Hybrid Structural Default Modeling

Hybrid Structural Default Modeling

Marat V. KraminDirectorFixed Income AnalyticsWells Fargo Securities

March 25, 2013UNCC

Stephen D. YoungChief Risk OfficerAffiliated Managers DivisionWells Fargo Asset Management

Disclosures

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Marat V. Kramin is a Director in the Fixed Income Analytics group at Wells Fargo Securities in Charlotte, North Carolina.

Stephen D. Young is the Chief Risk Officer for the Affiliated Managers Division of Wells Fargo Asset Management in Charlotte, North Carolina.

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Default Modeling

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Statistical Approaches• Discriminant analysis• Logit models• Probit models

Reduced Form Approach• Jump process

Structural Default Approaches• Firm Value• First Passage

Frequently used for default prediction offirms and individuals (i.e. “scoring models”).

Frequently used for valuing credit derivatives by taking observable CDS or bond data andsolving for market implied hazard rates or survival probabilities.

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).

Structural Default Modeling: Introduction

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• 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.

Structural Default Modeling: Introduction

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• Thus we have the value from the perspective of equity and debt holders:

VT

Payo

ff to

cla

imho

lder

s

M

MEquity

Debt

Structural Default Modeling: Firm Value Approach

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• Merton (1974) assumes that the asset value is given by the following stochastic differential equation in the risk-neutral measure:

• In Merton’s (1974) framework we have that the boundary condition for the firm’s equity holders at the terminal period is:

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:

Merton (1974) Model: Firm Value Approach

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• Following the Merton (1974) model, the value of the firm at time t is:

where N(.) is the standard normal cumulative density function and:

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:

And the following relationship between asset and equity volatility:

Merton (1974) Model: Firm Value Approach

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• 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 rateso Default never a “surprise” (i.e. no jump to default)

Geske (1977) Model: Firm Value Approach

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• 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:

where in the Geske (1977) model with two tranches of debt N2 is the bi-variate 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.

Geske Model: Firm Value Approach

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• Similar to Merton (1974) we have that:

• With Geske (1977) and two tranches of debt we can compute the following total, short, and forward risk-neutral probabilities of default:

• 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, …)

Structural Default Modeling: Extensions to Merton Framework

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• There have been numerous extensions to the original Merton (1974) framework including Geske (1977) and other works that account for:

o Default before maturityo Stochastic interest rateso Jumps in the firm-value processo 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.

Structural Default Modeling: Firm Value and First Passage

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• 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?

T

V

M

No default in either formulation

Default in both formulationsDefault probability(the area)

No default for firm value

Default for first passage

Merton (1976) Model: Option Pricing under Discontinuous Asset Returns

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• Merton (1976) models the asset price S as a combination of a Brownian motion and a compound Poisson process:

where is the Poisson process intensity,

the log price jump size

are all independent of each other. Jump Risk is diversifiable and earn no risk premium

Merton (1976) Model: Option Pricing under Discontinuous Asset Returns

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• Following Merton (1976), the value of the firm at time t is:

where EMerton is the standard Merton (1974) option value. We also have that:

and:

Thus the Merton (1976) model allows for jump diffusion. Similar to Merton (1974) and Geske (1977) we have that:

A Firm-Value Structural Default Lattice based Approach

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• 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.


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• 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:

which represent the up and down multipliers, the drift of log asset returns, the risk-neutral probability, and the time increment.


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• 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.

Backward Induction

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• 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:

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):

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.

Terminal and Exercise Conditions

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• The terminal conditions in the case of the option valuation and the computation of unconditional survival probability are respectively as follows:

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:

is the indicator variable, which is one when the option/company exists and zero if the option is optimal to

exercise

and are the continuation (i.e. backward-inducted) option and survival probability

and are corresponding values after the optimal option decision

tO tQ

tO tQ

Forward Induction

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• Standard Expectation:

• Expectation for Forward Induction:

• Standard Density:

• Adjusted Density:

Forward Induction

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• 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):

• 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:

where 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:

ij

Induction Process

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• 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.

Backward Induction

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EXHIBIT 1Backward induction in the lattice method.

O 44 >01

O 33 >01

O 22 >0 O 43 >01 1

O 11 >0 O 32 >01 1

O 00 >0 O 21 >0 O42 = 01 1 0

O 10 >0 O31 = 01 0

O20 = 0 O41 = 00 0

O30 = 00

O40 = 00

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: thefirm's equity value and the default indicator. The equity value is positive when the value of the firm's assetsexceeds 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. Whilethe example above has only four steps, backward induction is easily generalized for any number of steps.

Debt is due attimes 2, 3, and 4with the highlightedregion those nodeswhere the asset valueis less than the debt due

Forward Induction

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EXHIBIT 2Forward induction in the lattice method.

0.06251

0.1251

0.25 0.251 1

0.50 0.3751 1

1.00 0.50 0.3751 1 0

0.50 0.3751 0

0.25 0.2500 0

0.1250

0.06250

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.

From forward inductionwe can calculate the probabilityof reaching a particular node as the product of one-half and the sumof the probabilities of the two priornodes.

The time dependent survivalprobabilities are given by thesum of the products of the stateprices and indicator variables. Fromthese we can calculate defaultprobabilities and conditional measures

Merton (1974), Geske (1977), and Firm-Value Structural Default Lattice

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• 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.

Equity Value (E t ) $mm RNPD T

Black-Scholes-Merton 35,584 31.43%Geske 35,583 31.43%Lattice 35,584 31.21%

Case 2 - Two tranches of debt.

Equity Value (E t ) $mm RNPD T RNPD S RNPD F

Black-Scholes-Merton NA NA NA NAGeske 30,096 14.10% 0.85% 13.36%Lattice 30,095 13.90% 0.83% 13.18%

In the above exhibit we establish consistency between the lattice model and each of the Black-Scholes-Mertonand Geske approaches. For each of these models and the values presented in the table we reduce the termstructure of liabilities using a simple weighting scheme. The above exhibit includes equity values for each modelfollowed 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 ).

Firm-Value Structural Default Lattice based Model

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• 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 4Plot of risk neutral survival and default probabilities.

The above exhibit depicts the term structure of risk neutral survival and default probabilities based on thecomplete set of debt tranches and the lattice model.

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

90.00%

91.00%

92.00%

93.00%

94.00%

95.00%

96.00%

97.00%

98.00%

99.00%

100.00%

1 2 5 7 10 20 30

Ris

k N

eutr

al D

efau

lt P

roba

bili

ty

Ris

k N

eutr

al S

urvi

val P

roba

bili

ty

Tenor

Risk Neutral Survival and Default Probabilities

Risk Neutral Su rv ival Probability Risk Neutral Default Probability

A Hybrid Structural Default Model

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• Firm Valueo 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 Passageo 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.

Amin (1993) Model

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• 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:

where is the intensity of the jump process, which is the expectation of the distribution function of which are independent and identically distributed random variables corresponding to the Poisson jump magnitudes, is a standard Brownian motion under a risk neutral measure, and is the total number of jumps up to time .

Amin (1993) Model

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• With a partitioning of the trading interval into subintervals of length Amin (1993) derives the following magnitudes and risk-neutral probabilities for the discrete time approximating process:

where the above represent a move up, a jump, and down along with their respective probability measures with the drift of the logarithm of the asset value.

Amin (1993) Model

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• is an integer which spans all nodes at a particular time slice excluding a single move up or down. corresponds to the distribution of the probability mass associated with jumps over all states and for and all other integer is given by:

• For we have a move along the center of the lattice. • With local and non-local moves, while built upon a standard

binomial lattice with nodes at each time slice, Amin (1993) results in nodes at each time period which is consistent with many trinomial models and necessary to accommodate the case where . The number of total jumps emanating from all nodes at a time slice is equal to one for and .

Amin (1993) Model

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• To be consistent with Merton (1976) we specify a normal distribution for . To complete the specification we need only the risk-neutral measure with probability (see Amin (1993), Equation (25), p. 1847) given by :

• 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.

A Hybrid Structural Default Model

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• 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.

Backward Induction

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EXHIBIT 5Geometry and backward induction in the lattice method.

O24 > 01

O12 > 0 O23 > 01 1

O00 > 0 O11 > 0 O22 > 01 1 1

O10 = 0 O21 = 00 0

O20 = 00

t = 0 t = 1 t = T = 2

In the above exhibit the lattice geometry and backward induction procedure is presented. At each node thereare two variables: the firm's equity value and the default indicator. The equity value is positive when thevalue of the firm's assets exceeds the value of debt and/or a boundary, and zero otherwise. Here it is assumedthere is debt and/or a boundary at times t = {1, 2} (i.e. the shaded region). The lower cells in each node are forindicator 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 thelattice construction we point readers to Amin (1993).

Forward Induction

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EXHIBIT 6Geometry and forward induction in the lattice method.

S24

1

S12 S23

1 1

S00 S11 S22

1 1 1

S10 S21

0 0

S20

0

P 0 = S 00 P 1 = S 11 + S 12, S 10 = 0 P 2 = S 22 + S 23 + S 24, S 20 = 0, S 21 = 0t = 0 t = 1 t = T = 2

In the above exhibit the lattice geometry and forward induction procedure is presented. At each node thereare 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 itis assumed there is debt and/or a boundary at times t = {1, 2} (i.e. the shaded region). The lower cells in eachnode are for indicator variables and are zero where the asset value of the firm is less than the debt due and ordefault boundary. Solid lines represent local moves and dashed lines are used to indicate jumps. Below the lattice wehave the time-dependent firm survival probability which is the sum of the individual nodal survival probabilitieswhich are zero when the indicator variable is zero. Thus the time-dependent firm survival probability is calculatedvia a simple sum where adds all values conditional on the indicator variable being one. For a complete description ofthe lattice construction we point readers to Amin (1993).

Merton (1974), Geske (1977), Merton (1976) and Hybrid Structural Default Lattice

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• 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 7Merton (1974), Geske (1977), Merton (1976), and lattice equity values and risk-neutral default probabilities.

Case 1 - Single tranche of debt.


Merton (1974) 35,584 31.43%Geske (1977) 35,583 31.43%Merton (1976) 35,584 31.43%Lattice 35,584 31.64%

Case 2 - Two tranches of debt.

Equity Value (E t ) $mm RNPD T RNPD S RNPD F

Geske (1977) 30,096 14.10% 0.85% 13.36%Lattice 30,096 14.17% 0.83% 13.43%

Case 3 - Single tranche of debt with l = 1 (i.e. 1 jump per annum), k = 0, and u = .0484.


Merton (1976) 40,669 48.36%Lattice 40,667 48.55%

In the above we establish the consistency among the lattice model and the Merton (1974), Geske (1977) andthe Merton (1976) models where the structure of the tranches of debt have been transformed into one and twotranches for Merton (1974) and Merton (1976) and then Geske (1977) respectively. The above exhibit includesequity values for each model followed by risk-neutral default probabilities where for the Geske (1977) and latticemodels in Case 2 with two tranches of debt, we have a total, short, and forward measure (i.e. RNPD T , RNPD S ,and RNPD F ). For the third case we have l, k, and u which are the default intensity expressed in number of jumpsper annum, mean, and variance associated with the distribution of the jumps. For the lattice results, the numberof time steps is set equal to 6,000 (i.e. n = 6,000).

Hybrid Structural Default Lattice

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EXHIBIT 8Lattice equity values and risk-neutral default probabilities.Case 1 l = 0 (i.e. 0 jumps per annum)

Equity Value (E t ) $mm 28,556Time 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%

Case 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 32,462Time 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 = .0484 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%


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 = - .025 5.00 5.09% 0.91% 8.92% 2.17% 12.70% 3.58%u = .0484 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%


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 = .025 5.00 4.20% 0.77% 8.14% 2.00% 12.32% 3.52%u = .0484 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 results for the hybrid lattice for cases which include one, two, and three jumps per annum for the full term structure ofliabilities associated with the hypothetical firm. The results include the full term structure of risk neutral default probabilities and the correspondingunconditional denoted RNPD T and the conditional RNPD F values . In the model we allow for a functional default boundary (f). The functionalboundary may be a constant, related to asset of equity value, or some other specification which is up to the user.

Conclusions

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• 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.

Hybrid Structural Default Modeling

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