Hybrid Structural Default Modeling
Marat V. Kramin
Director
Fixed Income Analytics
Wells Fargo Securities
March 25, 2013
UNCC
Stephen D. Young
Chief Risk Officer
Affiliated Managers Division
Wells Fargo Asset Management
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Stephen D. Young is the Chief Risk Officer for the Affiliated Managers Division of Wells Fargo Asset
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2
Default Modeling
Statistical Approaches
• Discriminant analysis
• Logit models
• Probit models
Frequently used for default prediction of
firms and individuals (i.e. “scoring models”).
Reduced Form Approach
• Jump process
Frequently used for valuing credit derivatives
by taking observable CDS or bond data and
solving for market implied hazard rates
or survival probabilities.
Structural Default Approaches
• Firm Value
• First Passage
Frequently used for ordinal ranking of firm
credit quality. Also used for relative value trading
(e.g. debt, CDS, and equity trading by hedge funds).
3
Structural Default Modeling: Introduction
• The central distinguishing point of “structural models” is the view of debt,
equity, and other claims issued by a firm as contingent claims on the firm’s
asset value.
• Merton (1974) is the seminal work on structural default modeling and is
based on a firm defaulting if the value of its assets is below the value of debt
at the expiration date of current debt contracts.
• Key idea - M is the debt principal amount and VT is the value of the firm at
date T. Equity is a call option on firm value and the debt holder’s position is
analogous to long a risk-free bond and short a put option on the firm value.
4
Structural Default Modeling: Introduction
Payoff to claimholders
• Thus we have the value from the perspective of equity and debt
holders:
Equity
M
Debt
M
VT
5
Structural Default Modeling: Firm Value Approach
•
Merton (1974) assumes that the asset value 𝑉𝑇 is given by the following
stochastic differential equation in the risk-neutral measure:
𝑑𝑉𝑡 = 𝑉𝑡 𝑟𝑑𝑡 + 𝜎𝑣 𝑑𝑊
•
In Merton’s (1974) framework we have that the boundary condition for the
firm’s equity holders at the terminal period is:
𝐸𝑇 = 𝑀𝑎𝑥 𝑉𝑇 − 𝑀, 0
where ET represents the firm’s equity value at time T, VT is the value of the
firm’s assets at time T, and M is the face amount of a single pure discount
bond or an approximation to total liabilities.
•
Denoting a bond value as B(t,T) one can represent the payoff to the debt
holder as:
𝐵 𝑇, 𝑇 = 𝑉𝑇 − 𝐸𝑇 = 𝑀 − 𝑀𝑎𝑥 𝑀 − 𝑉𝑇 , 0 = 𝑀𝑖𝑛 𝑀, 𝑉𝑇
6
Merton (1974) Model: Firm Value Approach
•
Following the Merton (1974) model, the value of the firm at time t is:
𝐸𝑡 = 𝑉𝑡 ∙ 𝑁 𝑑1 − 𝑀 ∙ 𝑒 −𝑟
𝑇−𝑡
∙ 𝑁 𝑑2
where N(.) is the standard normal cumulative density function and:
𝑑1,2
𝑉
1
𝑙𝑛 𝑀𝑡 + 𝑟 ± 2 ∙ 𝜎𝑣2
=
𝜎𝑉 ∙ 𝑇 − 𝑡
𝑇−𝑡
with r the risk-free interest rate, σv the instantaneous standard deviation of
the return on the firm’s assets, t is the current time, and T is the maturity of
the debt. And, in this model there is one risk-neutral probability of default
given by:
𝑅𝑁𝑃𝐷𝐵𝑆𝑀 = 1 − 𝑁 𝑑2 = 𝑁 −𝑑2
And the following relationship between asset and equity volatility:
𝜎𝐸 = 𝑘𝐸∙ 𝜎𝑉 ,
𝑘𝐸 =
7
𝜕𝐸𝑡
𝜕𝑉𝑡
𝑉
𝑉
∙ 𝐸𝑡 = 𝑁 𝑑1 ∙ 𝐸𝑡
𝑡
𝑡
Merton (1974) Model: Firm Value Approach
• The obvious appeal of the Merton (1974) model is the economic
intuition whereby default occurs when the firm’s asset value (VT) is
less than the debt due (M).
• However, Merton’s model contains several restrictive assumptions
including:
o Simple debt structure (i.e. single debt due at time T where debt
maturity is chosen and debt payments are mapped to single
payment on debt maturity date in some manner)
o Default possible only at maturity of debt (i.e. time T)
o Constant interest rates
o Default never a “surprise” (i.e. no jump to default)
8
Geske (1977) Model: Firm Value Approach
•
•
Geske (1977) generalizes the Merton (1974) model to cases where the firm is
financed with coupon-paying debt or with debt maturity at different dates.
At each payment date, shareholders decide either to meet their obligation or
discontinue firm operations and leave firm assets to debt holders.
The value of the firm with two tranches of debt is given by:
𝐸𝑡 = 𝑉𝑡 ∙ 𝑁2 𝑘1 + 𝑠1 , 𝑘2 + 𝑠2 ; 𝜌 − 𝑀2 ∙ 𝑒 −𝑟
−𝑀1 ∙ 𝑒 −𝑟 𝑇1−𝑡 ∙ 𝑁 𝑘1
𝑇2 −𝑡
∙ 𝑁2 𝑘1 , 𝑘2 ; 𝜌
𝑉
1
𝑙𝑛 𝑌𝑡 + 𝑟 ∙ 𝑇𝑖 − 𝑡 − 2 𝑠𝑖2
𝑖
𝑘𝑖 =
, 𝑖 = 1, 2
𝑠𝑖
𝑠𝑖 = 𝜎𝑉 ∙ 𝑇𝑖 − 𝑡, 𝑖 = 1, 2
𝜌=
𝑇1 −𝑡
,𝑌
𝑇2 −𝑡 1
= 𝑉, 𝑌2 = 𝑀2
where in the Geske (1977) model with two tranches of debt N2 is the bivariate cumulative normal distribution function. Yi is related to M1 and M2
and “critical V” – (see paper) with M1 short-term debt at time T1 and M2
long-term debt at time T2. N2 is the bivariate cumulative normal distribution
and V is the critical firm value for bankruptcy at T1.
9
Geske Model: Firm Value Approach
•
Similar to Merton (1974) we have that:
𝜎𝐸 = 𝑁2 𝑘1 + 𝑠1 , 𝑘2 + 𝑠2 ; 𝜌
•
With Geske (1977) and two tranches of debt we can compute the following
total, short, and forward risk-neutral probabilities of default:
𝑅𝑁𝑃𝐷𝐺𝑇 = 1 − 𝑁2 𝑘1 , 𝑘2 ; 𝜌 ,
𝑅𝑁𝑃𝐷𝐺𝐹 = 1 −
•
•
𝑉𝑡
∙𝜎
𝐸𝑡 𝑉
𝑅𝑁𝑃𝐷𝐺𝑆 = 1 − 𝑁 𝑘1
𝑁2 𝑘1 , 𝑘2 ; 𝜌
𝑁 𝑘1
Delianedis and Geske (2003) find that the compound option formulation
provides additional information about migration and default relative to the
Merton framework.
However, one drawback to the compound option framework is that with
each additional tranche of debt considered one must evaluate a more
complex cumulative distribution function (e.g. three tranches tri-variate, …)
10
Structural Default Modeling: Extensions to Merton Framework
• There have been numerous extensions to the original Merton (1974)
framework including Geske (1977) and other works that account for:
o
o
o
o
Default before maturity
Stochastic interest rates
Jumps in the firm-value process
Many others
• While firm value approaches are based on analyzing a firm’s capital
structure and comparing asset value to debt, first passage models
allow for default when the asset value drops for the first time below
a pre-defined barrier, allowing for default at any time.
11
Structural Default Modeling: Firm Value and First Passage
• Given a particular firm and a firm value (e.g. Merton ((1974))
and first passage model (e.g. Black and Cox (1976)) what is the
probability that the firm will end up insolvent?
V
No default in either formulation
No default for firm value
M
Default for first passage
Default in both formulations
T
12
Default probability
(the area)
Merton (1976) Model: Option Pricing under Discontinuous Asset Returns
• Merton (1976) models the asset price S as a combination of a Brownian
motion and a compound Poisson process:
𝑑𝑆𝑡 = 𝑆𝑡
1
𝑟 − − 𝜆𝑘 𝑑𝑡 + 𝜎𝑑𝑊𝑡 + 𝑦𝑡 − 1 𝑑𝑁𝑡
2
where 𝜆 is the Poisson process intensity,
1
the log price jump size ln 𝑦𝑡 ~𝑁 𝜇, 𝛿 2 ∶ κ ≡ 𝐸 𝑦𝑡 − 1 = 𝑒𝑥𝑝 𝜇 + 2 𝛿 2
𝑦𝑡 , 𝑁𝑡 , 𝑊𝑡 are all independent of each other. Jump Risk is diversifiable and
earn no risk premium
13
Merton (1976) Model: Option Pricing under Discontinuous Asset Returns
•
Following Merton (1976), the value of the firm at time t is:
∞
𝐸𝑡 =
𝑛=0
′
eλ τ λ′ τ
n!
n
⋅ 𝐸𝑀𝑒𝑟𝑡𝑜𝑛 𝑉𝑡 , 𝑀, 𝑟𝑛 , 𝜎𝑛2 , 𝑇 − 𝑡
where EMerton is the standard Merton (1974) option value. We also have
that:
𝜏 = 𝑇 − 𝑡,
𝜆′ = 𝜆 ⋅ 1 + 𝜅
and:
𝜎𝑛2
=
𝜎𝑣2
𝑛𝛿 2
+
,
𝜏
𝑟𝑛 = 𝑟 − 𝜆 ⋅ 𝜅 +
𝑛 ⋅ 𝑙𝑛 1 + 𝜅
𝜏
Thus the Merton (1976) model allows for jump diffusion. Similar to Merton
(1974) and Geske (1977) we have that:
∞
𝜎𝐸 =
𝑛=0
′
eλ τ λ ′ τ
n!
n
⋅ 𝑁 𝑑1
𝑉𝑡
∙ ∙ 𝜎𝑉,
𝐸𝑡
14
∞
𝑅𝑁𝑃𝐷𝑀 = 1 −
𝑛=0
′
e λ τ λ′ τ
n!
n
⋅ 𝑁 𝑑2
A Firm-Value Structural Default Lattice based Approach
• A simple lattice based approach allows us to address shortcomings
of other formulations such as Merton (1974) and Geske (1977).
• The first lattice we present will allow for many tranches of debt,
coupons, interest payments, etc.
• This lattice will serve to “operationalize” the Geske framework so
that the implementation to many tranches of debt is quite simple.
• After we will see how one may use the lattice to create a hybrid firm
value/first passage approach and allow for jumps in the underlying
asset value-process.
15
A Firm-Value Structural Default Lattice based Approach
•
The lattice leverages both backward recursion and forward induction
whereby given an underlying process for the firm’s asset value and a
debt structure, using backward recursion one assumes that the
shareholder’s are rational and will pay off debt obligations if the asset
value is greater than the payment due at the time due.
•
Then, using simple forward induction one may compute not a single
risk-neutral probability of default but rather an entire term structure.
•
For this lattice we will use the following parameterization:
𝑢=𝑒
𝛼Δ𝑡+𝜎 Δ𝑡
,𝑑 = 𝑒
𝛼Δ𝑡−𝜎 Δ𝑡
1 2
1
𝑇
, 𝛼 = 𝑟 − 𝜎 , 𝑝 = , Δ𝑡 =
2
2
𝑛
which represent the up and down multipliers, the drift of log asset
returns, the risk-neutral probability, and the time increment.
16
A Firm-Value Structural Default Lattice based Approach
• Upon evolving the asset value V, we apply backward induction
(recursion) to populate the lattice with firm values where default
occurs should the asset value become insufficient to meet debt
obligations (i.e. Vt ≤ Mt). We carry and indicator variable back
through the lattice where one is to signify solvency and zero default.
• We then apply forward induction to calculate state prices and
calculate survival (default) probabilities from which we can calculate
conditional measures.
17
Backward Induction
•
•
•
Set the value of the variable as one at the given time and backward induct it
(with no discounting) to time zero resetting all the state (node) values where
the n-order compound option on the firm’s assets is exercised (does not
exist anymore) to zero along the way.
The calculated probability of option existence at the given time corresponds
to survival probability up to this time.
Standard backward induction the iterative equation is as follows:
𝑂𝑖,𝑗 = 𝑑𝑖,𝑗
•
𝑘∈𝐾𝑖𝑗
𝑝𝑘 ∙ 𝑂𝑖+1,𝑘
Ki,j is the set of numbers of lattice nodes at the next (i+1) time layer linked
to the node (i,j). The sum is a probability weighted average of the values of
the option at the corresponding nodes at the next time layer. Oi,j, di,j are an
option value and a local discount factor at node (i,j) respectively.
The backward induction to compute the survival probability Qi,j at node (i,j):
𝑄𝑖,𝑗 = 𝐼𝑖,𝑗
𝑘∈𝐾𝑖𝑗
𝑝𝑘 ∙ 𝑄𝑖+1,𝑘
Ii,j is a local indicator at the node (i,j), which is one when the
option/company exists and zero if the option is optimal to exercise.
18
Terminal and Exercise Conditions
•
The terminal conditions in the case of the option valuation and the
computation of unconditional survival probability are respectively as
follows:
𝑂𝑇 = 𝐽𝑇 𝑀𝑎𝑥 𝑉𝑇 − 𝑀𝑇 , 0 ,
•
𝑄𝑇 = 𝐽𝑇 𝐻 𝑉𝑇 − 𝑀𝑇
O, V, M are the values of the option (equity), the underlying (assets), and
the compound option strike (debt), and H is the Heaviside function.
At every time t where intermediate exercise may be optimal (i.e. a debt
tranche is due) the following reset calculations take place:
𝑂𝑡 = 𝐽𝑡 𝑀𝑎𝑥 𝑂𝑡 − 𝑀𝑡 , 0 = 𝐽𝑡 𝑂𝑡 − 𝑀𝑡 ∙ 𝐼𝑡 ,
𝑄𝑡 = 𝐽𝑡 𝑄𝑡 ∙ 𝐼𝑡
𝐼𝑡 = 𝐽𝑡 𝐻 𝑂𝑡 − 𝑀𝑡 is the indicator variable, which is one when the
option/company exists and zero if the option is optimal to exercise
O t andQ t are the continuation (i.e. backward-inducted) option and survival
probability
O t and Q t are corresponding values after the optimal option decision
19
Forward Induction
•
Standard Expectation:
𝐸 𝑓 𝑥𝑡
𝐸
•
= 𝐸 𝑓 𝑥𝑡
𝑓 𝑥 𝛿 𝑥 − 𝑥𝑡 𝑑𝑥 =
𝑓 𝑥𝑡 𝛿 𝑥 − 𝑥𝑡 𝑑𝑥 =
𝑓 𝑥 𝐸 𝛿 𝑥 − 𝑥𝑡 𝑑𝑥 =
𝑓 𝑥 𝐸 𝐼𝑡 𝛿 𝑥 − 𝑥𝑡 𝑑𝑥 =
Standard Density:
𝑝 𝑥, 𝑥𝑡 = 𝐸 𝛿 𝑥 − 𝑥𝑡
•
𝑓 𝑥 𝑝 𝑥, 𝑥𝑡 𝑑𝑥
Expectation for Forward Induction:
𝐸 𝑓 𝑥𝑡 𝐼𝑡 =
•
𝛿 𝑥 − 𝑥𝑡 𝑑𝑥 = 𝐸
Adjusted Density:
q 𝑥, 𝑥𝑡 = 𝐸 𝐼𝑡 𝛿 𝑥 − 𝑥𝑡
20
𝑓 𝑥 𝑞 𝑥, 𝑥𝑡 𝑑𝑥
Forward Induction
•
The probability P of the option existence at a given time can be alternatively
computed as a sum of state prices of security S that have the constant value
of one at this time if the option exists (and zero otherwise):
𝑃𝑖 =
𝑆𝑖,𝑗
𝑗
•
The state prices of such securities can be built forward along the lattice
taking into account the existence indicators Ii,j obtained as a by-product
during the backward induction used to value the option:
𝑆𝑖+1,𝑗 =
𝑝𝑘∙ 𝑆𝑖,𝑘 ∙ 𝐼𝑖,𝑘
𝑘∈Λij
•
where  ij is the set of numbers of lattice nodes at the previous (i) time
layer linked to the node (i+1,j) and the sum is a probability weighted
average of the values of the state prices at the corresponding nodes at the
previous time layer.
The initial condition when it is not optimal to exercise the option at time
zero:
𝑆0,0 = 1
21
Induction Process
•
We carry option values (i.e. equity) and indicator variables back through the
lattice with the backward recursion.
•
Under the assumption that the firm’s managers are rational, option values
are positive if the value of the firm’s assets exceeds the value of liabilities
due at points where the firm has debt and perhaps coupons to pay.
•
In the event the firm’s asset value is less than the liabilities, or if this value is
less than a pre-specified default boundary, default occurs and the equity
value goes to zero.
•
With the backward recursion we carry an indicator variable back through
the lattice where this variable is reset to zero in the case of default.
•
For the forward induction, we carry the state price adjusted with the
indicator variable through the lattice in order to derive survival probabilities
from which we can get unconditional and conditional default probabilities.
•
At any time slice, the survival probability is the sum of the adjusted state
prices S.
22
Backward Induction
EXHIBIT 1
Backward induction in the lattice method.
O 44 >0
1
O 33 >0
1
O 22 >0
1
O 11 >0
1
O 00 >0
O 43 >0
1
O 32 >0
1
O 21 >0
1
O42 = 0
1
0
O 10 >0
O31 = 0
1
0
O20 = 0
O41 = 0
0
0
O30 = 0
0
O40 = 0
0
t=0
t =1
t=2
t=3
t=T=4
In the above exhibit the backward induction procedure is presented. At each node there are two variables: the
firm's equity value and the default indicator. The equity value is positive when the value of the firm's assets
exceeds the value of debt, and zero otherwise. Here it is assumed that the debt is due at times t = {2, 3, 4}.
The bold cells represent the default area where the default indicator is zero; for the other cells it is one. While
the example above has only four steps, backward induction is easily generalized for any number of steps.
23
Debt is due at
times 2, 3, and 4
with the highlighted
region those nodes
where the asset value
is less than the debt due
Forward Induction
EXHIBIT 2
Forward induction in the lattice method.
From forward induction
we can calculate the probability
of reaching a particular node as
the product of one-half and the sum
of the probabilities of the two prior
nodes.
0.0625
1
0.125
1
0.25
1
0.50
1
1.00
1
0.25
1
0.375
1
0.50
1
0.50
1
The time dependent survival
probabilities are given by the
sum of the products of the state
prices and indicator variables. From
these we can calculate default
probabilities and conditional measures
0.375
0
0.375
0
0.25
0
0.250
0
0.125
0
0.0625
0
1.0
1.0
0.75
0.5
0.3125
t=0
t =1
t=2
t=3
t=T=4
In the above exhibit the forward induction procedure is presented. At each node there are two variables:
the state price and the default indicator. The bold cells represent the default area where the default indicator is
zero; for the other cells it is one. In the bottom the survival probability of the firm is given for every time period.
The above lattice is consistent with Exhibit 1 in terms of the number of steps and the default boundary.
24
Merton (1974), Geske (1977), and Firm-Value Structural Default Lattice
•
Results for a hypothetical firm that has an asset value of $70B (Billion),
volatility of assets of = 20%, seven tranches of liabilities due at each of 1, 2,
5, 7, 10, 20, and 30 years with amounts of $15B, $10B, $2.5B, $2.5B, $5B,
$10B and $15B respectively with risk-free rate is assumed to be 3.5%.
Liability structure is simplified to accommodate Merton (1974) and Geske
(1977) approaches.
EXIBIT 3
Equity values and risk-neutral default probabilities.
Case 1 - Single tranche of debt.
Black-Scholes-Merton
Geske
Lattice
Equity Value (E t ) $mm
35,584
35,583
35,584
RNPD T
31.43%
31.43%
31.21%
Equity Value (E t ) $mm
NA
30,096
30,095
RNPD T
NA
14.10%
13.90%
Case 2 - Two tranches of debt.
Black-Scholes-Merton
Geske
Lattice
RNPD S
NA
0.85%
0.83%
RNPD F
NA
13.36%
13.18%
In the above exhibit we establish consistency between the lattice model and each of the Black-Scholes-Merton
and Geske approaches. For each of these models and the values presented in the table we reduce the term
structure of liabilities using a simple weighting scheme. The above exhibit includes equity values for each model
followed by risk-neutral default probabilities where for the Geske and lattice models in Case 2 we have a total,
short, and forward measure (i.e. RNPD T , RNPD S , and RNPD F ).
25
Firm-Value Structural Default Lattice based Model
Benchmarking was done with simplifications to debt structure for Merton
(1974) and Geske (1977). Below is full term structure of risk-neutral
survival and default probabilities from the Firm-Value Structural Default
Lattice and full debt structure using all liabilities with respective times.
EXHIBIT 4
Plot of risk neutral survival and default probabilities.
Risk Neutral Survival and Default Probabilities
100.00%
7.00%
99.00%
6.00%
5.00%
97.00%
96.00%
4.00%
95.00%
3.00%
94.00%
93.00%
2.00%
92.00%
1.00%
91.00%
90.00%
0.00%
1
2
5
7
10
20
Risk Neutral Default Probability
98.00%
Risk Neutral Survival Probability
•
30
Tenor
Risk Neut ral Su rv ival Probabil ity
Risk Neut ral Defaul t Probabil ity
The above exhibit depicts the term structure of risk neutral survival and default probabilities based on the
complete set of debt tranches and the lattice model.
26
A Hybrid Structural Default Model
• Firm Value
o Default can only occur when debt is due.
o Empirically, fails to explain short-term credit spreads
largely as a result of default only likely when debt is due and
debt may not be due making the probability of default zero
and thus credit spreads should be zero.
• First Passage
o Artificial default boundary is typically some function of
asset value or debt.
o Less economic intuition than with Firm Value where asset
value versus debt are used to define default.
27
Amin (1993) Model
• Amin (1993) develops a discrete time model to value options when
the underlying follows a jump diffusion process.
o Multivariate jumps are superimposed on to a standard binomial
lattice.
o Consistent with Merton (1976) and his assumption of
diversifiable jump risk, in a risk neutral world the governing
process for the underlying is given by:
𝑉𝑡
=𝑒
𝑉0
𝑡
0
𝑡
1
𝑟−2∙𝜎𝑉2 −𝜆∙𝐾 𝑑𝑢+ 0 𝜎𝑑 𝑊 𝑢 + 𝑁𝑡
𝑗=1 𝑙𝑛 𝑌 𝑗
where 𝜆 is the intensity of the jump process, 𝐾 = 𝐸𝑌 𝑌 − 1 which is
the expectation of the distribution function of 𝑌 𝑗 ′𝑠 which are
independent and identically distributed random variables
corresponding to the Poisson jump magnitudes, 𝑊 𝑢 is a standard
Brownian motion under a risk neutral measure, and 𝑁𝑡 is the total
number of jumps up to time 𝑡.
28
Amin (1993) Model
• With a partitioning of the trading interval 0, 𝜏 into 𝑛 subintervals of
length ℎ𝑛 = 𝜏/𝑛 Amin (1993) derives the following magnitudes and
risk-neutral probabilities for the discrete time approximating
process:
𝑃𝑟𝑜𝑏 𝑋𝑛 𝑡 + ℎ𝑛 − 𝑋𝑛 𝑡 = 𝛼 ∙ ℎ𝑛 + 𝜎𝑉 ∙ ℎ𝑛 = 𝑞𝑛 ∙ 1 − 𝜆 ∙ ℎ𝑛
𝑃𝑟𝑜𝑏 𝑋𝑛 𝑡 + ℎ𝑛 − 𝑋𝑛 𝑡 = 𝛼 ∙ ℎ𝑛 + 𝑙 ∙ 𝜎𝑉 ∙ ℎ𝑛 ; 𝑙 ≠ ±1 = 𝜆 ∙ ℎ𝑛 ∙ 𝑑𝑁𝑛 𝑙
𝑃𝑟𝑜𝑏 𝑋𝑛 𝑡 + ℎ𝑛 − 𝑋𝑛 𝑡 = 𝛼 ∙ ℎ𝑛 − 𝜎𝑉 ∙ ℎ𝑛 = 1 − 𝑞𝑛 ∙ 1 − 𝜆 ∙ ℎ𝑛
where the above represent a move up, a jump, and down along with
1
their respective probability measures with 𝛼 = 𝑟 − ∙ 𝜎𝑉2 − 𝜆 ∙ 𝐾 the
2
drift of the logarithm of the asset value.
29
Amin (1993) Model
• 𝑙 is an integer which spans all nodes at a particular time slice
excluding a single move up or down. 𝑑𝑁𝑛 𝑙 corresponds to the
distribution of the probability mass associated with jumps over all
states and for 𝑙 = 0, 𝑙 ≠ ±1 and all other integer 𝑙 is given by:
𝑑𝑁𝑛 0 = 𝑁 𝛼 ∙ ℎ𝑛 + 1 +
𝑑𝑁𝑛 𝑙 = 𝑁 𝛼 ∙ ℎ𝑛 + 𝑙 +
1
1
∙ 𝜎𝑉 ∙ ℎ𝑛 − 𝑁 𝛼 ∙ ℎ𝑛 − 1 +
∙ 𝜎𝑉 ∙ ℎ𝑛
2
2
𝑑𝑁𝑛 ±1 = 0
1
1
∙ 𝜎𝑉 ∙ ℎ𝑛 − 𝑁 𝛼 ∙ ℎ𝑛 + 𝑙 −
∙ 𝜎𝑉 ∙ ℎ𝑛
2
2
• For 𝑙 = 0 we have a move along the center of the lattice.
• With local and non-local moves, while built upon a standard
binomial lattice with 𝑛 + 1 nodes at each time slice, Amin (1993)
results in 2 ∙ 𝑛 + 1 nodes at each time period which is consistent with
many trinomial models and necessary to accommodate the case
where 𝑙 = 0. The number of total jumps emanating from all nodes at
a time slice is equal to one for 𝑛 = 0 and 2 ∙ 𝑛 + 1 ∙ 3 ∀ 𝑛 > 0.
30
Amin (1993) Model
• To be consistent with Merton (1976) we specify a normal
distribution for 𝑙𝑛 𝑌 . To complete the specification we need only
the risk-neutral measure with probability (see Amin (1993),
Equation (25), p. 1847) given by 𝑞𝑛 :
𝑞𝑛 =
𝑒 𝑟∙ℎ𝑛 −𝜆∙ℎ𝑛 ∙𝐸𝑌𝑛 𝑌𝑛
1−𝜆∙ℎ𝑛
−𝑒 𝛼∙ℎ𝑛 −𝜎𝑉 ∙ ℎ𝑛
𝑒 𝛼∙ℎ𝑛 +𝜎𝑉 ∙ ℎ𝑛 −𝑒 𝛼∙ℎ𝑛 −𝜎𝑉 ∙ ℎ𝑛
• It should be noted that the lattice specification can easily be adjusted
to accommodate a constant dividend yield with minor modifications
to the magnitudes and risk-neutral probability of the lattice
equations.
• Practical implementation of the lattice entails a truncation of the
distribution of non-local moves as one is effectively integrating over
a specified density and therefore has to choose the upper and lower
limits accordingly.
31
A Hybrid Structural Default Model
• The hybrid structural default model lattice construction is based on
•
•
•
•
Amin (1993).
The hybrid lattice approach allows for local and non-local (i.e.
jumps) moves in the asset value thereby allowing for default to come
as a surprise.
The hybrid lattice allows for a full term structure of debt, coupons,
interest, etc and leverages the same backward and forward induction
from Jabbour, Kramin, and Young (2010) Journal of Derivatives,
Summer 2012.
In addition, we allow for tranches of debt as well as a default
boundary.
The Merton (1974), Geske (1977), and Merton (1976) models are
particular cases of the hybrid lattice. Many extant models are
particular cases of the hybrid lattice.
32
Backward Induction
EXHIBIT 5
Geometry and backward induction in the lattice method.
O24 > 0
1
O12 > 0
O23 > 0
1
1
O00 > 0
O11 > 0
O22 > 0
1
1
1
O10 = 0
O21 = 0
0
0
O20 = 0
0
t=0
t=1
t=T=2
In the above exhibit the lattice geometry and backward induction procedure is presented. At each node there
are two variables: the firm's equity value and the default indicator. The equity value is positive when the
value of the firm's assets exceeds the value of debt and/or a boundary, and zero otherwise. Here it is assumed
there is debt and/or a boundary at times t = {1, 2} (i.e. the shaded region). The lower cells in each node are for
indicator variables and are zero where the asset value of the firm is less than the debt due and/or default boundary.
Solid lines represent local moves and dashed lines are used to indicate jumps. For a complete description of the
lattice construction we point readers to Amin (1993).
33
Forward Induction
EXHIBIT 6
Geometry and forward induction in the lattice method.
S 24
1
S 12
S 23
1
1
S 00
S 11
S 22
1
1
1
S 10
S 21
0
0
S 20
0
P 0 = S 00
t=0
P 1 = S 11 + S 12, S 10 = 0
t=1
P 2 = S 22 + S 23 + S 24, S 20 = 0, S 21 = 0
t=T=2
In the above exhibit the lattice geometry and forward induction procedure is presented. At each node there
are two variables: the nodal survival probability and the default indicator. The surivival probability S is in the range
(0,1] when the value of the firm's assets exceeds the value of debt and/or a boundary, and zero otherwise. Here it
is assumed there is debt and/or a boundary at times t = {1, 2} (i.e. the shaded region). The lower cells in each
node are for indicator variables and are zero where the asset value of the firm is less than the debt due and or
default boundary. Solid lines represent local moves and dashed lines are used to indicate jumps. Below the lattice we
have the time-dependent firm survival probability which is the sum of the individual nodal survival probabilities
which are zero when the indicator variable is zero. Thus the time-dependent firm survival probability is calculated
via a simple sum where adds all values conditional on the indicator variable being one. For a complete description of
the lattice construction we point readers to Amin (1993).
34
Merton (1974), Geske (1977), Merton (1976) and Hybrid Structural Default Lattice
•
Benchmark results based on same inputs as prior set (see slide 23). Liability
structure is simplified to accommodate Merton (1974), Geske (1977), and
Merton (1976) models.
EXHIBIT 7
Merton (1974), Ges ke (1977), Merton (1976), and lattice equity values and ris k-neutral default probabilities .
Cas e 1 - Single tranche of debt.
Merton (1974)
Ges ke (1977)
Merton (1976)
Lattice
Equity Value (E t ) $mm
35,584
35,583
35,584
35,584
RNPD T
31.43%
31.43%
31.43%
31.64%
Equity Value (E t ) $mm
30,096
30,096
RNPD T
14.10%
14.17%
Cas e 2 - Two tranches of debt.
Ges ke (1977)
Lattice
RNPD S
0.85%
0.83%
RNPD F
13.36%
13.43%
Cas e 3 - Single tranche of debt with l = 1 (i.e. 1 jump per annum), k = 0, and u = .0484.
Merton (1976)
Lattice
Equity Value (E t ) $mm
40,669
40,667
RNPD T
48.36%
48.55%
In the above we es tablis h the cons is tency among the lattice model and the Merton (1974), Ges ke (1977) and
the Merton (1976) models where the s tructure of the tranches of debt have been trans formed into one and two
tranches for Merton (1974) and Merton (1976) and then Ges ke (1977) res pectively. The above exhibit includes
equity values for each model followed by ris k-neutral default probabilities where for the Ges ke (1977) and lattice
models in Cas e 2 with two tranches of debt, we have a total, s hort, and forward meas ure (i.e. RNPD T , RNPD S ,
and RNPD F ). For the third cas e we have l, k, and u which are the default intens ity expres s ed in number of jumps
per annum, mean, and variance as s ociated with the dis tribution of the jumps . For the lattice res ults , the number
of time s teps is s et equal to 6,000 (i.e. n = 6,000).
35
Hybrid Structural Default Lattice
EXHIBIT 8
Lattice equity values and ris k-neutral default probabilities .
Cas e 1
l = 0 (i.e. 0 jumps per annum)
Equity Value (E t ) $mm
28,556
Time Period
RNPD T
RNPD F
1.00
0.53%
0.53%
2.00
0.54%
0.02%
k = 0
5.00
0.59%
0.05%
u = 0
7.00
0.72%
0.12%
f= 0
10.00
1.20%
0.49%
20.00
4.83%
3.67%
30.00
6.38%
1.63%
Cas e 2
l = 1 (i.e. 1 jump per annum)
l = 2 (i.e. 2 jumps per annum)
l = 3 (i.e. 3 jump per annum)
Equity Value (E t ) $mm
29,675
Equity Value (E t ) $mm
31,105
Equity Value (E t ) $mm
32,462
Time Period
RNPD T
RNPD F
RNPD T
RNPD F
RNPD T
RNPD F
1.00
3.45%
3.45%
5.66%
5.66%
7.34%
7.34%
2.00
3.90%
0.46%
6.73%
1.13%
9.05%
1.84%
k = 0
5.00
4.72%
0.85%
8.59%
1.99%
12.17%
3.43%
u = .0 4 8 4
7.00
5.95%
1.29%
11.16%
2.81%
16.11%
4.49%
f= 0
10.00
8.90%
3.14%
16.94%
6.51%
24.33%
9.79%
20.00
23.10%
15.58%
38.24%
25.64%
49.60%
33.40%
30.00
30.69%
9.87%
49.68%
18.54%
62.87%
26.32%
Cas e 3
l = 1 (i.e. 1 jump per annum)
l = 2 (i.e. 2 jumps per annum)
l = 3 (i.e. 3 jump per annum)
Equity Value (E t ) $mm
29,837
Equity Value (E t ) $mm
31,401
Equity Value (E t ) $mm
32,894
Time Period
RNPD T
RNPD F
RNPD T
RNPD F
RNPD T
RNPD F
1.00
3.67%
3.67%
5.78%
5.78%
7.70%
7.70%
2.00
4.21%
0.56%
6.90%
1.18%
9.46%
1.91%
k = -.0 2 5
5.00
5.09%
0.91%
8.92%
2.17%
12.70%
3.58%
u = .0 4 8 4
7.00
6.32%
1.30%
11.57%
2.91%
16.59%
4.46%
f= 0
10.00
9.39%
3.27%
17.28%
6.46%
24.42%
9.39%
20.00
23.18%
15.22%
37.86%
24.88%
49.21%
32.79%
30.00
30.63%
9.70%
49.11%
18.10%
62.30%
25.78%
Cas e 4
l = 1 (i.e. 1 jump per annum)
l = 2 (i.e. 2 jumps per annum)
l = 3 (i.e. 3 jump per annum)
Equity Value (E t ) $mm
29,547
Equity Value (E t ) $mm
30,875
Equity Value (E t ) $mm
32,116
Time Period
RNPD T
RNPD F
RNPD T
RNPD F
RNPD T
RNPD F
1.00
3.04%
3.04%
5.26%
5.26%
7.58%
7.58%
2.00
3.46%
0.44%
6.26%
1.06%
9.12%
1.66%
k = .0 2 5
5.00
4.20%
0.77%
8.14%
2.00%
12.32%
3.52%
u = .0 4 8 4
7.00
5.42%
1.27%
10.79%
2.88%
16.45%
4.71%
f= 0
10.00
8.49%
3.25%
16.85%
6.79%
24.77%
9.96%
20.00
23.15%
16.02%
39.03%
26.68%
51.05%
34.93%
30.00
31.07%
10.31%
50.81%
19.32%
64.42%
27.32%
In the above we provide res ults for the hybrid lattice for cas es which include one, two, and three jumps per annum for the full term s tructure of
liabilities as s ociated with the hypothetical firm. The res ults include the full term s tructure of ris k neutral default probabilities and the corres ponding
unconditional denoted RNPD T and the conditional RNPD F values . In the model we allow for a functional default boundary (f). The functional
boundary may be a cons tant, related to as s et of equity value, or s ome other s pecification which is up to the us er.
36
Conclusions
•
In this presentation we present a lattice based approach to structural default
modeling.
•
The lattice is flexible and may accommodate complex capital structures and
serves to “operationalize” the Geske (1997) model which is used in practice but
typically limited to two tranches of debt as a result of the necessary integration.
•
As shown, the lattice may be extended to include jumps in the asset value
process. We implement a hybrid based structural default model by leveraging
Amin (1993). With jumps, and an assumption around the distribution when a
jump occurs (we assumed that the log of jump magnitude is normally distributed
– could be double exponential, etc.), and a more complete liability structure one
may better model default and capture short-term spread behavior. With the
introduction of a default boundary one may create a hybrid based structural
default model that is flexible, computationally efficient, and could produce a
wealth of term structures of default probabilities.
•
Structural default models have been proven to be useful. The proposed hybrid
based approach should serve to further enhance their usefulness for ordinal
ranking of credit worthiness, relative value trading, and, perhaps relative pricing
in the future.
37