!
Open Interest on Put contracts with strike prices K = 2.5 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 2009
4/1
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 2009
5/1
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 2009
5/1
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 2009
5/1
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 2009
6/1
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 2009
6/1
Our Contributions
We introduce a new analytically tractable class of credit-equity
models.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
7/1
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 2009
7/1
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 2009
7/1
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 2009
7/1
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 V arrival rates of large jumps increase
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
7/1
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 V arrival rates of large jumps increase
As stock price rises V arrival rate of large jumps decrease
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
7/1
Our Contributions (cont.)
In our model architecture, time changes of diffusions have the following
effects:
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
8/1
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 2009
8/1
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 2009
8/1
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 2009
9/1
Jump to Default Extended Diffusions (JDED)
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
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 2009
10 / 1
Jump to Default Extended Diffusions (JDED)
.
Stock Price
..
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
0, ζ ≤ t
.
.
100
S(t)
80
60
40
20
(ζ default time)
0
0.0
0.5
1.0
Time (yrs)
1.5
.
..
Model the pre-default stock dynamics under an EMM Q as:
d S̃t = [ µ + h(S̃t ) ]S̃t dt + σ(S̃t ) S̃t dBt
|{z}
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
2.0
.
.
.
.
.
..
10 / 1
Jump to Default Extended Diffusions (JDED)
.
Stock Price
..
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
0, ζ ≤ t
.
.
100
S(t)
80
60
40
20
(ζ default time)
0
0.0
0.5
1.0
Time (yrs)
1.5
.
..
Model the pre-default stock dynamics under an EMM Q as:
d S̃t = [ µ + h(S̃t ) ]S̃t dt + σ(S̃t ) S̃t dBt
|{z}
2.0
.
.
.
.
.
..
⇒ µ = r − q. Drift: short rate r minus the dividend yield q
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
10 / 1
Jump to Default Extended Diffusions (JDED)
.
Stock Price
..
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
0, ζ ≤ t
.
.
100
S(t)
80
60
40
20
(ζ default time)
0
0.0
0.5
1.0
Time (yrs)
1.5
.
..
Model the pre-default stock dynamics under an EMM Q as:
d S̃t = [ µ + h(S̃t ) ]S̃t dt + σ(S̃t ) S̃t dBt
|{z}
| {z }
2.0
.
.
.
.
.
..
⇒ σ(S). State dependent volatility
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
10 / 1
Jump to Default Extended Diffusions (JDED)
.
Stock Price
..
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
0, ζ ≤ t
.
.
100
S(t)
80
60
40
20
(ζ default time)
0
0.0
0.5
1.0
Time (yrs)
1.5
.
..
Model the pre-default stock dynamics under an EMM Q as:
d S̃t = [ µ + h(S̃t ) ]S̃t dt + σ(S̃t ) S̃t dBt
|{z}
| {z }
2.0
.
.
.
.
.
..
⇒ h(S). State dependent default intensity
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
10 / 1
Jump to Default Extended Diffusions (JDED)
.
Stock Price
..
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
0, ζ ≤ t
.
.
100
S(t)
80
60
40
20
(ζ default time)
0
0.0
0.5
1.0
Time (yrs)
1.5
.
..
Model the pre-default stock dynamics under an EMM Q as:
d S̃t = [ µ + h(S̃t ) ]S̃t dt + σ(S̃t ) S̃t dBt
|{z}
| {z }
2.0
.
.
.
.
.
..
⇒ 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 2009
10 / 1
Jump to Default Extended Diffusions (JDED)
.
Stock Price
..
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
0, ζ ≤ t
.
.
100
S(t)
80
60
40
20
(ζ default time)
0
0.0
.
..
0.5
1.0
Time (yrs)
1.5
2.0
.
.
.
.
.
..
If the diffusion S̃t can hit zero:
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
11 / 1
Jump to Default Extended Diffusions (JDED)
.
Stock Price
..
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
0, ζ ≤ t
.
.
100
S(t)
80
60
40
20
τ0
.
0
0.0
0.5
.
..
1.0
Time (yrs)
1.5
2.0
.
.
(ζ default time)
.
.
..
If the diffusion S̃t can hit zero:
V Bankruptcy at the first hitting time of zero,
{
}
τ0 = inf t : S̃t = 0
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
11 / 1
Jump to Default Extended Diffusions (JDED)
.
Stock Price
..
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
0, ζ ≤ t
.
100
S(t)
80
60
40
.
.
(ζ default time)
0
0.0
0.5
1.0
Time (yrs)
τ0
1.5
2.0
.
..
.
Prior to τ0 default could also arrive by a jump-to-default ζ̃ with
default intensity h(S̃),
{
}
∫t
ζ̃ = inf t ∈ [0, τ0 ] : 0 h(S̃u ) ≥ e , e ≈ Exp(1)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
ζ̃
20
.
..
.
12 / 1
Jump to Default Extended Diffusions (JDED)
.
Stock Price
..
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
0, ζ ≤ t
.
100
S(t)
80
60
40
ζ̃
20
(ζ default time)
.
.
.
..
.
0
0.0
0.5
1.0
Time (yrs)
τ0
1.5
2.0
.
.
..
.
Prior to τ0 default could also arrive by a jump-to-default ζ̃ with
default intensity h(S̃),
{
}
∫t
ζ̃ = inf t ∈ [0, τ0 ] : 0 h(S̃u ) ≥ e , e ≈ Exp(1)
V At time ζ̃ the stock price St jumps to zero and the firm defaults
on its debt
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
12 / 1
Jump to Default Extended Diffusions (JDED)
.
Stock Price
..
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
0, ζ ≤ t
.
100
S(t)
80
60
40
.
.
(ζ default time)
0
0.0
.
..
The default time ζ is the earliest of:
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
0.5
1.0
Time (yrs)
τ0
1.5
2.0
.
Credit Risk 2009
.
ζ̃
20
.
..
.
13 / 1
Jump to Default Extended Diffusions (JDED)
.
Stock Price
..
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
0, ζ ≤ t
.
100
S(t)
80
60
40
.
.
0
0.0
.
..
The default time ζ is the earliest of:
1
0.5
1.0
Time (yrs)
τ0
1.5
2.0
.
.
ζ̃
20
(ζ default time)
.
..
.
The stock hits level zero by diffusion: τ0
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
13 / 1
Jump to Default Extended Diffusions (JDED)
.
Stock Price
..
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
0, ζ ≤ t
.
100
S(t)
80
60
40
.
.
0
0.0
.
..
The default time ζ is the earliest of:
1
2
0.5
1.0
Time (yrs)
τ0
1.5
2.0
.
.
ζ̃
20
(ζ default time)
.
..
.
The stock hits level zero by diffusion: τ0
The stock jumps to zero from a positive value: ζ̃
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
13 / 1
Jump to Default Extended Diffusions (JDED)
.
Stock Price
..
.
Defaultable Stock Price
..
{
S̃t , ζ > t
St =
0, ζ ≤ t
.
100
Default Time
ζ:
ζ = min ζ̃ , τ0
S(t)
80
60
40
.
.
0
0.0
.
..
The default time ζ is the earliest of:
1
2
0.5
1.0
Time (yrs)
τ0
1.5
2.0
.
.
ζ̃
20
(ζ default time)
.
..
.
The stock hits level zero by diffusion: τ0
The stock jumps to zero from a positive value: ζ̃
)
(
ζ = min ζ̃, τ0
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
13 / 1
Contingent Claims
.
Risk Neutral Survival Probability (no default by time T)
[
Q (S, t; T ) = E 1{ζ>T }
[ R
]
T
h(Su )du
= E e| − t {z
1
0 >T }
} | {τ{z
}
(
)
Recall: Default time ζ = min ζ̃, τ0 .
.
..
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
.
]
.
Credit Risk 2009
.
..
14 / 1
Contingent Claims
.
Risk Neutral Survival Probability (no default by time T)
[
Q (S, t; T ) = E 1{ζ>T }
[ R
]
T
h(Su )du
= E e| − t {z
1
0 >T }
} | {τ{z
}
(
)
Recall: Default time ζ = min ζ̃, τ0 .
.
..
1
.
]
.
.
..
No jump-to-default before maturity T,
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
14 / 1
Contingent Claims
.
Risk Neutral Survival Probability (no default by time T)
[
Q (S, t; T ) = E 1{ζ>T }
[ R
]
T
h(Su )du
= E e| − t {z
1
0 >T }
} | {τ{z
}
(
)
Recall: Default time ζ = min ζ̃, 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 2009
.
..
14 / 1
Contingent Claims
.
Defaultable Zero Coupon Bond (at time t)
.
B (S, t; T ) = e −r (T −t) Q (S, t; T ) + e −r (T −t) R [1 − Q (S, t; T )]
|
{z
}
|
{z
}
Disc. Dollar if
No Default occurs
prior to maturity
Disc. recovery R ∈ [0, 1]
if Default occurs
before maturity T
Recall:Q (S, t; T ) is the risk neutral survival probability
.R is a fraction of a dollar paid at maturity.
..
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
.
Credit Risk 2009
.
..
15 / 1
Contingent Claims
.
Defaultable Zero Coupon Bond (at time t)
.
B (S, t; T ) = e −r (T −t) Q (S, t; T ) + e −r (T −t) R [1 − Q (S, t; T )]
|
{z
}
|
{z
}
Disc. Dollar if
No Default occurs
prior to maturity
Disc. recovery R ∈ [0, 1]
if Default occurs
before maturity
Recall:Q (S, t; T ) is the risk neutral survival probability
.R is a fraction of a dollar paid at maturity.
..
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
.
Credit Risk 2009
.
..
15 / 1
Contingent Claims
.
Defaultable Zero Coupon Bond (at time t)
.
B (S, t; T ) = e −r (T −t) Q (S, t; T ) + e −r (T −t) R [1 − Q (S, t; T )]
|
{z
}
|
{z
}
Disc. Dollar if
No Default occurs
prior to maturity
Disc. recovery R ∈ [0, 1]
if Default occurs
before maturity
Recall:Q (S, t; T ) is the risk neutral survival probability
.R is a fraction of a dollar paid at maturity.
..
.
.
..
Defaultable bonds with coupons are valued as portfolios of
zero-coupon bonds
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
15 / 1
Contingent Claims
.
Defaultable Zero Coupon Bond (at time t)
.
B (S, t; T ) = e −r (T −t) Q (S, t; T ) + e −r (T −t) R [1 − Q (S, t; T )]
|
{z
}
|
{z
}
Disc. Dollar if
No Default occurs
prior to maturity
Disc. recovery R ∈ [0, 1]
if Default occurs
before maturity
Recall:Q (S, t; T ) is the risk neutral survival probability
.R is a fraction of a dollar paid at maturity.
..
.
.
..
Defaultable bonds with coupons are valued as portfolios of
zero-coupon bonds
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
.
.
.
Call Option
..
[ RT
]
C (S, t; K , T ) = e −r (T −t) E e − t h(Su )du (ST − K )+ 1{τ0 >T }
.
..
15 / 1
Contingent Claims
.
Put Payoff (Strike Price K > 0)
.
..
(K − ST )+ 1{ζ>T } +
|
{z
}
Put Payoff
given no default
by time T
.
K 1{ζ≤T }
| {z }
Recovery amount K
if default occurs
before maturity T
.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
..
16 / 1
Contingent Claims
.
Put Payoff (Strike Price K > 0)
.
..
(K − ST )+ 1{ζ>T } +
|
{z
}
Put Payoff
given no default
by time T
.
K 1{ζ≤T }
| {z }
Recovery amount K
if default occurs
before maturity T
.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
..
16 / 1
Contingent Claims
.
Put Payoff (Strike Price K > 0)
Put Payoff
given no default
by time T
K 1{ζ≤T }
| {z }
Recovery amount K
if default occurs
before maturity T
.
.
Put Option Price
..
[ RT
]
P (S, t; K , T ) = e −r (T −t) E e − t h(Su )du (K − ST )+ 1{τ0 >T }
+ Ke −r (T −t) [1 − Q (S, t; T )]
.
..
.
.
..
(K − ST )+ 1{ζ>T } +
|
{z
}
.
.
.
.
..
NOTE. A default claim is embedded in the Put Option
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
16 / 1
Jump-to-Default Extended Constant Elasticity of
Variance (JDCEV) Model
.
The JDCEV process (Carr and Linetsky (2006))
.
..
.
..
σ(S) = aS β
h(S) = b + c σ 2 (S)
CEV Volatility
(Power function of S)
Default Intensity
(Affine function of Variance)
.
a>0
β<0
b≥0
c≥0
.
dSt = [µ + h(St )]St dt + σ(St )St dBt , S0 = S > 0
⇒ 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 2009
17 / 1
Jump-to-Default Extended Constant Elasticity of
Variance (JDCEV) Model
.
The JDCEV process (Carr and Linetsky (2006))
.
..
.
..
σ(S) = aS β
h(S) = b + c σ 2 (S)
CEV Volatility
(Power function of S)
Default Intensity
(Affine function of Variance)
.
.
dSt = [µ + h(St )]St dt + σ(St )St dBt , S0 = S > 0
The model is consistent with:
leverage effect V S ⇓→ σ(S) ⇑
stock volatility–credit spreads linkage V σ(S) ⇑↔ h(S) ⇑
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
17 / 1
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 2009
18 / 1
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 2009
19 / 1
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 2009
19 / 1
Equity Default Swaps (EDS)
Credit-Type Instrument to bring protection in case of a Credit Event
Credit Events:
1
2
Reference Entity Defaults
Reference Stock Price drops significantly (L = 30%S0 )
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
19 / 1
Equity Default Swaps (EDS)
Credit-Type Instrument to bring protection in case of a Credit Event
Credit Events:
1
2
Reference Entity Defaults
Reference Stock Price drops significantly (L = 30%S0 )
Similar to CDS
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
19 / 1
Equity Default Swaps (EDS)
Credit-Type Instrument to bring protection in case of a Credit Event
Credit Events:
1
2
Reference Entity Defaults
Reference Stock Price drops significantly (L = 30%S0 )
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 2009
19 / 1
Equity Default Swaps (EDS)
Credit-Type Instrument to bring protection in case of a Credit Event
Credit Events:
1
2
Reference Entity Defaults
Reference Stock Price drops significantly (L = 30%S0 )
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 2009
19 / 1
Equity Default Swaps (EDS)
Equity Default Swap
(EDS)
Protection
Payment
Protection Seller
t
T
Hitting Level
Hits Level (L)
(L)
100
80
S(t)
60
40
20
L
Protection Buyer
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (yrs)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
20 / 1
Equity Default Swaps (EDS)
Equity Default Swap
(EDS)
Protection
Payment
Default Event
(or)
Hitting Level
(L)
100
80
60
S(t)
Hits Level (L)
Or
Default Occurs
Protection Seller
t
T
40
20
L
Protection Buyer
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (yrs)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
20 / 1
Equity Default Swaps (EDS)
t
Protection
Payment
T
Default Event
(or)
Hitting Level
(L)
100
80
60
S(t)
Premium Payments
+
Accrued Interest
Protection Seller
Hits Level (L)
Or
Default Occurs
∆t
Premium
Payment
Equity Default Swap
(EDS)
40
20
L
Protection Buyer
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (yrs)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
20 / 1
Equity Default Swaps (EDS)
t
Protection
Payment
T
Default Event
(or)
Hitting Level
(L)
100
80
60
S(t)
Premium Payments
+
Accrued Interest
Hits Level (L)
Or
Default Occurs
Protection Seller
Acc.
Interest
∆t
Premium
Payment
Equity Default Swap
(EDS)
40
20
L
Protection Buyer
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (yrs)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
20 / 1
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 V TL∆ = min{first hitting time to L, Default Time}
PV(Protection Payment)
PV(Premium Payments)
PV(Accrued Interests)
[
]
∆
(1 − r) · E e −r · TL 1{TL∆ ≤ T }
[
]
∑N
ϱ · ∆t · i=1 e −r · ti E 1{TL∆ ≥ ti }
(
]
[
[ ∆ ])
TL
−r · TL∆
∆
ϱ·E e
TL − ∆t · ∆t
1{TL∆ ≤ 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 2009
21 / 1
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 2009
22 / 1
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 2009
22 / 1
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 2009
22 / 1
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 2009
23 / 1
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 2009
23 / 1
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 2009
24 / 1
General Panorama
Continuous
Markov Process
w/ Default Intensity
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
25 / 1
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 2009
25 / 1
General Panorama
Time Changes
Continuous
Markov Process
w/ Default Intensity
Bochner
Levy Subordination
Levy
Subordination
&
Absolute Continuous
Time Changes
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Absolute
Continuous
Time Changes
Credit Risk 2009
25 / 1
General Panorama
Time Changes
Continuous
Markov Process
w/ Default Intensity
Bochner
Levy Subordination
Levy
Subordination
&
Jump-Diffusion
Process w/
Stochastic Volatility
Default Intensity
Rafael Mendoza (McCombs)
Absolute Continuous
Time Changes
Unified Credit-Equity Modeling
Absolute
Continuous
Time Changes
Credit Risk 2009
25 / 1
General Panorama
Time Changes
Continuous
Markov Process
w/ Default Intensity
Bochner
Levy Subordination
Levy
Subordination
&
Jump-Diffusion
Process w/
Stochastic Volatility
Default Intensity
Absolute Continuous
Time Changes
Absolute
Continuous
Time Changes
Analytical Unified Credit and Equity Option Pricing Formulas
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
25 / 1
General Panorama
Time Changes
Continuous
Markov Process
w/ Default Intensity
Bochner
Levy Subordination
Levy
Subordination
&
Jump-Diffusion
Process w/
Stochastic Volatility
Default Intensity
Absolute Continuous
Time Changes
Absolute
Continuous
Time Changes
Analytical Unified Credit and Equity Option Pricing Formulas
f (x) ∈
/ L2
Laplace Transform
Approach
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
25 / 1
General Panorama
Time Changes
Continuous
Markov Process
w/ Default Intensity
Bochner
Levy Subordination
Levy
Subordination
&
Jump-Diffusion
Process w/
Stochastic Volatility
Default Intensity
Absolute
Continuous
Time Changes
Absolute Continuous
Time Changes
Analytical Unified Credit and Equity Option Pricing Formulas
f (x) ∈
/ L2
Laplace Transform
Approach
Rafael Mendoza (McCombs)
f (x) ∈ L2
Spectral Expansion
Approach
Unified Credit-Equity Modeling
Credit Risk 2009
25 / 1
Time-Changed Process Yt = XTt
.
Time Changed Process Construction
..
Yt = XTt
.
.
..
Tt is a random clock process independent of Xt
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
.
Credit Risk 2009
.
Xt is a background process (e.g. JDCEV)
26 / 1
Time-Changed Process Yt = XTt
.
Time Changed Process Construction
..
Yt = XTt
.
Tt is a random clock process independent of Xt
.
Random Clock {Tt , t ≥ 0}
.
..
Non-decreasing RCLL process starting at T0 = 0 and E [Tt ] < ∞.
.
..
We are interested in T.C. with analytically
[ −λT ]tractable Laplace Transform (LT):
L(t, λ) = E e
Rafael Mendoza (McCombs)
t
<∞
Unified Credit-Equity Modeling
.
Credit Risk 2009
.
.
..
.
.
Xt is a background process (e.g. JDCEV)
26 / 1
Time-Changed Process Yt = XTt
.
Time Changed Process Construction
..
Yt = XTt
.
Tt is a random clock process independent of Xt
.
..
.
.
Random Clock {Tt , t ≥ 0}
.
Xt is a background process (e.g. JDCEV)
.
..
Non-decreasing RCLL process starting at T0 = 0 and E [Tt ] < ∞.
L(t, λ) = E e
.
..
1
t
<∞
Lévy Subordinators with L.T. L(t, λ) = e −ϕ(λ)t V induce jumps
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
.
We are interested in T.C. with analytically
[ −λT ]tractable Laplace Transform (LT):
26 / 1
Time-Changed Process Yt = XTt
.
Time Changed Process Construction
..
Yt = XTt
.
Tt is a random clock process independent of Xt
.
..
.
.
Random Clock {Tt , t ≥ 0}
.
Xt is a background process (e.g. JDCEV)
.
..
Non-decreasing RCLL process starting at T0 = 0 and E [Tt ] < ∞.
L(t, λ) = E e
.
..
1
2
t
<∞
Lévy Subordinators with L.T. L(t, λ) = e −ϕ(λ)t V induce jumps
.
.
We are interested in T.C. with analytically
[ −λT ]tractable Laplace Transform (LT):
Absolutely Continuous (A.C.) time changes V induce stochastic
volatility
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
26 / 1
Time-Changed Process Yt = XTt
.
Time Changed Process Construction
..
Yt = XTt
.
Tt is a random clock process independent of Xt
.
..
.
.
Random Clock {Tt , t ≥ 0}
.
Xt is a background process (e.g. JDCEV)
.
..
Non-decreasing RCLL process starting at T0 = 0 and E [Tt ] < ∞.
L(t, λ) = E e
.
..
1
2
3
t
<∞
Lévy Subordinators with L.T. L(t, λ) = e −ϕ(λ)t V induce jumps
.
.
We are interested in T.C. with analytically
[ −λT ]tractable Laplace Transform (LT):
Absolutely Continuous (A.C.) time changes V induce stochastic
volatility
Composite Time Changes V induce jumps & stochastic volatility
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
26 / 1
Illustration of Lévy Subordinators
.
..
Y = XTt where Xt = Bt and Tt = t+ Compound Poisson Process .
with Exponential Jumps
Background Process X(t)
0.6
0.5
Time Process
0.4
1.8
0.3
X(t)
2
1.6
0.2
Time T(t)
1.4
0.1
1.2
0
0
1
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
-0.1
Time T(t)
0.8
Time Changed Process Y(t)=X(T(t))
0.6
0.6
0.4
0.5
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time t
Y(t)=X (T(t))
0.4
0
0.3
0.2
0.1
0
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
-0.1
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/ηRafael
= 0.1 Mendoza
yrs.
(McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
27 / 1
Illustration of Lévy Subordinators
.
..
Y = XTt where Xt = Bt and Tt = t+ Compound Poisson Process .
with Exponential Jumps
Background Process X(t)
0.6
0.5
Time Process
0.4
1.8
0.3
X(t)
2
1.6
0.2
Time T(t)
1.4
0.1
1.2
0
0
1
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
-0.1
Time T(t)
0.8
Time Changed Process Y(t)=X(T(t))
0.6
0.6
0.4
0.5
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time t
Y(t)=X (T(t))
0.4
0
0.3
0.2
0.1
0
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
-0.1
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/ηRafael
= 0.1 Mendoza
yrs.
(McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
27 / 1
Illustration of Lévy Subordinators
.
..
Y = XTt where Xt = Bt and Tt = t+ Compound Poisson Process .
with Exponential Jumps
Background Process X(t)
0.6
0.5
Time Process
0.4
1.8
0.3
X(t)
2
1.6
0.2
Time T(t)
1.4
0.1
1.2
0
0
1
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
-0.1
Time T(t)
0.8
Time Changed Process Y(t)=X(T(t))
0.6
0.6
0.4
0.5
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time t
Y(t)=X (T(t))
0.4
0
0.3
0.2
0.1
0
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
-0.1
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/ηRafael
= 0.1 Mendoza
yrs.
(McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
27 / 1
Illustration of Lévy Subordinators
.
..
Y = XTt where Xt = Bt and Tt = t+ Compound Poisson Process .
with Exponential Jumps
Background Process X(t)
0.6
0.5
Time Process
0.4
1.8
0.3
X(t)
2
1.6
0.2
Time T(t)
1.4
0.1
1.2
0
0
1
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
-0.1
Time T(t)
0.8
Time Changed Process Y(t)=X(T(t))
0.6
0.6
0.4
0.5
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time t
Y(t)=X (T(t))
0.4
0
0.3
0.2
0.1
0
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
-0.1
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/ηRafael
= 0.1 Mendoza
yrs.
(McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
27 / 1
Illustration of Lévy Subordinators
.
..
Y = XTt where Xt = Bt and Tt = t+ Compound Poisson Process .
with Exponential Jumps
Background Process X(t)
0.6
0.5
Time Process
0.4
1.8
0.3
X(t)
2
1.6
0.2
Time T(t)
1.4
0.1
1.2
0
0
1
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
-0.1
Time T(t)
Time Changed Process Y(t)=X(T(t))
0.8
0.6
0.6
0.4
0.5
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time t
Y(t)=X (T(t))
0.4
0
0.3
0.2
0.1
0
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
-0.1
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/ηRafael
= 0.1 Mendoza
yrs.
(McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
27 / 1
Illustration of Lévy Subordinators
.
..
Y = XTt where Xt = Bt and Tt = t+ Compound Poisson Process .
with Exponential Jumps
Background Process X(t)
0.6
0.5
Time Process
0.4
1.8
0.3
X(t)
2
1.6
0.2
Time T(t)
1.4
0.1
1.2
0
0
1
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
-0.1
Time T(t)
Time Changed Process Y(t)=X(T(t))
0.8
0.6
0.6
0.4
0.5
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time t
Y(t)=X (T(t))
0.4
0
0.3
0.2
0.1
0
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
-0.1
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/ηRafael
= 0.1 Mendoza
yrs.
(McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
27 / 1
Illustration of Lévy Subordinators
.
..
Y = XTt where Xt = Bt and Tt = t+ Compound Poisson Process .
with Exponential Jumps
Background Process X(t)
0.6
0.5
Time Process
0.4
1.8
0.3
X(t)
2
1.6
0.2
Time T(t)
1.4
0.1
1.2
0
0
1
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
-0.1
Time T(t)
Time Changed Process Y(t)=X(T(t))
0.8
0.6
0.6
0.4
0.5
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time t
Y(t)=X (T(t))
0.4
0
0.3
0.2
0.1
0
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
-0.1
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/ηRafael
= 0.1 Mendoza
yrs.
(McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
27 / 1
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 2009
.
.
..
.
28 / 1
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 − C Γ(−Y )[(λ + η)Y − η Y ], Y ̸= 0
ϕ(λ) =
.
..

γλ + C ln(1 + λ/η),
Y =0
.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
where
28 / 1
Absolutely Continuous Time Changes
.
Absolutely Continuous Time Changes (A.C)
..
An A.C. Time change is the time integral of some positive function V (z)
of a Markov process {Zt , t ≥ 0},
Tt =
0
V (Zu )du
We are interested in cases with Laplace Transform in closed form:
[
]
Rt
Lz (t, λ) = Ez e −λ 0 V (Zu )du
.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
.
..
∫t
.
29 / 1
Absolutely Continuous Time Changes
.
Absolutely Continuous Time Changes (A.C)
..
An A.C. Time change is the time integral of some positive function V (z)
of a Markov process {Zt , t ≥ 0},
Tt =
0
V (Zu )du
We are interested in cases with Laplace Transform in closed form:
[
]
Rt
Lz (t, λ) = Ez e −λ 0 V (Zu )du
.
.
.
..
∫t
.
Example: The Cox-Ingersoll-Ross (CIR) process:
√
dVt = κ(θ − Vt )dt + σV Vt dWt
with V0 = v > 0, rate of mean reversion κ > 0, long-run level θ > 0,
and volatility σV > 0.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
29 / 1
Absolutely Continuous Time Changes
The Laplace Transform of the Integrated CIR process:
[
]
Rt
Lv (t, λ) = Ev e −λ 0 Vu du = A(t, λ)e −B(t,λ)v
A=
2ϖe (ϖ+κ)t/2
(ϖ + κ)(e ϖt − 1) + 2ϖ
Rafael Mendoza (McCombs)
! 2κθ
σ2
V
, B=
q
2λ(e ϖt − 1)
,
ϖ
=
2σV2 λ + κ2
(ϖ + κ)(e ϖt − 1) + 2ϖ
Unified Credit-Equity Modeling
Credit Risk 2009
30 / 1
Absolutely Continuous Time Changes
The Laplace Transform of the Integrated CIR process:
[
]
Rt
Lv (t, λ) = Ev e −λ 0 Vu du = A(t, λ)e −B(t,λ)v
A=
2ϖe (ϖ+κ)t/2
(ϖ + κ)(e ϖt − 1) + 2ϖ
! 2κθ
σ2
V
, B=
q
2λ(e ϖt − 1)
,
ϖ
=
2σV2 λ + κ2
(ϖ + κ)(e ϖt − 1) + 2ϖ
This is the Zero Coupon Bond formula under the CIR interest rate
rt = λVt .
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
30 / 1
Illustration of Absolutely Continuous Time Changes
.
CIR parameters κ = 7, θ = 2, V0 = 0.5 and σv =
√
..
.
2
Time Process
CIR Process
3
1
V(t)
0.9
0.8
2.5
0.7
Time T(t)
2
1.5
1
0.6
0.5
0.4
0.3
0.2
0.5
0.1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (t)
0.7
0.8
0.9
0
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Time t
P ro c e s s e s X (t) v s Y (t)= X (T (t))
1
X (t)
0 .5
0 .4
0 .3
0 .2
0 .1
0
0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
1 .1
T im e ( t)
.
.
.
..
-0 .1
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 2009
31 / 1
Illustration of Absolutely Continuous Time Changes
.
CIR parameters κ = 7, θ = 2, V0 = 0.5 and σv =
√
..
.
2
Time Process
CIR Process
3
1
V(t)
0.9
2.5
0.8
0.7
Time T(t)
2
1.5
1
0.6
0.5
0.4
0.3
0.2
0.5
0.1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (t)
0.7
0.8
0.9
0
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Time t
1
X (t)
X (T (t))
P ro c e s s e s X (t) v s Y (t)= X (T (t))
0 .5
0 .4
0 .3
0 .2
0 .1
0
0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
1 .1
T im e ( t )
.
.
.
..
-0 .1
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 2009
31 / 1
Illustration of Absolutely Continuous Time Changes
.
CIR parameters κ = 7, θ = 2, V0 = 0.5 and σv =
√
..
.
2
Time Process
CIR Process
3
1
V(t)
0.9
2.5
0.8
0.7
Time T(t)
2
1.5
1
0.6
Slow
0.5
Fast
0.4
0.3
0.2
0.5
0.1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (t)
0.7
0.8
0.9
0
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Time t
1
X (t)
X (T (t))
P ro c e s s e s X (t) v s Y (t)= X (T (t))
0 .5
0 .4
0 .3
Slow
Fast
0 .1
0 .2
0 .2
0 .1
0
0
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
1 .1
T im e ( t )
.
.
.
..
-0 .1
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 2009
31 / 1
Composite Time Changes
.
Composite Time Changes
..
A Composite Time Change induces both jumps and stochastic volatility
.
Tt = TT1 2
t
.
..
T22 is and A.C time change
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
.
Credit Risk 2009
.
Tt1 is a Lévy Subordinator
32 / 1
Composite Time Changes
.
Composite Time Changes
..
A Composite Time Change induces both jumps and stochastic volatility
.
Tt = TT1 2
t
T22 is and A.C time change
.
.
Laplace Transform of the Composite Time Change
..
It is obtained by first conditioning w.r.t. the A.C. time change
E[e −λTt ] = E[e −Tt
2
.
..
Rafael Mendoza (McCombs)
ϕ(λ) ]
= Lz (t, ϕ(λ))
Unified Credit-Equity Modeling
.
.
Credit Risk 2009
.
.
..
.
Tt1 is a Lévy Subordinator
32 / 1
Quick Summary
We have:
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
33 / 1
Quick Summary
We have:
1
A Jump-to-Default Extended Diffusion process:
[ Rt
]
[
]
E f (Xt ) 1{ζ>t} = E e − 0 h(Xu )du f (Xt ) 1{τ0 >t}
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
33 / 1
Quick Summary
We have:
1
A Jump-to-Default Extended Diffusion process:
[ Rt
]
[
]
E f (Xt ) 1{ζ>t} = E e − 0 h(Xu )du f (Xt ) 1{τ0 >t}
2
A time-changed process Yt = XTt with the Laplace transform for the
time change Tt given in closed form,
[
]
E e −λTt = L (t, λ)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
33 / 1
Quick Summary
We have:
1
A Jump-to-Default Extended Diffusion process:
[ Rt
]
[
]
E f (Xt ) 1{ζ>t} = E e − 0 h(Xu )du f (Xt ) 1{τ0 >t}
2
A time-changed process Yt = XTt with the Laplace transform for the
time change Tt given in closed form,
[
]
E e −λTt = L (t, λ)
How do we evaluate contingent claims written on the time-changed
process Yt ?
[
]
E f (Yt ) 1{ζ>Tt }
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
33 / 1
Contingent Claims for the Time-Changed Process
.
Valuing contingent claims written on Yt = XTt
..
]]
[
]
[ [
E 1{ζ>Tt } f (Yt ) = E Ex 1{ζ>Tt } f (XTt ) Tt
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
.
Credit Risk 2009
.
Conditioning
since Xt and Tt are independent
.
..
.
34 / 1
Contingent Claims for the Time-Changed Process
.
Valuing contingent claims written on Yt = XTt
..
]]
[
]
[ [
E 1{ζ>Tt } f (Yt ) = E Ex 1{ζ>Tt } f (XTt ) Tt
.
Unified Credit-Equity Modeling
Credit Risk 2009
.
.
It is equivalent to pricing a contingent claim written on the process Xt
maturing
at time Tt
.
..
.
Rafael Mendoza (McCombs)
.
Conditioning
since Xt and Tt are independent
.
..
.
Conditional Expectation
..
]
[
E 1{ζ>Tt } f (XTt ) Tt
.
34 / 1
Contingent Claims for the Time-Changed Process
.
Valuing contingent claims written on Yt = XTt
..
]]
[
]
[ [
E 1{ζ>Tt } f (Yt ) = E Ex 1{ζ>Tt } f (XTt ) Tt
.
Unified Credit-Equity Modeling
Credit Risk 2009
.
.
It is equivalent to pricing a contingent claim written on the process Xt
maturing
at time Tt
.
..
.
We employ two methodologies to evaluate the expectations and do
the pricing in closed form:
Rafael Mendoza (McCombs)
.
Conditioning
since Xt and Tt are independent
.
..
.
Conditional Expectation
..
]
[
E 1{ζ>Tt } f (XTt ) Tt
.
34 / 1
Contingent Claims for the Time-Changed Process
.
Valuing contingent claims written on Yt = XTt
..
]]
[
]
[ [
E 1{ζ>Tt } f (Yt ) = E Ex 1{ζ>Tt } f (XTt ) Tt
.
.
.
It is equivalent to pricing a contingent claim written on the process Xt
maturing
at time Tt
.
..
.
We employ two methodologies to evaluate the expectations and do
the pricing in closed form:
1
.
Conditioning
since Xt and Tt are independent
.
..
.
Conditional Expectation
..
]
[
E 1{ζ>Tt } f (XTt ) Tt
.
Resolvent Operator: general methodology.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
34 / 1
Contingent Claims for the Time-Changed Process
.
Valuing contingent claims written on Yt = XTt
..
]]
[
]
[ [
E 1{ζ>Tt } f (Yt ) = E Ex 1{ζ>Tt } f (XTt ) Tt
.
2
.
.
It is equivalent to pricing a contingent claim written on the process Xt
maturing
at time Tt
.
..
.
We employ two methodologies to evaluate the expectations and do
the pricing in closed form:
1
.
Conditioning
since Xt and Tt are independent
.
..
.
Conditional Expectation
..
]
[
E 1{ζ>Tt } f (XTt ) Tt
.
Resolvent Operator: general methodology.
Spectral Representation: for square-integrable payoffs.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
34 / 1
Resolvent Operator
.
Resolvent Operator:
..
The Laplace Transform of the Expectation Operator:
(Rλ f )(x) :=
Rafael Mendoza (McCombs)
∫∞
0
[
]
e −λt Ex 1{ζ>t} f (Xt ) dt
Unified Credit-Equity Modeling
.
Credit Risk 2009
.
.
..
.
35 / 1
Resolvent Operator
.
Resolvent Operator:
..
The Laplace Transform of the Expectation Operator:
(Rλ f )(x) :=
∫∞
0
[
]
e −λt Ex 1{ζ>t} f (Xt ) dt
.
.
.
..
.
We recover the Expectation via the Bromwich Laplace Inversion
formula:
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
35 / 1
Resolvent Operator
.
Resolvent Operator:
..
The Laplace Transform of the Expectation Operator:
(Rλ f )(x) :=
∫∞
0
[
]
e −λt Ex 1{ζ>t} f (Xt ) dt
.
.
.
..
.
We recover the Expectation via the Bromwich Laplace Inversion
formula:
Rafael Mendoza (McCombs)
1
2πi
∫ ϵ+i∞
ϵ−i∞
.
e λ t (Rλ f )(x)dλ
Unified Credit-Equity Modeling
.
Credit Risk 2009
.
.
Bromwich Laplace Inversion
..
[
]
E
1
f
(X
)
=
x
t
{ζ>t}
.
..
35 / 1
Resolvent Operator
.
Resolvent Operator:
..
The Laplace Transform of the Expectation Operator:
(Rλ f )(x) :=
∫∞
0
[
]
e −λt Ex 1{ζ>t} f (Xt ) dt
.
.
.
..
.
We recover the Expectation via the Bromwich Laplace Inversion
formula:
1
2πi
∫ ϵ+i∞
ϵ−i∞
.
e λ t (Rλ f )(x)dλ
.
.
.
Bromwich Laplace Inversion
..
[
]
E
1
f
(X
)
=
x
t
{ζ>t}
.
..
NOTE. The time t enters in this expression only through the
exponential e λ t
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
35 / 1
Spectral Expansion
1
.. Details
If the infinitesimal generator G of the diffusion process X is
self-adjoint
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
36 / 1
Spectral Expansion
1
.. Details
If the infinitesimal generator G of the diffusion process X is
self-adjoint
If X is a 1D diffusion process then G is self-adjoint in
H = L2 ((0, ∞), m) with respect to the speed measure m(dx)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
36 / 1
Spectral Expansion
1
.. Details
If the infinitesimal generator G of the diffusion process X is
self-adjoint
If X is a 1D diffusion process then G is self-adjoint in
H = L2 ((0, ∞), m) with respect to the speed measure m(dx)
2
If f ∈ H
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
36 / 1
Spectral Expansion
1
.. Details
If the infinitesimal generator G of the diffusion process X is
self-adjoint
If X is a 1D diffusion process then G is self-adjoint in
H = L2 ((0, ∞), m) with respect to the speed measure m(dx)
2
If f ∈ H
VThen we can use the Spectral Representation Theorem in order to
obtain the Expectation
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
36 / 1
Spectral Expansion
1
.. Details
If the infinitesimal generator G of the diffusion process X is
self-adjoint
If X is a 1D diffusion process then G is self-adjoint in
H = L2 ((0, ∞), m) with respect to the speed measure m(dx)
2
If f ∈ H
VThen we can use the Spectral Representation Theorem in order to
obtain the Expectation
.
Eigenfunction Expansion (when the spectrum of G is discrete):
.
..
.
[
]
∑
−λn t c φ (x)
Ex 1{ζ>t} f (Xt ) = ∞
n n
n=1 e
where cn = ⟨f , φ⟩ are the expansion coefficients and, λn are the
eigenvalues, φn (x) the eigenfunctions solving Gφn (x) = λn φn (x)
.
.
..
NOTE. The time t enters in this expression only through the
exponential e −λn t
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
36 / 1
Valuing contingent claims written on Yt = XTt
.
Rafael Mendoza (McCombs)
.
.
..
.
.
Spectral Expansion
..
[
]
E 1{ζ>Tt } f (Yt )
.
..
Unified Credit-Equity Modeling
.
.
Credit Risk 2009
.
.
Resolvent Opertator
..
[
]
E 1{ζ>Tt } f (Yt )
37 / 1
Valuing contingent claims written on Yt = XTt
.
Resolvent Opertator
..
[
]
E 1{ζ>Tt } f (Yt )
.
.
Spectral Expansion
..
[
]
E 1{ζ>Tt } f (Yt )
]]
[ [
= E Ex 1{ζ>Tt } f (XTt ) Tt
.
Rafael Mendoza (McCombs)
.
..
Unified Credit-Equity Modeling
.
Credit Risk 2009
.
]]
[ [
= E Ex 1{ζ>Tt } f (XTt ) Tt
.
.
..
.
37 / 1
Valuing contingent claims written on Yt = XTt
.
Resolvent Opertator
..
[
]
E 1{ζ>Tt } f (Yt )
.
.
Spectral Expansion
..
[
]
E 1{ζ>Tt } f (Yt )
]]
[ [
= E Ex 1{ζ>Tt } f (XTt ) Tt
ϵ+i∞ λ Tt
dλ
(Rλ f )(x) 2πi
ϵ−i∞ e
.
..
]
.
Rafael Mendoza (McCombs)
=E
[∑∞
n=1 e
−λn Tt c
.
..
Unified Credit-Equity Modeling
]
n φn (x)
.
Credit Risk 2009
.
[∫
]]
[ [
= E Ex 1{ζ>Tt } f (XTt ) Tt
.
=E
.
37 / 1
Valuing contingent claims written on Yt = XTt
.
Resolvent Opertator
..
[
]
E 1{ζ>Tt } f (Yt )
.
.
Spectral Expansion
..
[
]
E 1{ζ>Tt } f (Yt )
]]
[ [
= E Ex 1{ζ>Tt } f (XTt ) Tt
ϵ+i∞ λ Tt
dλ
(Rλ f )(x) 2πi
ϵ−i∞ e
∫ ϵ+i∞
ϵ−i∞
]
=E
[
]
dλ
E e λ Tt (Rλ f )(x) 2πi
.
..
=
.
Rafael Mendoza (McCombs)
[∑∞
n=1 e
∑∞
n=1 E
−λn Tt c
[ −λ T ]
e n t cn φn (x)
.
..
Unified Credit-Equity Modeling
]
n φn (x)
.
Credit Risk 2009
.
=
[∫
]]
[ [
= E Ex 1{ζ>Tt } f (XTt ) Tt
.
=E
.
37 / 1
Valuing contingent claims written on Yt = XTt
.
Resolvent Opertator
..
[
]
E 1{ζ>Tt } f (Yt )
.
.
Spectral Expansion
..
[
]
E 1{ζ>Tt } f (Yt )
]]
[ [
= E Ex 1{ζ>Tt } f (XTt ) Tt
. =
..
ϵ+i∞ λ Tt
dλ
(Rλ f )(x) 2πi
ϵ−i∞ e
∫ ϵ+i∞
ϵ−i∞
∫ ϵ+i∞
ϵ−i∞
]
=E
[
]
dλ
E e λ Tt (Rλ f )(x) 2πi
dλ
L(t, −λ)(Rλ f )(x) 2πi
Rafael Mendoza (McCombs)
=
.
.
..
[∑∞
n=1 e
∑∞
=
Unified Credit-Equity Modeling
n=1 E
−λn Tt c
]
n φn (x)
[ −λ T ]
e n t cn φn (x)
∑∞
n=1 L(t, λn )cn φn (x)
Credit Risk 2009
.
.
=
[∫
]]
[ [
= E Ex 1{ζ>Tt } f (XTt ) Tt
.
=E
.
37 / 1
A new class of Credit-Equity Models with
state-dependent jumps, S.V. and default intensity
.
Model Architecture for the Defaultable Stock
..
St = 1{t<τd } e ρt XTt .
.
..
.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
.
38 / 1
A new class of Credit-Equity Models with
state-dependent jumps, S.V. and default intensity
.
Model Architecture for the Defaultable Stock
..
St = 1{t<τd } e ρt XTt .
.
Xt V Jump-to-Default Extended Diffusion; e.g. JDCEV Process:
.
..
.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
dXt = [µ + h(Xt )]Xt dt + σ(Xt )Xt dBt , X0 = x > 0,
σ(x) = ax β , h(x) = b + c σ 2 (x)
38 / 1
A new class of Credit-Equity Models with
state-dependent jumps, S.V. and default intensity
.
Model Architecture for the Defaultable Stock
..
St = 1{t<τd } e ρt XTt .
.
Xt V Jump-to-Default Extended Diffusion; e.g. JDCEV Process:
dXt = [µ + h(Xt )]Xt dt + σ(Xt )Xt dBt , X0 = x > 0,
σ(x) = ax β , h(x) = b + c σ 2 (x)
.
..
.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
Tt V Random Clock: Lévy Subordinator, A.C. Time Change, or
Composite T.C.
38 / 1
A new class of Credit-Equity Models with
state-dependent jumps, S.V. and default intensity
.
Model Architecture for the Defaultable Stock
..
St = 1{t<τd } e ρt XTt .
.
Xt V Jump-to-Default Extended Diffusion; e.g. JDCEV Process:
dXt = [µ + h(Xt )]Xt dt + σ(Xt )Xt dBt , X0 = x > 0,
σ(x) = ax β , h(x) = b + c σ 2 (x)
Tt V Random Clock: Lévy Subordinator, A.C. Time Change, or
Composite T.C.
.. Details
.
..
.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
ρ V Compensation Parameter (discounted martingale)
38 / 1
A new class of Credit-Equity Models with
state-dependent jumps, S.V. and default intensity
.
Model Architecture for the Defaultable Stock
..
St = 1{t<τd } e ρt XTt .
.
Xt V Jump-to-Default Extended Diffusion; e.g. JDCEV Process:
dXt = [µ + h(Xt )]Xt dt + σ(Xt )Xt dBt , X0 = x > 0,
σ(x) = ax β , h(x) = b + c σ 2 (x)
Tt V Random Clock: Lévy Subordinator, A.C. Time Change, or
Composite T.C.
ρ V Compensation Parameter (discounted martingale)
.. Details
.
..
.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
τd V Default Time
38 / 1
A new class of Credit-Equity Models with
state-dependent jumps, S.V. and default intensity
.
Model Architecture for the Defaultable Stock
..
St = 1{t<τd } e ρt XTt .
.
Xt V Jump-to-Default Extended Diffusion; e.g. JDCEV Process:
dXt = [µ + h(Xt )]Xt dt + σ(Xt )Xt dBt , X0 = x > 0,
σ(x) = ax β , h(x) = b + c σ 2 (x)
Tt V Random Clock: Lévy Subordinator, A.C. Time Change, or
Composite T.C.
ρ V Compensation Parameter (discounted martingale)
.. Details
.
..
If ζ = min(τ0 , ζ̃) is the lifetime of X , then
τd = inf{t ≥ 0 : ζ ≤ Tt }
At τd the stock drops to zero (Bankruptcy)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
.
Credit Risk 2009
.
τd V Default Time
38 / 1
Survival Probability and Defaultable Zero Bonds
.
Survival Probability
..
=
.
Q(τd > t) = Q(ζ > Tt )
∑∞
n=0 L (t, (b
1
× A 2|β| xe −Ax
−2β
+ ωn)) Γ(1+c/|β|)Γ(n+1/(2|β|))
Γ(ν+1)Γ(1/(2|β|))n!
1 F1 (1
− n + c/|β|, ν + 1, Ax −2β )
.
..
Where 1 F1 (a, b, z) is the Kummer Confluent Hypergeometric function;
µ+b
and ω = 2|β|(µ + b), ν = 1+2c
2|β| , and A = a2 |β| .
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
.
.. Details
39 / 1
Survival Probability and Defaultable Zero Bonds
.
Survival Probability
..
=
.
Q(τd > t) = Q(ζ > Tt )
∑∞
n=0 L (t, (b
1
× A 2|β| xe −Ax
−2β
+ ωn)) Γ(1+c/|β|)Γ(n+1/(2|β|))
Γ(ν+1)Γ(1/(2|β|))n!
1 F1 (1
− n + c/|β|, ν + 1, Ax −2β )
.
Defaultable Zero Coupon Bond
..
BR (x, t) = e −rt Q(τd > t) + Re −rt [1 − Q(τd > t)]
Recovery fraction R ∈ [0, 1]
.
..
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
.
.
.
.
..
Where 1 F1 (a, b, z) is the Kummer Confluent Hypergeometric function;
µ+b
and ω = 2|β|(µ + b), ν = 1+2c
2|β| , and A = a2 |β| .
.
.. Details
39 / 1
Put Options
.
Put Option
..
.
P(x; K , t) = PD (x; K , t) + P0 (x; K , t),
where:
PD (x; K , t) = Ke −rt [1 − Q(τd > t)],
[
]
P0 (x; K , t) = e −rt Ex (K − e ρt XTt )+ 1{τd >t}
Rafael Mendoza (McCombs)
No Default before t
∑∞
n=1 L(t, λn )cn φn (x)
Unified Credit-Equity Modeling
.
Credit Risk 2009
.
= e −(r −ρ)t
.
..
Default before t
40 / 1
Put Options
.
Put Option
..
.
P(x; K , t) = PD (x; K , t) + P0 (x; K , t),
where:
PD (x; K , t) = Ke −rt [1 − Q(τd > t)],
[
]
P0 (x; K , t) = e −rt Ex (K − e ρt XTt )+ 1{τd >t}
No Default before t
∑∞
n=1 L(t, λn )cn φn (x)
.
.
= e −(r −ρ)t
.
..
Default before t
The default claim PD (x; K , t) is directly calculated from the Survival
Probability Q(τd > t) previously computed
The claim, P0 (x; K , t), is calculated by means of the Spectral
Expansion since f (x) = (K − x)+ ∈ L2 ((0, ∞), m)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
40 / 1
Put Options
.
Put claim conditional on no default event before maturity
..
∞
∑
P0 (x; K , t) = e −(r −ρ)t
L(t, λn )cn φn (x)
.
where k = K e −ρt and,
Eigenvalues V λn = ωn + 2c(µ + b) + b, with ω = 2|β|(µ + b)
√
ν
−2β (ν)
Eigenfunctions V φn (x) = A 2 (n−1)!(µ+b)(2c+1)
xe −Ax Ln−1 (Ax −2β )
Γ(ν+n)
√
Aν/2+1 k 2c+1−2β Γ(ν+n)
√
Expansion Coefficients V cn =
Γ(ν+1) (µ+b)(2c+1)(n−1)!
)
{
(
}
c
)
1
−
n,
+
1
Γ(ν+1)(n−1)! ν+1 (
|β|
|β|
−2β
−2β
; Ak
− Γ(ν+n+1) Ln−1 Ak
× c+|β| 2 F2
,
c
+2
ν + 1, |β|
.
..
.
(ν)
2 F2 : generalized hypergeometric function, Ln : generalized Laguerre
polynomials.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
n=1
41 / 1
Numerical Examples
Assume a background process {Xt , t > 0} following a JDCEV, and a
composite time change Inverse Gaussian Process & CIR:
.
Parameters
..
.
.
..
50
10
−1
0.5
0.01
0.05
0
CIR
IG
V
θ
σV
κ
γ
η
C
1
1
1
4
0
√8
2 2/π
.
.
JDCEV
S
a
β
c
b
r
q
Note that γ = 0, thus the time changed process is a pure jump
process!
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
42 / 1
Infinitesimal Generator of the Time Changed
Process (Yt = XTt , Vt )
Gf (x, v ) =
(
)
1 2 2β+2 ∂ 2 f
∂f
2 2β
2 2 2β
γv
a x
(x, v ) + (b + ca x )x (x, v ) − (b + c a x )f (x, v )
2
∂x 2
∂x
|
{z
}
Gx f (x,v ) JDCEV’s infinitesimal generator
(∫
)
(f (y , v ) − f (x, v )) π(x, y )dy − k (x) f (x, v )
+v
|
(0,∞)
R
{z
0,∞ (Ps f −f )ν(ds) Subordination component
}
σ2 ∂ 2f
∂f
+ V v 2 (x, v ) + κ(θ − v ) (x, v )
∂v
| 2 ∂v
{z
}
Gv f (x,v ) CIR’s infinitesimal generator
∫
State dependent jump measure π( x, y ) = (0,∞) p(s; x, y )ν(ds)
∫
.. Details
Additional killing rate k(x) = (0,∞) Ps (x, {0})ν(ds)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
43 / 1
Infinitesimal Generator of the Time Changed
Process (Yt = XTt , Vt )
Gf (x, v ) =
(
)
1 2 2β+2 ∂ 2 f
∂f
2 2β
2 2 2β
γv
a x
(x, v ) + (b + ca x )x (x, v ) − (b + c a x )f (x, v )
2
∂x 2
∂x
|
{z
}
Gx f (x,v ) JDCEV’s infinitesimal generator
(∫
)
(f (y , v ) − f (x, v )) π(x, y )dy − k (x) f (x, v )
+v
|
(0,∞)
R
{z
(0,∞) (Ps f −f )ν(ds) Subordination component
}
σ2 ∂ 2f
∂f
+ V v 2 (x, v ) + κ(θ − v ) (x, v )
∂v
| 2 ∂v
{z
}
Gv f (x,v ) CIR’s infinitesimal generator
∫
State dependent jump measure π( x, y ) = (0,∞) p(s; x, y )ν(ds)
∫
.. Details
Additional killing rate k(x) = (0,∞) Ps (x, {0})ν(ds)
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
43 / 1
Numerical Examples (Cont.)
.
Implied Volatility
..
.
Implied Volatility
61%
1/4
Implied Volatility
1/2
1
49%
2
3
37%
25%
ATM
Volatility
13%
30
35
40
45
50
55
60
65
Strike
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
45
26.19
26.39
27.53
29.61
31.34
50
21.41
22.64
24.30
26.72
28.64
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.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
β
Credit Risk 2009
.
.
Time/Strike
1/4
1/2
1
2
3
.
44 / 1
Numerical Examples (Cont.)
.
Credit Spreads and Default Probability
..
.
Credit Spreads
Default Probability
80.0%
8.5%
70.0%
6.5%
Probability of default
Credit spreads
40
50
60
5.5%
70
4.5%
3.5%
60.0%
50.0%
40.0%
30
40
30.0%
50
20.0%
2.5%
10.0%
1.5%
0.0%
0
5
10
15
20
25
30
Time to maturity (years)
35
40
45
50
60
70
0
5
10
15
20
25
30
Time to maturity (years)
35
40
45
50
.
..
.
Credit spreads and default probabilities as functions of time to maturity for
current stock price levels S = 30, 40, 50, 60, 70.
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
.
30
7.5%
45 / 1
Summary of Features and Practical Benefits of Our
Modeling Framework
Our Stock price is a jump-diffusion process with stochastic volatility
and default intensity,
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
46 / 1
Summary of Features and Practical Benefits of Our
Modeling Framework
Our Stock price is a jump-diffusion process with stochastic volatility
and default intensity,
The Default intensity explicitly depends on the stock price and
volatility
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
46 / 1
Summary of Features and Practical Benefits of Our
Modeling Framework
Our Stock price is a jump-diffusion process with stochastic volatility
and default intensity,
The Default intensity explicitly depends on the stock price and
volatility
The leverage effect is introduced in the diffusion and in jumps
components - as the stock falls, the diffusion volatility and arrival
rates of large jumps increase
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
46 / 1
Summary of Features and Practical Benefits of Our
Modeling Framework (cont.)
Stochastic volatility affects the diffusion and jump components
.. Finish
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
47 / 1
Summary of Features and Practical Benefits of Our
Modeling Framework (cont.)
Stochastic volatility affects the diffusion and jump components
Unified credit-equity framework V consistency in the pricing and
hedging of credit and equity derivatives
.. Finish
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
47 / 1
Summary of Features and Practical Benefits of Our
Modeling Framework (cont.)
Stochastic volatility affects the diffusion and jump components
Unified credit-equity framework V consistency in the pricing and
hedging of credit and equity derivatives
We obtain analytical solutions V faster computation of prices and
Greeks, and faster calibration
.. Finish
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
47 / 1
Questions?
Thank you!
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
48 / 1
Appendix
Appendix
Lévy Subordinators
.
Lévy subordinator
..
Non-decreasing Lévy process {Tt , t ≥ 0} with positive jumps and
non-negative drift
Laplace Transform (LT):
L (t, λ) = E[e −λTt ] = e −tϕ(λ)
.
The Laplace exponent ϕ (λ) is given by the Lévy-Khintchine formula:
(
∫
(0,∞)
)
1 − e −λs ν (ds)
.
.
.
..
ϕ (λ) = γλ +
Lévy Subordinators
.
Lévy subordinator
..
Non-decreasing Lévy process {Tt , t ≥ 0} with positive jumps and
non-negative drift
Laplace Transform (LT):
L (t, λ) = E[e −λTt ] = e −tϕ(λ)
.
The Laplace exponent ϕ (λ) is given by the Lévy-Khintchine formula:
.
..
(
∫
(0,∞)
)
1 − e −λs ν (ds)
.
γ≥0
V positive drift
.
ϕ (λ) = γλ +
Lévy Subordinators
.
Lévy subordinator
..
Non-decreasing Lévy process {Tt , t ≥ 0} with positive jumps and
non-negative drift
Laplace Transform (LT):
L (t, λ) = E[e −λTt ] = e −tϕ(λ)
.
The Laplace exponent ϕ (λ) is given by the Lévy-Khintchine formula:
.
..
(
∫
(0,∞)
)
1 − e −λs ν (ds)
.
γ≥0
ν (ds)
V positive drift
∫
V Lévy measure which satisfies
(0,∞)
(s ∧ 1) ν (ds) < ∞
.
ϕ (λ) = γλ +
Lévy Subordinators
.
Lévy subordinator
..
Non-decreasing Lévy process {Tt , t ≥ 0} with positive jumps and
non-negative drift
Laplace Transform (LT):
L (t, λ) = E[e −λTt ] = e −tϕ(λ)
.
The Laplace exponent ϕ (λ) is given by the Lévy-Khintchine formula:
.
..
(
∫
(0,∞)
)
1 − e −λs ν (ds)
.
γ≥0
ν (ds)
V positive drift
∫
V Lévy measure which satisfies
(0,∞)
(s ∧ 1) ν (ds) < ∞
transition probability ∫πt (ds) is obtained by:
−λs π (ds) = e −tϕ(λ)
[0,∞) e
.
ϕ (λ) = γλ +
Examples of Lévy Subordinators (cont.)
The processes Tt is a Compound Poisson processes with gamma
distributed jump sizes if Y < 0
Compound Poisson process with exponential jumps (Y = −1)
ν(ds) = αηe −ηs ds,
ϕ(λ) = γλ +
αλ
λ+η
Tempered Stable Subordinators (Y ∈ (0, 1))
Inverse Gaussian process (Y = 1/2)
Gamma process (Y → 0)
The processes with Y ∈ [0, 1) are of infinite activity.
.. Go Back
Appendix
.. Go Back
Martingale Property
Intensity h(S) 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 time-changed 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.
.. Go Back
Appendix
.. Go Back
Survival Probability
1
Condition w.r.t the Random Clock Tt
[ [ Ru
]]
Q(τd > t) = Q(ζ > Tt ) = E E e − 0 λ(Sv )dv 1{T0 >u} Tt = u
2
Since the Function f (x) = 1 is NOT in L2 (D, m), we use the
resolvent operator Rλ
∫ ε+i∞
1
Q(ζ > Tt ) =
L(t, −λ)(Rλ 1)(x)dλ,
2πi ε−i∞
3
The resolvent is available in closed form
∫ ∞
Rλ f (x) =
Gλ (x, y )f (y )dy
0
Gλ (x, y ) is the Resolvent Kernel or Green’s Function
.. Go Back
Survival Probability
4
Gλ (x, y ) is known in closed form (µ + b > 0):
Γ(ν/2 + 1/2 − k(λ))
Gλ (x, y ) =
(µ + b)Γ(1 + ν)y
( )c+1/2−β
x
−2β
−2β
e A(y −x )
y
× Mk(λ), ν2 (A(x ∧ y )−2β )Wk(λ), ν2 (A(x ∨ y )−2β )
where ν =
1+2c
2|β| ,
k(λ) =
ν−1
2
−
λ
2|β|(µ+b) ,
A=
µ+b
a2 |β|
and,
Mk,m (z) and Wk,m (z) are the first and second Whittaker functions.
5
Using the Cauchy Residue Theorem to invert the Resolvent we obtain
the Survival Probability
.. Go Back
Appendix
.. Go Back
Spectral Expansion
Assume ∃m on D with full support (i.e. SSup(m) = D) s.t. the
(bounded) contraction semigroup Pt (e.g. Pt f (x) = Ex [f (Xt )1{ζ>t} ])
are symmetric on H = L2 (D, m)
∫
∫
⟨Pt f , g ⟩m = D Pt f g dm = D f Pt g dm = ⟨f , Pt g ⟩m
Then the infinitesimal generator G is (generally unbounded)
self-adjoint operator in H, i.e., G is symmetric,
⟨Gf , g ⟩m = ⟨f , Gg ⟩m,
∀f , g ∈ Dom(G)
The domains of G and its adjoint G ∗ coincide in H, i.e.
Dom(G) = Dom(G ∗ ) ⊂ H
The infinitesimal operator G is non-positive in H, i.e. ⟨Gf , f ⟩m < 0
for all f ∈ Dom(G).
.. Go Back
Spectral Representation Theorem
.
Spectral Representation Theorem
..
.
Let H be a separable real Hilbert space and let {Pt , t ≥ 0} be a strongly
continuous self-adjoint contraction semigroup in H with the non-positive
self-adjoint infinitesimal generator G. Then there exists a unique integral
representation of {Pt , t ≥ 0} of the form
∫
Pt f = e tG f = [0,∞) e −λt E (dλ)f , f ∈ H, t ≥ 0,
.
..
Hille and Phillips (1957, Theorem 22.3.1) and Reed and Simon (1980,
Theorem VIII.6)
.
.. Go Back
.
where E is the spectral measure of the negative −G of the infinitesimal generator
G of P with the support of the spectral measure (the spectrum of −G)
Supp(E ) ⊂ [0, ∞), namely,
∫
−Gf = [0,∞) λE (dλ)f , f ∈ Dom(G),
{
}
∫
Dom(G) = f ∈ H : [0,∞) λ2 (E (dλ)f , f ) < ∞ .
Discrete Case Spectral Representation
Things simplify further when the generator has a purely discrete spectrum.
Let −G be a self-adjoint non-negative operator with purely discrete
spectrum σd (−G) ⊂ [0, ∞). Then the spectral measure can be defined by
∑
E (B) = λ∈B P(λ),
where P(λ) is the orthogonal projection onto the eigenspace corresponding
to the eigenvalue λ ∈ σd (−G). Then the spectral theorem for the
self-adjoint semigroup takes the simpler form:
∑
Pt f = e tG f = λ∈σd (−G) e −λt P(λ)f , t ≥ 0, f ∈ H,
−Gf =
∑
λ∈σd (−G) λP(λ)f ,
f ∈ Dom(G).
(e.g. P(λ)f = c(λ)ϕλ = ⟨f , ϕλ ⟩m ϕλ )
.. Go Back
Appendix
.. Go Back
Notes on Calibration and the Implied Measure
(Cont & Tankov, 2004)
Exponential Lévy and jump-diffusion models correspond to incomplete
market models
V No perfect hedges can be found
V The (equivalent) martingale measure cannot be defined in a unique
way
Any arbitrage-free market prices of securities can be represented as
discounted conditional expectations w.r.t. a risk-neutral measure Q
under which discounted asset prices are martingales
V Model Calibration. Find a risk-neutral model Q which matches the
prices of the observed market prices V{i∈I } (S) of securities i ∈ I at
time t = 0,
∀i ∈ I , Vi (S) = e −rti EQ [f (Sti )]
.. Go Back
Notes on Calibration and the Implied Measure
(Cont & Tankov, 2004)
Least Square Calibration.
2
∑
θ∗ = arg minQθ ∈Q i∈I ωi Viθ (S, ti ) − Vi (S)
where Q is the set of martingale measures
V The objective functional is non-convex.
V Since the number of observable prices is finite there are multiple
Lévy measures giving the same error level (multiple local minimum)
To obtain a unique solution in a stable manner we need to introduce
a penalty functional (regularization) F
2
∑
θ∗ = arg minQθ ∈Q i∈I ωi Viθ (S, ti ) − Vi (S) + αF (Qθ |P0 )
where P0 is the historical measure at t = 0 and F is a convex
function which penalizes the objective if Q deviates much from P0
and ensures uniqueness (v.g. F relative entropy)
.. Go Back
Appendix
.. Go Back
Jump measure and killing rate
( )c− 1 −(2β+1)
2
y
(Jump measure) π (x, y ) = 2|β|AC yx
{
( −2β ωs −2β )} (
)
ων
∫
−3/2 e ( 2 −ξ−η )s
A(xy )−β
x
e +y
× (0,∞) s
exp
−A
I
ν sinh(ωs/2) ds.
e ωs −1
e ωs −1
and
(killing rate) k (x) =



C (0,∞) 1 −

∫
where τ (s) :=
Γ
“
c
+1
|β|
0
c
”
1
(τ (s)) 2|β| e −τ (s)−bs 1 F1 @ |β|
+1
ν+1
Γ(ν+1)
1
;τ (s)A 
 −3/2 −ηs
e ds
s

ω x −2β
,
2|β|2 a2 (1−e −ω s )
.. Go Back
Appendix
.. Go Back
Appendix
.. Go Back
Protection Payment under JDCEV
h
i
∆
PV(Protection Payment) = (1 − r) E e −r · TL 1{T ∆ ≤ T }
L
8
9
>
>
>
>
>
–
< »
=
h Ru
i >
R TL
R
T
= (1 − r) E e −r · TL − 0 h(Xu )du 1{TL ≤ T } + 0 e −r · u E e − 0 h(Xv )dv h (Xu ) 1{TL > u} du
>
>
>
|
{z
}>
>|
>
{z
}
:
;
Jump Term
Diffusion Term
Recall that the first hitting time to L is given by TL = inf {t : Xt = L},
and that{ the first jump time to ∆ is given
} by
∫t
ζ = inf t ∈ [0, ∞] : 0 h (Xu ) du ≥ e
The default intensity is the power function:
h(Xt ) = b + ca2 Xt2β
Notice. Since Re is an exponentially distributed r.v. with
unit mean, then
Rv
∫t
t
P[ζ > t] = e − 0 h(Xu )du and P[ζ < t] = 0 h (Xv ) e − 0 h(Xu )du dv
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
40 / 1
Premium Payment under JDCEV
[
]
∑N
PV(Premium Payment) = ϱ · ∆t · i=1 e −r · ti E 1{TL∆ ≥ ti }
[ R ti
]
∑N
= ϱ · ∆t · i=1 e −r · ti E e − 0 h(Xu )du 1{TL ≥ ti }
|
{z
}
NO jump to default & NO hitting level
The premium is paid at times ti conditional on No default and that the stock
price did Not drop to level L by time ti
The default intensity is the power function:
h(Xt ) = b + ca2 Xt2β
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
41 / 1
Accrued Interests under JDCEV
[
(
]
[ ∆ ])
TL
−r · TL∆
∆
TL − ∆t · ∆t
PV(Acc. Interest) = ϱ · E e
1{TL∆ ≤ T }
[
]
)
∑N−1
∆ (
= ϱ i=0 E e −r · TL TL∆ − ∆t · i 1{TL∆ ∈ (ti ,ti+1 )}
Expressed in terms of Diffusion and Jump components:
=ϱ·
8
>
>
>
<R
>
>
>
:
T
0
h Ru
i
u e −r · u E e − 0 h(Xv )dv h (Xu ) 1{TL ≥ u} du
|
{z
}
Jump Term
»
–
R TL
+ E e −r · TL − 0 h(Xu )du TL 1{TL ≤ T }
|
{z
}
Diffusion
Term
h
i
R
R ti+1 −r · u
P
u
e
E e − 0 h(Xv )dv h (Xu ) 1{TL ≥ u} du
− N−1
i=1 (i · ∆t ) ti
{z
}
|
Jump Term
19
>
>
–
–
»
»
=
B
C>
RT
RT
PN−1
B
C
−r · TL − 0 L h(Xu )du
−r · TL − 0 L h(Xu )du
− i=1 (i · ∆t ) BE e
1{TL ≤ ti +1 } − E e
1{TL ≤ ti } C
@
A>
>
|
{z
} |
{z
} >
;
0
Diffusion Term
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Diffusion Term
.. Go Back
Credit Risk 2009
42 / 1
Expectations to Solve: Jump Term and Diffusion Term
Jump Term.
[ Ru
]
E e − 0 h(Xv )dv h (Xu ) 1{TL > u}
Since the default intensity is given by a power function,
h(Xt ) = b + ca2 Xt2β , we can solve, more generally, for a given p the
expectation which we name truncated p-Moment
[ Ru
]
p
E e − 0 h(Xv )dv (Xu ) 1{TL > u}
Diffusion Term.1 This term can be seen as the Expected Discount (given no
default) up to the first hitting time to level L
[
]
R TL
E e −r · TL − 0 h(Xu )du 1{TL ≤ T }
.. Go Back
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
43 / 1
Solving the Expectations: the truncated p-Moment
The truncated p-Moment for L > 0 and µ + b > 0 is given by
=
P∞
0
n=0
@
h Rt
i
Ex e − 0 h(Xu )du 1{TL >t} (Xt )p
A
1−2c−2p
−1
2
4|β|
2
“
”
+n− 2c+p
, ν2
2|β|
0
1−p
2|β|
”
` −2β ´
Ax
−
“
”
Γ 1+ 2c+p
2|β|
M ν−1
+n−
„
2c+p
2|β|
1 −c+β
2
2|β| A 2 L−2β−1+p Γ(ν)
2 F2
(2|β|−1+p)
“
”
A L−2β
(A L−2β )
W ν−1
2
“
”
+n− 2c+p
, ν2
2|β|
31
` −2β ´7C
Ax
5A
1+ν
−A
x −2β
2
2
!
1−p
, 1−ν
− κn
2|β|
2
; A L−2β
1−p
− 2|β| , 1 − ν
!#!
1 + 2c+p
, 1+ν
− κn
2|β|
2
−2β
;AL
2 + 2c+p
, 1+ν
2|β|
1−
2|β| A 2 L2c+p−2β Γ(−ν)Γ( 1+ν
−κn )
2”
“
−
2 F2
(2|β|+2c+p)Γ 1−ν
−κ
n
2
n
«
e (p(µ+b)−(b+ω n))t
“
”
“
”
Mκ , ν A L−2β Wκ , ν Ax −2β
n 2
n 2
»
–˛
x
e
˛
d W
−2β ) ˛
Γ(1+ν) dκ
κ, ν (A L
˛
2
κ=κn
"
“
” “
” “
”
1−p
2c+p
2c+p
ν−1
1−2c−2p
1
−κ
−
Γ
1−
Γ
1+
Γ
n
−2
2
2|β|
2|β|
2|β|
4|β|
“
”
× A
Γ 1−ν
−κn
2
“ “
” ”
ω κn − ν−1
+ξ t
2
1−ν
where κn =
A −2β
1
x 2 −c+β e − 2 x
,ν
2
2
„
«
W ν−1
2c+p
+n−
,ν
2
2
2|β|
P
B −
+ ∞
n=1 @e
−
n
n! Γ(1+ν)
2
6
× 4M ν−1
“
o
`
´
κ| Wκ, ν A L−2β = 0
2
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
Credit Risk 2009
44 / 1
Solving the Expectations: the truncated p-Moment
The truncated p-Moment for L = 0 (CDS case) and µ + b > 0 is given by
[ Rt
]
Ex e − 0 h(Xu )du 1{TL >t} (Xt )p
=
∑∞
A
1−2c−2p
−1
2
4|β|
n=0
1−p
)n Γ(1+ 2c+p
( 2|β|
2|β| )
2
{
(
where κn = κ| Wκ, ν2 A L
Rafael Mendoza (McCombs)
−2β
x 2 −c+β e − 2 x
(
)
×M ν−1 +n−( 2c+p ), ν Ax −2β
1
A
n! Γ(1+ν)
)
−2β
2|β|
e (p(µ+b)−(b+ω n))t
2
}
=0
Unified Credit-Equity Modeling
Credit Risk 2009
45 / 1
Solving the Expectations: Diffusion Term
The Diffusion Term
[
] ( )1
R TL
−c+β ϵ A (x −2β −L−2β )
Ex e −r · TL − 0 h(Xu )du 1{TL ≤ T } = xL 2
e 2
×
[
−2β
W 1−ν r +ξ ν (Ax
) ∑∞ ω e −(ω(κn −ϵ 1−ν
+r +ξ)T
Wκn , ν (Ax −2β )
2 )
ϵ
−
,
2
ω
2
2
i˛
+ n=1 ω κ −ϵ 1−ν +r +ξ h ∂
W 1−ν r +ξ ν (AL−2β )
−2β ) ˛
( ( n
)
ν
˛
ϵ
−
,
2 )
∂κ Wκ, 2 (AL
2
ω
2
Rafael Mendoza (McCombs)
Unified Credit-Equity Modeling
]
κ=κn
Credit Risk 2009
46 / 1
Numerical Example 1: the effect of the sensitivity to
variance “c”
Default Intensity function: h(Xt ) = b + ca2 Xt2β . We choose a = σ/S0β = 10
CDS (vol.=20%, c=0, b=0%)
EDS (L=30%S, vol.=20%, c=0, b=0%)
EDS (L=50%S, vol.=20%, c=0, b=0%)
CDS (vol.=20%, c=1, b=2%)
EDS (L=30%S, vol.=20%, c=1, b=2%)
EDS (L=50%S, vol.=20%, c=1, b=2%)
CDS (vol.=20%, c=2, b=2%)
EDS (L=30%S, vol.=20%, c=2, b=2%)
EDS (L=50%S, vol.=20%, c=2, b=2%)
500
450
400
c=2
b=2%
c=1
b=2%
EDS Rates
(bps)
Spreads (bps)
350
300
250
200
150
CEV
c=b=0
100
50
0
0
1
2
3
4
5
6
7
8
9
10
Time (years)
Time
(yrs)
∆t
0.25
r
0.5
Rafael Mendoza (McCombs)
L
{0, 15, 25}
S0
50
r
0.05
q
0
b
{0, 0.02}
Unified Credit-Equity Modeling
c
{0, 1, 2}
β
-1
σ
0.20
Credit Risk 2009
47 / 1
Numerical Example 2: the effect of volatility “σ”
Default Intensity function: h(Xt ) = b + ca2 Xt2β . We choose a = σ/S0β = 20
CDS (vol.=40%, c=0, b=0%)
EDS (L=30%S, vol.=40%, c=0, b=0%)
EDS (L=50%S, vol.=40%, c=0, b=0%)
CDS (vol.=40%, c=1, b=2%)
EDS (L=30%S, vol.=40%, c=1, b=2%)
EDS (L=50%S, vol.=40%, c=1, b=2%)
1300
1200
1100
1000
EDS Rates
(bps)
Spreds (bps)
900
800
c=1
b=2%
700
600
500
CEV
c=b=0
400
300
200
100
0
0
1
2
3
4
5
6
7
8
9
10
Time (years)
Time
(yrs)
∆t
0.25
r
0.5
L
{0, 15, 25}
S0
50
r
0.05
q
0
b
{0, 0.02}
c
{0, 1}
β
-1
σ
0.40
.. Go Back
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Unified Credit-Equity Modeling
Credit Risk 2009
48 / 1