Ryerson Applied Mathematics Laboratory.

Technical Report.

Title: An intensity-based approach for equity modeling.

Author: M. Escobar, T. Friederich, M. Krayzler, L. Seco, R. Zagst.

2010: # 5

1

An intensity-based approach

for equity modeling

Authors: M. Escobar, T. Friederich, M. Krayzler, L. Seco, R. Zagst.

Abstract

This paper analyzes an intensity-based approach for equity modeling.We use the Cox-Ingersoll-Ross (CIR) process to describe the intensity of thefirm’s default process. The intensity is purposely linked to the assets of thefirm and consequently is also used to explain the equity. We examine twodifferent approaches to link assets and intensity and derive closed-form ex-pressions for the firms’ equity in both models. We use the Kalman filter toestimate the parameters of the unobservable intensity process. The applica-bility of the presented methods is demonstrated on real data working withhistorical time series of Merrill Lynch.

Key Words: Equity modeling, reduced-form model, default process, Kalman

filter.

Marcos Escobar is Assistant Professor at Ryerson University, Toronto, Canada.

Tim Friederich is a PhD student at the HVB Institute for Mathematical Finance

of the Technische Universitat Munchen, Germany.

Mikhail Krayzler is a PhD student at the HVB Institute for Mathematical Fi-

nance of the Technische Universitat Munchen, Germany.

Luis Seco is the Director of RiskLab Toronto, and Professor at the University of

Toronto, Canada.

Rudi Zagst is the Director of the HVB Institute for Mathematical Finance, and

Professor at the Technische Universitat Munchen, Germany.

2 1 Introduction

1 Introduction

In the last decades several approaches for equity modeling were proposed by aca-

demics and practitioners in the financial industry. Starting from a standard Geo-

metric Brownian Motion and the assumption of normality of log-returns as in [1],

a large number of models were developed. Some famous examples include models

capturing different volatility structures (see e.g. [2] or [3]) and models allowing

jumps in the equity price processes as in [4]. One of the characteristics of this type

of models is the fact that most of them concentrate on modeling equity returns

ignoring the possibility of bankruptcy of the underlying firm.

An alternative approach that provides a link to the underlying firm’s financial

quality can be found in the credit literature and is often referred to as a structural

model. This type of models, according to the first structural model provided by

Merton in [5], has the economic interpretation that default occurs when the value

of the assets falls below the value of the company’s debt at the time of servicing

the debt. Equity is then modeled as a call option on the underlying firm’s assets.

Thereafter, a lot of extensions were developed, such as by Black and Cox in [6]

allowing the default at any time before the debt matures, or by Finkelstein et al.

in [7] where the default threshold is modeled as a random variable.

Structural models in their turn have often been criticized for the underestimation

of credit spreads (see e.g. [8]) and the assumption of complete information, which

leads to a predictable default time [9]. To overcome these issues the so-called

reduced-form (or intensity-based) models have been developed, where default is

determined as the first jump of an exogenously given jump process. Thus, a firm’s

default time is inaccessible and driven by a default intensity, a function of some

latent state variables. However, as none of the models is perfect, the absence of

the economic and financial interpretation of the intensity-based models remains to

be their major drawback.

This paper aims to bridge the gap and combine the reduced-form model with the

Merton approach, and therefore gain from their advantages and try to eliminate

their disadvantages at the same time. In the presented setup equity is modeled as a

derivative on the firm’s assets that are purposely linked to the company’s economic

strengths with respect to its default risk. The default is triggered by the underlying

intensity process. This basic model is then extended to include an additional source

of default, which occurs when the assets fall below some predefined debt threshold.

3

Both models presented shed some light into an interesting fact that could be ob-

served during the recent financial crisis, the apparently high leverage and its effect.

Up to now, there are still a lot of discussions about the reasons that led to the

very severe problems for almost all market participants. It was often argued that

“the end of the investment banking”, as this period was called in several financial

magazines, was caused to a large extent by the so-called leverage effect. Due to

the nature of business former investment banks always used to have a very high

leverage, which we define in this work as a portion of debt respectively to the assets

of the company. According to companies’ annual reports this leverage was above

95% for most of the investment banks in the time of the crisis.

However, an interesting observation can be made if one compares these balance

sheets over a longer period. The leverage, though relatively high, was fluctuat-

ing around the same level throughout the decade. Thus, it becomes questionable

whether the leverage alone can explain such dramatic worsening of the firms’ credit

qualities, as its percentage with respect to the firm’s assets did not increase sub-

stantially during the crisis. To answer this question, we analyze an explanatory

effect of both our models, where in the first case the default is only triggered by an

unobservable intensity and the second approach explicitly includes the influence of

leverage as well.

This paper is organized as follows. In Section 2 we present our two general models

for pricing a firm’s equity. In the first model we do not include the debt of the firm

explicitly, whereas in the second one the payoff to the shareholders depends directly

on the company’s assets and debt. We use the theory of ordinary differential

equations to derive a solution in the first model and the mechanism of Green’s

function in the second. In Section 3 we concentrate on the numerical methods

in our analysis, especially on the Kalman Filter used to estimate the parameters

of the CIR process. Thereafter, in Section 4.1, we describe the data used in this

study. Section 4.2 is dedicated to the parameter estimation and the application

of the presented models on real data. Finally, we summarize the most important

findings in Section 5.

2 Model setup

In this section we present two models that allow us to express the equity of a

company as a derivative on its assets within a reduced-form methodology. We will

start with a simplified version and then proceed to the more general case, capturing

4 2 Model setup

the leverage effect and the complete behavior of the equity price.

For a mathematical formulation, let T ∗ be an arbitrary but finite planning horizon

and (Ω,F ,F,P) be a filtered probability space. Furthermore, let N = N(t)t≥0

be a counting process, i.e. a non-decreasing, integer-valued process with N(0) =

0 , and let the default time τD be defined as the first jump of N , i.e. τD =

inf t > 0 : N(t) > 0 . We assume that the filtration F is generated by the count-

ing process N and satisfies the usual conditions of completeness and right-continuity.

Setting F = Ftt≥0 , we assume that the counting process N = N(t)t≥0 admits

a non-negative Ft -predictable intensity λ = λ(t)t≤0 . For a detailed definition and

a proof of the existence see [10]. Following Bremaud et al. [10] and Duffie et al.

[11], the conditional default probability PD (t, T |Ft) for the maturity time T at

time t is given by

PD (t, T |Ft) = 1− IEP

[e−

∫ Tt λ(s)ds

∣∣∣Ft] . (1)

The corresponding conditional survival probability

PS (t, T |Ft) = P (N(T ) = 0|Ft) = P (τD > t|Ft)

is given by

PS (t, T |Ft) = IEP

[e−

∫ Tt λ(s)ds

∣∣∣Ft] .We assume the CIR process as in [12] for the intensity time series

dλ(t) = α(β − λ(t))dt+ σ√λ(t)dW (t) (2)

and that the interest rate r is constant. The choice of the CIR process to describe

the behavior of the default intensity was motivated by the fact that it leads to a

so-called affine model. One of the most important features of these models that

have been thoroughly investigated in the literature (see for example [13] or [14]) is

the existence of the analytical formulas for the default probabilities. This fact helps

to avoid time-consuming simulations and thus simplifies the calibration procedure.

Having originally been proposed by Cox, Ingersoll and Ross in [12] to describe the

short-rate dynamics, the square-root process (CIR process) is nowadays often used

in the reduced-form models (see e.g. [15]). The square root term in the volatil-

ity part assures positive intensity given that the parameters satisfy the condition

2αβ ≥ σ2 . Furthermore, the possibility of direct sampling from the non-central

5

chi-squared distribution without the need of discretization makes the CIR model

even more appealing.

For the pricing of financial derivatives, we furthermore assume the existence of a

risk-neutral probability measure Q . We introduce the risk premium as γ(t) =

γσ√λ(t) and change the Wiener process via dW (t) = dW (t) + γ(t)dt as well as

the measure from IP to Q using the Radon-Nikodym derivative

dQdP

∣∣∣Ft = exp

(−∫ t

0

γ(s)dW (s)− 1

2

∫ t

0

γ2(s)ds

).

Thus, the intensity dynamics under Q can be described by

dλ(t) = (αβ − λ(t)(α + γσ2))dt+ σ√λ(t)dW (t).

Denoting η = γσ2 , we get

dλ(t) = (αβ − λ(t)(α + η))dt+ σ√λ(t)dW (t). (3)

Hence, besides the real-world parameters (α, β, σ) one additional parameter η has

to be estimated. Equation (3) still represents the CIR process and can be rewritten

in more natural form:

dλ(t) = α(β − λ(t))dt+ σ√λ(t)dW (t), (4)

where α = α+η and β = αβα+η

are the risk-neutral parameters of the CIR process.

We treat the firm’s equity as a derivative on the value A of the firms’ assets with

payoff f (A(T )) at time T and some well-defined function f . Expectations under

the risk-neutral measure are denoted by E , i.e. we skip the subscript Q of EQ .

Thus, we conclude (see, e.g. [13]):

S(t) = E[e−r(T−t)1τD>Tf(A(T ))|Ft] = e−r(T−t)E[e−∫ Tt λ(s)dsf(A(T ))|Ft]. (5)

To specify the model, we link the expression f(A(T )) to the underlying intensity.

We differentiate between two models: one with and one without leverage constraint.

6 2 Model setup

2.1 Model without leverage constraint

For simplification purposes we first assume

f(A(T )) = A(T ) = a · e−bλ(T ),

where a and b are two positive constants, representing the fact that the higher

the intensity (the higher default probability) the lower the value of the assets.

In this model we do not explicitly take the firm’s liabilities into account. We

apply the framework of Duffie et al. [13], to derive the corresponding closed-form

expressions for the necessary expectations. Setting τ = T − t , we rewrite the

equations provided in [13] in our notation and evaluate for the firm’s equity by

S(t) = a · eθ0(τ)+θ1(τ)·λ(t) · e−rτ (6)

where θ0(τ) and θ1(τ) satisfy the following ODEs:

θ1(τ) = −1− αθ1 +1

2θ2

1σ2,

θ0(τ) = αβθ1,

with boundary conditions θ1(0) = −b and θ0(0) = 0 . Solving this system of

differential equations we get:

θ0(τ) = αβ · α− hσ2

τ − 2αβ

σ2· ln(c+ e−hτ

c+ 1

)(7)

and

θ1(τ) =cehτ (α− h) + (α + h)

σ2(cehτ + 1), (8)

where

h =√α2 + 2σ2 (9)

and

c =h+ σ2b+ α

h− σ2b− α. (10)

We furthermore show in Appendix 6.1 that these formulas represent the generalized

case of a well-known pricing formula for zero-coupon bonds when the short-term

interest rate follows a CIR process.

2.2 Model with leverage constraint 7

Using Equation (6), the firm’s equity and its assets have the following reverse

dependence:

λ(t) =rτ + ln(S(t)/a)− θ0(τ)

θ1(τ).

Another important result can be perceived from Equations (5) and (6). We let

a = 1, b = 0 and assume no discounting factor. Then, the closed-form solution for

the risk-neutral probability of default (compare with (1)) can be derived via

PD(t, T ) = 1− eθ0(τ)+θ1(τ)·λ(t), (11)

where θ0(τ) and θ1(τ) can be calculated as above but with c defined as:

c =h+ α

h− α.

Corresponding real-world probabilities are given by the same formula where we

take the real-world parameters instead of the risk-neutral ones.

2.2 Model with leverage constraint

In this section we extend the previous model by introducing the leverage constraint

X . We define f(A(T )) by

f (A(T )) = max (A(T )−X, 0) ,

where

A(T ) = a · e−λ(T )b

as in the first model corresponds to the assets of the company and X to the

company’s debt that is assumed to be constant. The structure of this payoff has a

key financial implication. It mixes a Merton style structural payoff with a reduced

form default scenario. Therefore, it implies that equity prices are affected not only

by an intermediate default, which could be defined exogenously, but also in the

event that assets fall below the debt level at the time of servicing the debt. Hence,

8 2 Model setup

we can calculate the firm’s equity by taking the following expectation

S(t) = e−rτE[e−∫ Tt λ(s)dsmax(ae−λ(T )b −X, 0)|Ft] (12)

Using Green’s function, G(λ, t, λ′, T ) , we derive the closed-form solution for the

firm’s equity defined above. The explicit form of Green’s function for the CIR

process, as presented in [16], is

GCIR(λ, t, λ′, T ) = P (λ, τ)2p(τ)H(2p(τ)λ′|2k + 2, f(λ, τ)), (13)

with

τ = T − t,

h =2αβ

σ2− 1,

p(τ) =d2(τ)

σ2d1(τ),

f(λ, τ) =8h2ehτλ

σ2d2(τ)d1(τ),

P (λ, τ) = B1(τ)e−B2(τ)λ,

B1(τ) =

(2he(α+h) τ

2

d2(τ)

) 2αβ

σ2

, (14)

B2(τ) =2d1(τ)

d2(τ),

h =√α2 + 2σ2,

d1(τ) = ehτ − 1,

d2(τ) = (h+ α)ehτ + (h− α),

H(x|v, w) =1

2

( xw

)(v−2)/4

e−(w+x)/2I(v−2)/2(√wx)

where Iy(·) denotes the modified Bessel function of the first kind of order y .

Given this definition of the Green’s function, Equation (12) can be represented via

2.2 Model with leverage constraint 9

two integrals over the Green’s function and f(A(T )) :

S(t) = e−rτ∫ λ∗

0

GCIR(λ, t, λ′, T )ae−λ′bdλ′

−e−rτX∫ λ∗

0

GCIR(λ, t, λ′, T )dλ′

= e−rτ (I1 − I2) .

The integral limit λ∗ is defined such that ae−λ∗b − X = 0 , i.e. λ∗ = 1

bln(aX

).

Substituting (13) in the expression for the second integral, we get

I2 = X

∫ λ∗

0

P (λ, τ)2p(τ)H(2p(τ)λ′|2k + 2, f(λ, τ))dλ′

= XP (λ, τ)

∫ 2p(τ)λ∗

0

H(y|2k + 2, f(λ, τ))dy

= XP (λ, τ)χ2(2p(τ)λ∗|2k + 2, f(λ, τ))

where we use the fact that H(x|v, w) is the density function of a non-central

Chi-square distribution χ2(x|v, w) . Analogously, for the first integral we have

I1 = aP (λ, τ)

∫ λ∗

0

2p(τ)H(2p(τ)λ′|2k + 2, f(λ, τ))e−λ′bdλ′

= aP (λ, τ)

∫ 2p(τ)λ∗

0

H(y|2k + 2, f(λ, τ))e−yb

2p(τ)dy

= aP (λ, τ)

∫ 2p(τ)λ∗

0

1

2

(y

f(λ, τ)

)k/2e−

f(λ,τ)2− y

2p(τ)+bp(τ) Ik(

√f(λ, τ)y)dy

Setting m(λ, τ) = f(λ, τ) · p(τ)p(τ)+b

and l = y · p(τ)+bp(τ)

, we get

I1 = aP (λ, τ)

(p(τ)

p(τ) + b

)k+1

e−f(λ,τ)

2b

p(τ)+b

∫ 2(p(τ)+b)λ∗

0

1

2

(l

m

)k/2e−

m+l2 Ik(

√ml)dl

= aP (λ, τ)

(p(τ)

p(τ) + b

)k+1

e−f(λ,τ)

2b

p(τ)+bχ2(2(p(τ) + b)λ∗|2k + 2,m(λ, τ)).

10 3 Procedure

Summarizing our findings, the firm’s equity is given by

S(t) = e−rτP (λ, τ)

(a

(p(τ)

b+ p(τ)

)k+1

e−f2

bp(τ)+bχ2(2(p(τ) + b)λ∗|2k + 2,m(λ, τ))

−Xχ2(2p(τ)λ∗|2k + 2, f(λ, τ))

)(15)

with k = 2αβσ2 − 1 and λ∗ = 1

bln(aX

).

The model derived above also includes the first model (without leverage constraint).

To see this, let X = 0 and receive f(A(T )) = a ·e−bλ(T ) . The closed-form solution

of the second model then simplifies to

S(t) = a · e−rτP (λ, τ)

(p(τ)

b+ p(τ)

)k+1

e−f2

bp(τ)+b . (16)

Here, we used the fact that λ∗ tends towards infinity when X → 0 and that

the probability function of the Chi-square distribution equals 1 when the argu-

ment reaches infinity. In Appendix 6.2 we show that for this case both analytical

expressions for the first and the second model coincide.

3 Procedure

In the previous section we have shown how we can model the firm’s equity within

a reduced-form model. The remaining question is how to estimate the unknown

parameters including the three coefficients from the CIR process, α, β and σ , as

well as the coefficients a and b that we use to link the assets with the intensity.

Hereby, we assume the leverage level to be known and constant. For this estimation

we use an extended Kalman filter. We first formulate the state-space problem where

the equity time series is an observable process and the underlying intensity is an

unobservable state process that should be estimated. We base our derivation on

the state-space approach described by Geyer and Pichler in [17].

We use real-world observations of the equity process which is why we describe the

intensity dynamics under the real-world measure P . However, the pricing should

be done under the risk-neutral measure Q . Hence, besides the real-world param-

eters (α, β, σ) one additional parameter η must be estimated. To summarize, we

11

will use the real-world measure P and Equation (2) in the prediction step of the

Kalman filter and the risk-neutral probability measure Q and Equation (3) or (4)

for the correction step. Since the use of the CIR process leads to a non-Gaussian

model, the standard Kalman filter is no longer optimal and cannot be applied in

its original form. To overcome this issue we replace the exact transition density

(non-central χ2 ) by a normal density (as in [17]):

λt|λt−1 ∼ N(µt, Qt).

We define µt and Qt within a moment matching method, so that the first two

moments of the exact and approximated density function coincide:

µt = β(1− e−α∆t) + e−α∆tλt−1

and

Qt = σ2 1− e−α∆t

α

[β

1− e−α∆t

2+ e−α∆tλt−1

]+ e−2α∆tQt−1.

Both equations help to construct the so-called prediction step. The value of the

unobservable factor λ at time t based on its value at time t− 1 is given by

λt|t−1 = β(1− e−α∆t) + e−α∆tλt−1|t−1,

with ∆t representing the time step between t− 1 and t . Analogously, we get for

the variance:

Qt|t−1 = σ2 1− e−α∆t

α

[β

1− e−α∆t

2+ e−α∆tλt−1|t−1

]+ e−2α∆tQt−1|t−1.

Observed and unobserved variables are linked by the so-called observation density

p(yt|λt) that can be described as a measurement equation

yt = At +Btλt + εt, εt ∼ N(0, q) (17)

Thus, the expected values for the state space variables conditional on the informa-

tion at time t− 1 are calculated as

yt = At +Btλt|t−1.

The formulas for these two coefficients At and Bt can be derived from the pricing

formulas for the firm equity, an observable process in our case, as will be shown in

12 3 Procedure

the sequel.

Model without leverage constraint. In the first model we use Equation (6)

to find the value of the equity when the intensity is given. We can transform this

equation so that we get an observable process linearly depending on the intensity.

We take the logarithm of both sides and get

ln(S(t)) = ln(a) + θ0(τ)− rτ + θ1(τ)λ(t),

where θ0 and θ1 depend on the risk-neutral parameters α , β and σ . Hence,

in terms of the parameters of the measurement Equation (17), yt = ln(S(t)) ,

At = ln(a) + θ0(τ)− rτ and Bt = θ1(τ) .

Model with leverage constraint. For the second model the whole procedure

is slightly more complicated because of the non-linear dependence between the

observable and the hidden process. For this purpose we apply an extended Kalman

filter and linearize Equation (15) by means of a first-order Taylor approximation

around the best prediction λt|t−1 :

S(t, λ(t)) = S(t, λt|t−1) +∂

∂λ(t)S(t, λ(t))|λ(t)=λt|t−1

· (λ(t)− λt|t−1). (18)

Comparing Equation (18) and the measurement Equation (17), we define At and

Bt as

At = S(t, λt|t−1)− ∂

∂λ(t)S(t, λ(t))|λ(t)=λt|t−1

and

Bt =∂

∂λ(t)S(t, λ(t))|λ(t)=λt|t−1

.

Thus, in both models we can calculate the state space variables yt based on the

predicted values and formulas derived above. In the next step we can use an

additional information provided by the observed variables yt via the so-called

correction step and update our estimations for λ(t) .

λt|t = max(λt|t−1 +K · [yt − yt], 0) (19)

13

and for the variance

Qt|t = [1−KBt]Qt|t−1,

where K is the so-called Kalman gain matrix which is given as

K =Qt|t−1Bt

Qt|t−1B2t + q

.

The update Equation (19) of the state estimate for the CIR model differs from the

standard Kalman filter because of the non-negativity condition for the underlying

process.

Finally, we calculate the quasi log-likelihood function for each parameter set

(α, β, σ, η, a, b, q) for both models:

logL = −1

2(N − 1)log(2π)− 1

2

N∑t=2

log|Ft| −1

2

N∑t=2

v2t

Ft,

with Ft = B2tQt + q and vt = yt − yt representing the error term. This function

is further maximized with respect to the unknown parameters.

4 Empirical results

As an application of the models presented in the previous sections we’ve chosen one

of the former investment banks, Merrill Lynch. For a very long period the company

was one of the leading and most renowned banks on Wall Street. However, due to

the sub-prime crises, as many other banks, Merrill Lynch had enormous losses, in

total about 40 billion USD. The company was even among the three banks that

suffered most from the financial crisis. As a result, in September 2008, Merrill

Lynch was taken over by the Bank of America. This acquisition seemed to be the

only possibility for Merrill Lynch not to follow the tragic fate of Lehman Brothers

which defaulted almost the same day.

14 4 Empirical results

4.1 Data

For our analysis we use the equity time series of Merrill Lynch as provided by

Bloomberg in the time span from September 2001 to September 2008. Figure 1

shows the equity values of Merrill Lynch in the period from September 2001 to

September 2008. Until the beginning of 2007 there were almost no signs for the

coming financial market crisis and the company’s equity was fluctuating around a

positive trend. Since then we can see the negative dynamics in the firm’s value

that became very strong after May 2007.

40

60

80

100

120

0

20

40

60

80

100

120

ML Equity

Figure 1: Equity time series of Merrill Lynch (September 2001 - September 2008).

The examined period consists of 1770 daily observations. Moreover, we assume that

the average maturity of Merrill Lynch’s debt equals T = 5 years. We also assume a

flat interest rate at the level of 4% , which corresponds to the average of 5-year US

Government interest rates in the considered time span. Furthermore, we scale the

observable equity time series by dividing it by the initial asset value A(0) . For the

second model we assume a constant leverage of X = D(0)/A(0) . This information

was extracted from the consolidated financial data of Merrill Lynch, as presented

in Table 11 which shows some main figures characterizing the creditworthiness of

the company and its overall economic and financial conditions for the time period

between 2001 and 2008. This table justifies our assumption of constant leverage2

1 Extracted from annual reports of Merrill Lynch for the corresponding years.2 Fluctuations in leverage are less than 3% .

4.2 Application of the model 15

which we set to the initial value X = 0.95 (which also corresponds to the average

level in the examined period).

Main figure 2001 2002 2003 2004 2005 2006 2007 2008(in bn USD)EBIT 1.38 3.76 5.65 5.84 7.23 10.43 -12.83 -41.83Net Revenues 21.88 18.61 20.15 22.02 26.01 34.66 11.25 -12.59Total Assets 419.42 447.93 494.52 648.06 681.02 841.30 1,020.05 667.54Total Liabilities 396.72 422.40 464.20 616.69 645.42 802.26 988.12 647.54Stockholders’ Equity 20.01 22.88 27.65 31.37 35.60 39.04 31.93 20.00Leverage 0.95 0.94 0.94 0.95 0.95 0.95 0.97 0.97

Table 1: Consolidated financial data of Merrill Lynch for 2001-2008

4.2 Application of the model

The equity time series in our model without leverage constraint is calculated via

(6) with the following parameter set Θ1 = (α, β, σ, η, a, b) . Estimating these pa-

rameters via the Kalman filter method described in Section 3, we get

α = 0.3143

β = 0.0126

σ = 0.0560

a = 6.9968

b = 506.0191

η = −0.1096.

We use the same technique of the Kalman filter for the model with leverage con-

straint X that is set to the constant value of 0.95 and get the following parameters

α = 0.2585

β = 0.0140

σ = 0.0505

a = 4.5633

b = 336.8172

η = −0.0108.

One of the main advantages of the Kalman filter is that, additional to parameter

estimations, it gives the values of the unknown state process, i.e. the intensity time

16 4 Empirical results

series in our case. Knowing the intensity and the corresponding parameters, it is

possible to calculate the default probabilities by applying Equation (11). We use

both parameter sets for our models and the corresponding intensity time series to

calculate the default probabilities and compare them in Figure 2. We see similar

evolutions of both time series. However, the default probabilities derived in the

second model are slightly higher.

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

First Model Second Model

Figure 2: 5-year default probabilities derived in both models.

Inappropriately high leverage that major investment banks ran in the last couple of

years was often named as one of the major reasons of subsequent default as in the

case of Lehman Brothers or fire sales as of Merrill Lynch and Bear Sterns. However,

if one just looks at the balance sheet information as in Table 1, there was actually

no acute reason for that: though the leverage ratio was quite high, it remained

almost constant over the entire examined period starting from 2001. At the same

time we can observe changing default probabilities (see Figure 2) and even their

triplication from beginning of 2007 to September 2008 when Lehman collapsed (on

both models). The reason for this at first glance odd phenomenon can be seen

in the fact that these changes in the firm’s creditworthiness cannot be attributed

to the leverage effect only. Some other economic factors might play an even more

essential role and should necessarily be taken into consideration. In the presented

modeling approach an unobservable intensity accounts for this exogenous effect.

The resulting relatively small differences between default probabilities implied by

both models can be seen as an evidence of this major impact of the unobservable

default intensity process on the firm’s credit quality.

4.2 Application of the model 17

We furthermore want to compare the empirical distribution of Merrill Lynch equity

returns with those implied by our models. For this purpose we use the parameter

sets determined above, simulate the underlying intensity process and transform

it to an equity time series. On Figure 3 the histogram presents the empirical

density function of Merrill Lynch returns as well as the density functions from

our simulation with 100 scenarios with 1770 time steps for the first (light grey

line - without leverage constraint) and the second (dark grey line - with leverage

constraint) model.

10

15

20

25

30

35

40

0

5

10

15

20

25

30

35

40

Hisorical First Model Second Model Normal Distribution

Figure 3: Empirical histogram and model density functions.

As we can see from the graph, simulated distributions in both cases are very

similar to the observed distribution. This fact can be verified via a two-sample

Kolmogorov-Smirnov test as described in [18]. The null hypothesis in our case

is that empirical log-returns of Merrill Lynch and simulated returns are samples

from the same continuous distribution. We test this hypothesis at the 1% and 5%

significance level. The corresponding statistics can be found in Table 2. The first

column (ksstat) represents the value of the Kolmogorov-Smirnov statistic itself,

which is the maximum difference between two cumulative distribution functions.

The second column gives the p-value for the corresponding test. In order to reject

the null hypothesis the p-value should be below some predefined confidence level.

We can conclude from Table 2 that in both models the null hypothesis can not be

rejected at the 1% significance level. At the same time, we reject the first model

at the 5% significance level.

18 5 Conclusion

Model ksstat pvalue 1% 5%First Model 0.0358 0.0217 not rejected rejectedSecond Model 0.0304 0.0766 not rejected not rejectedNormal Distribution 0.122 0 rejected rejected

Table 2: Kolmogorov-Smirnov Test

It is often claimed that equity returns are normally distributed. Thus, in order to

test this assumption, we add the normal distribution to our comparison analysis.

For this purpose we calculate mean and standard deviation of the Merrill Lynch

equity log-returns and then plot the density function of the corresponding normal

distribution. As before, the fitting quality can be analyzed on the histogram (see

the dotted line on Figure 3) and via Kolmogorov-Smirnov test. This time we

sample from the normal distribution with a given mean and standard deviation.

The corresponding test results are presented in the last row of Table 2. We can

observe that the value of the test statistic (ksstat column) doubled in comparison

with distributions provided by our models. In this case the corresponding null

hypothesis (normal distribution of the Merrill Lynch log-returns) can be rejected

at the 5% and 1% significance level.

5 Conclusion

In this paper we proposed two modeling approaches for the firm’s equity within an

intensity setup. In both cases, the company’s equity is treated as a credit derivative

with two different payoff structures - with and without leverage constraint. For the

first model, the payment of the derivative only depends on the corresponding value

of the intensity. The second model takes into account the liabilities of the firm by

including them into the payoff function. For both cases we were able to obtain

analytical formulas based on the paper of Duffie et al. [13] for the first model and

Buettler, Waldvogel [16] for the second one. We showed that the second model is a

generalization of the first one and thus we provide another mechanism for deriving

a closed-form solution for this type of affine models - Green’s function instead of

ODEs.

We used the Cox-Ingersoll-Ross process to describe the stochastic dynamics of

the intensity. However, other common modeling frameworks such as Vasicek [19]

or Hull-White [20] and even more general affine models are also applicable and

represent possible directions for further research. At this point one should bear in

19

mind, that some of these models might lead to negative intensities, which are not

accessible in our approach. Furthermore, the closed-form solution in case of a more

complicated model, allowing jumps for example, might not be achieved anymore.

This becomes a particularly complicated task for the second modeling approach,

where an expectation of a maximum function should be calculated explicitly. At

this stage we refer the Duffie’s paper mentioned above and his joint work with

Filipovic and Schachermayer [14] for more detailed information on the theory of

affine models.

Another way to generalize our approach and overcome the limitation of modeling

the firm’s assets depending only on the default intensity process is to include

some additional stochastic factor. This would lead to the two factor model, where

one factor accounts for the idiosyncratic part of the assets volatility and other

represents the external influence of the financial market in general. This paper can

be seen as a first step in the direction of such a generalized model that can better

capture the firm’s equity behavior and its credit quality.

We applied both presented methods to the case of Merrill Lynch and used its

equity time series to fit the parameters of the model. The estimation procedure is

based on the extended Kalman filter. We then were able to calculate the default

probabilities, which provide investors with additional insight on the firm’s credit

quality. This approach is especially valuable for those companies that do not have

liquid CDS or bond spreads and the only source of information is the equity time

series. Furthermore, using estimated parameters we ran a simulation with both

our models in order to compare historical equity returns with those implied by

the models. We observed that the resulting distributions are much closer to the

historical distribution than a normal one. This conclusion is also supported by the

Kolmogorov-Smirnov test that we performed at the end.

6 Appendix

6.1 Note on the model without constraint

It is well known that the value of a zero-coupon bond with stochastic interest rate

r is given by

ZB(t, T ) = E[e−

∫ Tt r(s)ds|Ft

].

20 6 Appendix

Moreover, there is a closed-form solution (see e.g. [21]) when the underlying stochas-

tic process follows a CIR process:

ZB(t, T ) = A(τ)e−B(τ)r(t),

where τ = T − t ,

A(τ) =

(2h e0.5(α+h)τ

2h+ (α + h)(ehτ − 1)

) 2αβ

σ2

and

B(τ) =2(ehτ − 1)

2h+ (α + h)(ehτ − 1). (20)

It can be shown that these formulas are a particular case of those derived in Section

2.1. There we got:

E[e−∫ Tt λ(s)dse−bλ(T )|Ft] = eθ0(τ)+θ1(τ)·λ(t), (21)

where θ0 and θ1 were defined in Equations (7) and (8). Setting b = 0 and

replacing λ by r , we receive:

θ1(τ) =c(α− h) + (α + h)e−hτ

σ2(c+ e−hτ ), (22)

where

c =h+ α

h− α(23)

and

h =√α2 + 2σ2.

Substituting (23) in (22) results in:

θ1(τ) =(α + h)(1− e−hτ )(α− h)

σ2(α + h+ e−hτh− e−hτ α)

which after some transformations and using the fact that

α2 − h2 = −2σ2

6.2 Note on the model with constraint 21

leads to:

θ1(τ) = − 2(ehτ − 1)

2h+ (α + h)(ehτ − 1).

The expression above coincides with −B(τ) from (20). Analogously, for θ0(τ) ,

we first calculate the expression

eθ0 = eαβα−hσ2 τ

(c+ 1

c+ e−hτ

) 2αβ

σ2

,

which after some simple transformations gives

eθ0 =

((c+ 1)e0.5(α+h)τ

cehτ + 1

) 2αβ

σ2

.

Substituting

c =h+ α

h− αleads us to

eθ0 =

(2he0.5(α+h)τ

2h+ (α + h)(ehτ − 1)

) 2αβ

σ2

.

Hence, eθ0(τ) = A(τ) and, consequently, Equation (20) is a particular case of the

generalized Equation (21).

6.2 Note on the model with constraint

As we have already mentioned in the Section 2, the closed-form solution from

the second model with leverage constraint represents a generalized version of the

first model. In the following we show that both analytical formulas coincide when

X = 0 . We recall the closed-form solution for the second model - Equation (16):

S(t) = a · e−rτP (λ, τ)

(p

b+ p

)k+1

e−f2

bp+b .

22 6 Appendix

We rewrite this equation using Equation set (14) in order to combine all con-

stituents with and without λ :

S(t) = ae−rτQ0(τ)e−Q1(τ)λ,

where

Q0(τ) =

(2he(α+h)τ/2

d2

) 2αβ

σ2(

p

p+ b

) 2αβ

σ2

and

Q1(τ) =4h2ehτ

σ2d2d1

b

p+ b+

2d1

d2

For the first model without leverage constraint (Equation (6)) we have

S(t) = ae−rτ θ0(τ)e−θ1(τ)λ,

where, according to Equation (7),

θ0(τ) = eθ0 =

((c+ 1)e0.5(α+h)τ

cehτ + 1

) 2αβ

σ2

,

and, according to Equation (8),

θ1(τ) = −cehτ (α− h) + (α + h)

σ2(cehτ + 1).

In order to verify the equivalence of both equations for S(t) , we compare the

parts including λ and not-including λ . We start with the terms without λ .

Substituting p = d2σ2d1

, we rewrite the expression for Q0 as

Q0(τ) =

(2he0.5(α+h)τ

d2

· d2

σ2d1b+ d2

) 2αβ

σ2

=

(2he0.5(α+h)τ

σ2d1b+ d2

) 2αβ

σ2

.

6.2 Note on the model with constraint 23

Using d1 = ehτ − 1 , d2 = (h + α)ehτ + (h − α) , and c = h+σ2b+αh−σ2b−α we further

conclude that

Q0(τ) =

(2he0.5(α+h)τ

σ2d1b+ d2

) 2αβ

σ2

=

(2he0.5(α+h)τ

(h+ α)ehτ + (h− α) + σ2b(ehτ − 1)

) 2αβ

σ2

=

(2he0.5(α+h)τ

(h+ σ2b+ α)ehτ + (h− σ2b− α)

) 2αβ

σ2

=

((c+ 1)e0.5(α+h)τ

cehτ + 1

) 2αβ

σ2

= θ0(τ).

In the next step we compare the coefficients of λ : θ0 and Q1 . We start again

from the one of the second model and use the fact, that p(τ) = d2σ2d1

:

Q1(τ) =4h2ehτ

σ2d2d1

b

p+ b+

2d1

d2

=4h2ehτ

d2

b

d2 + σ2d1b+

2d1

d2

(24)

=4h2ehτb+ 2d2

1σ2b+ 2d1d2

d2(d2 + σ2d1b).

Using Equations (9) and (10) we can also simplify the expression for θ1 :

24 References

θ1(τ) = −cehτ (α− h) + (α + h)

σ2(cehτ + 1)

= −(h+ σ2b+ α)ehτ (α− h) + (α + h)(h− σ2b− α)

σ2((h+ σ2b+ α)ehτ + (h− σ2b− α))

= −σ2b(ehτ (α− h)− (α + h))− (h2 − α2)ehτ + (h2 − α2)

σ2((h+ σ2b+ α)ehτ + (h− σ2b− α))

= − b(ehτ (α− h)− (α + h))− 2ehτ + 2

ehτ (h+ α) + (h− α) + σ2b(ehτ − 1)

=−b(ehτ (α− h)− (α + h)) + 2d1

d2 + σ2d1b

=−b(ehτ (α− h)− (α + h))d2 + 2d1d2

d2(d2 + σ2d1b).

Now, using Equation set (14), we conclude

−(ehτ (α− h)− (α + h))d2 = (ehτ (h− α) + (h+ α))((ehτ (h+ α) + (h− α))

= e2hτ (h2 − α2) + (h+ α)2ehτ + ehτ (h− α)2 + (h2 − α2)

= 2e2hτσ2 + 4α2ehτ + 4σ2ehτ + 2σ2

= 4α2ehτ + 8σ2ehτ + 2e2hτσ2 − 4σ2ehτ + 2σ2

= 4(α2 + 2σ2)ehτ + 2(ehτ − 1)2σ2

= 4h2ehτ + 2d21σ

2

and thus Q1(τ) = θ1(τ) , i.e. we have shown that both expression for the firm’s

equity coincide, and consequently, the first model represents a particular case of

the second model, when the leverage constraint is set to zero.

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