# The Merton’s Default Risk Model for Public Company

Battulga Gankhuu111 Department of Applied Mathematics, National University of
Mongolia, Ulaanbaatar, Mongolia; E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we developed the Merton’s structural model for public companies
under an assumption that liabilities of the companies are observed. Using
Campbell Shiller’s approximation method, we obtain formulas of risk–neutral
equity and liability values and default probabilities for the public
companies. Also, the paper provides ML estimators of suggested model’s
parameters.

Keywords: Public companies, Merton’s structural model, ML estimators.

##  1 Introduction

Dividend discount models (DDMs), first introduced by Williams38 are common
methods for equity valuation. The basic idea is that the market value of an
equity of a firm is equal to the present value of a sum of dividend paid by
the firm and market value of the firm, which correspond to the next period. If
the firm is financed by liabilities, which are publicly traded in the
exchanges, the same idea can be used to value the liabilities, see
Battulga23b. As the outcome of DDMs depends crucially on dividend payment
forecasts, most research in the last few decades has been around the proper
estimations of dividend development. Also, parameter estimation of DDMs is a
challenging task. Recently, Battulga22a introduced parameter estimation
methods for practically popular DDMs. To estimate parameters of the required
rate of return, Battulga23b used the maximum likelihood method and Kalman
filtering. Reviews of some existing DDMs that include deterministic and
stochastic models can be found in dAmico20a and Battulga22a.

Existing stochastic DDMs have one common disadvantage: If dividend and debt
payments have chances to take negative values, then the market values of the
firm’s equity and liabilities can take negative values with a positive
probability, which is the undesirable property for the market values. A log
version of the stochastic DDM, which is called by dynamic Gordon growth model
was introduced by Campbell88, who derived a connection between log price, log
dividend, and log return by approximation. Since their model is in a log
framework, the stock prices and dividends get positive values. For this
reason, by augmenting the dynamic Gordon growth model, Battulga24d obtained
pricing and hedging formulas of European options and equity–linked life
insurance products for public companies. For private companies, using the log
private company valuation model, based on the dynamic Gordon growth model,
Battulga24c developed closed–form pricing and hedging formulas for the
European options and equity–linked life insurance products and valuation
formula.

Sudden and dramatic changes in the financial market and economy are caused by
events such as wars, market panics, or significant changes in government
policies. To model those events, some authors used regime–switching models.
The regime–switching model was introduced by seminal works of
Hamilton89,Hamilton90 (see also books of Hamilton94 and Krolzig97) and the
model is hidden Markov model with dependencies, see Zucchini16. However,
Markov regime–switching models have been introduced before Hamilton (1989),
see, for example, Goldfeld73, Quandt58, and Tong83. The regime–switching model
assumes that a discrete unobservable Markov process generates switches among a
finite set of regimes randomly and that each regime is defined by a particular
parameter set. The model is a good fit for some financial data and has become
popular in financial modeling including equity options, bond prices, and
others. Recently, under normal framework, Battulga22b obtained pricing and
hedging formulas for the European options and equity–linked life insurance
products by introducing a DDM with regime–switching process. Also, Battulga24a
developed option pricing formulas for some frequently used options by using
Markov–Switching Vector Autoregressive process. To model required rate of
return on stock, Battulga23b applied a three–regime model. The result of the
paper reveals that the regimeswitching model is good fit for the required rate
of return.

Default risk is a possibility that a borrower fails to make full and timely
payments of principal and interest, which are stated in the debt contract. The
structural model of default risk relates to option pricing. In this model, a
default threshold, which represents the liabilities of the company is seen as
a strike price and a asset value of the company is seen as underlying asset of
the European option. For this reason, this approach is also referred to as the
firm–value approach or the option–theoretic approach. Original idea of the
structural model goes back to Black73 and Merton74. Black73 developed a
closed–form formula for evaluating the European option and Merton74 obtained
pricing formula for the liabilities of a company under Black–Scholes
framework. Battulga22b tried to estimate default probability using
regime–switching process.

This paper is organized as follows. In Section 2 of the paper, we develop
stochastic DDM for market values of equities and liabilities of companies
using the Campbell Shiller’s 1988 approximation method. Then, we model the
market values of assets of the companies using the approximation method once
again. In Section 3, we obtain pricing formulas of the European call and put
options on the market value of the asset. After that, we develop formulas of
risk–neutral equity and debt values, and default probability. In Section 4, we
study ML estimators of suggested model’s parameters. In Section 5, we conclude
the study. Finally, in Section 6, we provide Lemmas, which is used in the
paper.

##  2 Market Value Model of Equity and Liability

Let $(\Omega,\mathcal{H}_{T},\mathbb{P})$ be a complete probability space,
where $\mathbb{P}$ is a given physical or real–world probability measure and
$\mathcal{H}_{T}$ will be defined below. To introduce a regime–switching in
Merton’s default risk model, we assume that $\\{s_{t}\\}_{t=1}^{T}$ is a
homogeneous Markov chain with $N$ state and
$\mathsf{P}:=\\{p_{ij}\\}_{i=0,j=1}^{N}$ is a random transition probability
matrix, where $(p_{01},\dots,p_{0N})^{\prime}$ is an initial probability
vector. In this paper, we assume that market values of equities and
liabilities of companies are observed. For a case that market values of
equities and liabilities are both unobserved, we refer to Battulga24e.

Dividend discount models (DDMs), first introduced by Williams38, are a popular
tool for equity valuation. The basic idea of all DDMs is that the market value
of equity at time $t-1$ of the firm equals the sum of the market value of
equity at time $t$ and dividend payment at time $t$ discounted at
risk–adjusted rate (required rate of return on stock). Let us assume there are
$n$ companies. Therefore, for successive market values of equity of $i$–th
company, the following relation holds

$V_{i,t}^{e}=(1+k_{i,t}^{e})V_{i,t-1}^{e}-p_{i,t}^{e},~{}~{}~{}t=1,\dots,T,$
(1)

where $k_{i,t}^{e}$ is the required rate of return on the equity (investors)
at regime $s_{t}$, $V_{i,t}^{e}$ is the market value of equity, and
$p_{i,t}^{e}$ is the dividend payment for investors, respectively, at time $t$
of $i$–th company. On the other hand, to model market values of liabilities of
the company, it is the well known fact that successive values of a debt of
company or individual is given by the following equation

$D_{t}=(1+i)D_{t-1}-d_{t}$ (2)

where $D_{t}$ is a debt value at time $t$, $d_{t}$ is a debt payment at time
$t$, and $i$ is a interest rate of the debt, see, e.g., Gerber97. Note that
$D_{t}$ represents the principal outstanding, that is, the remaining debt
immediately after $r_{t}$ has been paid and debt equation (2) shares same
formula with market value of equity given in equation (1). The idea of
equation (2) can be used to model a value of liabilities of the company,
namely,

$V_{i,t}^{\ell}=(1+k_{i,t}^{\ell})V_{i,t-1}^{\ell}-p_{i,t}^{\ell},~{}~{}~{}t=1,\dots,T,$
(3)

where $k_{i,t}^{\ell}$ is the required rate of return on the liability
(debtholders) at regime $s_{t}$, $V_{i,t}^{\ell}$ is the market value of the
liability, and $p_{i,t}^{\ell}$ is a debt payment, which includes interest
payment for debtholders, respectively, at time $t$ of the company, see
Battulga23b.

To keep notations simple, let
$V_{t}^{e}:=(V_{1,t}^{e},\dots,V_{n,t}^{e})^{\prime}$ be an $(n\times 1)$
vector of market values of equities,
$V_{t}^{\ell}:=(V_{1,t}^{\ell},\dots,V_{n,t}^{\ell})^{\prime}$ be an $(n\times
1)$ vector of market values of liabilities,
$k_{t}^{e}:=(k_{1,t}^{e},\dots,k_{n,t}^{e})^{\prime}$ be an $(n\times 1)$
vector of required rate of returns on equities,
$k_{t}^{\ell}:=(k_{1,t}^{\ell},\dots,k_{n,t}^{\ell})^{\prime}$ be an $(n\times
1)$ vector of required rate of returns on liabilities,
$p_{t}^{e}:=(p_{1,t}^{e},\dots,p_{n,t}^{e})^{\prime}$ be an $(n\times 1)$
vector of the dividend payments, and
$p_{t}^{\ell}:=(p_{1,t}^{\ell},\dots,p_{n,t}^{\ell})^{\prime}$ be an $(n\times
1)$ vector of the debt payments, respectively, at time $t$, $I_{n}$ be an
$(n\times n)$ identity matrix, $i_{n}:=(1,\dots,1)^{\prime}$ be an $(n\times
1)$ vector, whose all elements equal one.

If payments of dividend and debt have chances to take negative values, then
the market values of equity and liability of a company can take negative
values with a positive probability, which is the undesirable property for
market values of equity and liability. That is why, we follow the idea in
Campbell88. As a result, the market values of equity and liability of the
company take positive values. Following the idea in Campbell88, one can obtain
the following approximation

$\exp\\{\tilde{k}_{t}\\}=(V_{t}+p_{t})\oslash
V_{t-1}\approx\exp\Big{\\{}\tilde{V}_{t}-\tilde{V}_{t-1}+\ln(g_{t})+G_{t}^{-1}(G_{t}-I_{2n})\big{(}\tilde{p}_{t}-\tilde{V}_{t}-\mu_{t}\big{)}\Big{\\}},$
(4)

where $\oslash$ is a component–wise division of two vectors,
$\tilde{k}_{t}:=\big{(}(\ln(i_{n}+k_{t}^{e}))^{\prime},(\ln(i_{n}+k_{t}^{\ell}))^{\prime}\big{)}^{\prime}$
is a ($2n\times 1$) log required rate of return process at time $t$,
$V_{t}:=\big{(}(V_{t}^{e})^{\prime},(V_{t}^{\ell})^{\prime}\big{)}^{\prime}$
is a $(2n\times 1)$ market value process at time $t$,
$p_{t}:=\big{(}(p_{t}^{e})^{\prime},(p_{t}^{\ell})^{\prime}\big{)}^{\prime}$
is a $(2n\times 1)$ payment process at time $t$, $\tilde{V}_{t}:=\ln(V_{t})$
is a $(2n\times 1)$ log market value process at time $t$,
$\tilde{p}_{t}:=\ln(p_{t})$ is a $(2n\times 1)$ log payment process at time
$t$,
$\mu_{t}:=\mathbb{E}\big{[}\tilde{p}_{t}-\tilde{V}_{t}\big{|}\mathcal{F}_{0}\big{]}$
is a $(2n\times 1)$ mean log payment–to–market value process at time $t$ of
the companies and $\mathcal{F}_{0}$ is initial information, which will be
defined below, $g_{t}:=i_{2n}+\exp\\{\mu_{t}\\}$ is a $(2n\times 1)$
linearization parameter, and $G_{t}:=\text{diag}\\{g_{t}\\}$ is a $(2n\times
2n)$ diagonal matrix, whose diagonal elements are $g_{t}$. As a result, for
the log market value process at time $t$, the following approximation holds

$\tilde{V}_{t}\approx
G_{t}(\tilde{V}_{t-1}-\tilde{p}_{t}+\tilde{k}_{t})+\tilde{p}_{t}-h_{t}.$ (5)

where $h_{t}:=G_{t}\big{(}\ln(g_{t})-\mu_{t}\big{)}+\mu_{t}$ is a
linearization parameter and the model is called by dynamic Gordon growth
model, see Campbell88. We assume that values of the log payment process
$\tilde{p}_{t}$ are known at time 0. For a quality of the approximation, we
refer to Campbell97. To estimate parameters of the dynamic Gordon growth model
and to price the Black–Scholes call and put options on asset value of the
company, we suppose that the log required rate of return process at time $t$
is represented by a sum of deterministic process and white noise process,
namely,

$\tilde{k}_{t}=C_{k,s_{t}}\psi_{t}+\delta\tilde{r}_{t}+u_{t},$ (6)

where $\psi_{t}:=(\psi_{1,t},\dots,\psi_{l_{k},t})^{\prime}$ is an $(l\times
1)$ vector, which consists of exogenous variables, $C_{k,s_{t}}$ is an
$(n\times l)$ random matrix at regime $s_{t}$,
$\delta:=(0,i_{n}^{\prime})^{\prime}$ is a $(2n\times 1)$ vector, whose first
$n$ elements are zero and others are one, $\tilde{r}_{t}:=\ln(1+r_{t})$ is a
log spot interest rate at time $t$, $r_{t}$ is a spot interest rate for
borrowing and lending over a period $(t,t+1]$, $u_{t}$ is an $(n\times 1)$
white noise process with random covariance matrix $\Sigma_{uu,s_{t}}$ at
regime $s_{t}$. In this case, equation (5) becomes

$\tilde{V}_{t}=G_{t}(\tilde{V}_{t-1}-\tilde{p}_{t}+C_{k,s_{t}}\psi_{t}+\delta\tilde{r}_{t})+\tilde{p}_{t}-h_{t}+G_{t}u_{t}.$
(7)

Now, we model spot interest rate $r_{t}$. By using the Dickey–Fuller test, it
can be confirm that quarterly log spot interest rate is the unit–root process
with drift, see data IRX of Yahoo Finance. Also, due to the fact that log
return of quarterly S&P 500 index is stationary process, see data SPX of Yahoo
Finance, we model the log required rate of return on equity by a trend
stationary process. Moreover, there may exist a cointegration between the log
required rate of return on debtholders and the spot interest rate. For this
reason, we model the required rate of return process by equation (6).
Consequently, we model the log spot rate by the following equation

$\tilde{r}_{t}=c_{r,s_{t}}^{\prime}\psi_{t}+\tilde{r}_{t-1}+v_{t},$ (8)

where $c_{r,s_{t}}$ is an $(l\times 1)$ random vector at regime $s_{t}$,
$v_{t}$ is a white noise process with random variance $\Sigma_{vv,s_{t}}$ at
regime $s_{t}$.

As a result, by combining equations (7) and (8), we arrive the following
system

$\begin{cases}\tilde{V}_{t}=\nu_{V,t}+G_{t}\tilde{V}_{t-1}+G_{t}\delta\tilde{r}_{t-1}+G_{t}u_{t}\\\
\tilde{r}_{t}=\nu_{r,t}+\tilde{r}_{t-1}+v_{t},\end{cases}~{}~{}~{}\text{for}~{}t=1,\dots,T$
(9)

under the real probability measure $\mathbb{P}$, where
$\nu_{V,t}:=G_{t}C_{k,s_{t}}\psi_{t}-(G_{t}-I_{2n})\tilde{p}_{t}-h_{t}$ is an
$(2n\times 1)$ intercept process of the log market value process
$\tilde{V}_{t}$ and $\nu_{r,t}:=c_{r,s_{t}}^{\prime}\psi_{t}$ is an $(1\times
1)$ intercept process of the log spot rate process $r_{t}$. Let us denote a
dimension of system (11) by $\tilde{n}$, that is, $\tilde{n}:=2n+1$.

Finally, we model the market values of the assets of the companies. Since the
market values of the assets equal sums of the market values of equities and
liabilities of the companies, we have

$V_{t}^{a}=V_{t}^{e}+V_{t}^{\ell},$

where $V_{t}^{a}$ is an $(n\times 1)$ asset value process at time $t$ of the
companies. Using the same approximation method, a log asset value process of
the companies is approximated by the following equation

$\displaystyle\tilde{V}_{t}^{a}:=\ln(V_{t}^{e}+V_{t}^{\ell})$
$\displaystyle\approx$
$\displaystyle(G_{t}^{a})^{-1}\tilde{V}_{t}^{e}+\big{(}I_{n}-(G_{t}^{a})^{-1}\big{)}\tilde{V}_{t}^{\ell}+(G_{t}^{a})^{-1}h_{t}^{a}$
(10) $\displaystyle=$ $\displaystyle
W_{t}^{a}\tilde{V}_{t}+(G_{t}^{a})^{-1}h_{t}^{a}$

where
$\mu_{t}^{a}:=\mathbb{E}[\tilde{V}_{t}^{\ell}-\tilde{V}_{t}^{e}|\mathcal{F}_{0}]$
is a mean log liability value–to–equity value ratio,
$g_{t}^{a}:=i_{n}+\exp\\{\mu_{t}^{a}\\}$ and
$h_{t}^{a}:=G_{t}^{a}(\ln(g_{t}^{a})-\mu_{t}^{a})+\mu_{t}^{a}$ are
linearization parameters for the log asset process,
$G_{t}^{a}:=\text{diag}\\{g_{t}^{a}\\}$ is a diagonal matrix, and
$W_{t}^{a}:=\big{[}(G_{t}^{a})^{-1}:I_{n}-(G_{t}^{a})^{-1}\big{]}$ is a weight
matrix of the approximation, respectively, at time $t$ of the company.

The stochastic properties of system (9) is governed by the random vectors
$\\{u_{1},\dots,u_{T},v_{1},\dots,v_{T}\\}.$ We assume that for $t=1,\dots,T$,
the white noise process $\xi_{t}:=(u_{t}^{\prime},v_{t})^{\prime}$ follows
normal distribution, namely,

$\xi_{t}\sim\mathcal{N}(0,\Sigma_{s_{t}})$ (11)

under the real probability measure $\mathbb{P}$, where

$\Sigma_{s_{t}}=\begin{bmatrix}\Sigma_{uu,s_{t}}&\Sigma_{uv,s_{t}}\\\
\Sigma_{vu,s_{t}}&\Sigma_{vv,s_{t}}\end{bmatrix}.$ (12)

is a covariance matrix of the $(\tilde{n}\times 1)$ white noise process
$\xi_{t}$.

##  3 Merton’s Structural Model

The Merton’s model 1974 is one of the structural models used to measure credit
risk. Merton74 was aim to value the liabilities of a specific company. As
mentioned above the model connects the European call and put options. The
European call and put options are contracts that give their owner the right,
but not the obligation, to buy or sell shares of a stock of a company at a
predetermined price by a specified date. Let us start this Section by
considering the valuation method of the European options on the asset values
of the companies.

Let $T$ be a time to maturity of the European call and put options,
$x_{t}:=(\tilde{V}_{t}^{\prime},\tilde{r}_{t})^{\prime}$ be $(\tilde{n}\times
1)$ process at time $t$ of endogenous variables, and
$C_{s_{t}}:=\big{[}C_{k,s_{t}}^{\prime}:c_{r,s_{t}}\big{]}^{\prime}$ be random
coefficient vector at regime $s_{t}$. We introduce stacked vectors and
matrices: $x:=(x_{1}^{\prime},\dots,x_{T}^{\prime})^{\prime}$,
$s:=(s_{1},\dots,s_{T})^{\prime}$, $C_{s}:=[C_{s_{1}}:\dots:C_{s_{T}}]$, and
$\Gamma_{s}:=[\Sigma_{s_{1}}:\dots:\Sigma_{s_{T}}]$. We suppose that the white
noise process $\\{\xi_{t}\\}_{t=1}^{T}$ is independent of the random
coefficient matrix $C_{s}$, random covariance matrix $\Gamma_{s}$, random
transition matrix $\mathsf{P}$, and regime–switching vector $s$ conditional on
initial information $\mathcal{F}_{0}:=\sigma(x_{0},\psi_{1},\dots,\psi_{T})$.
Here for a generic random vector $X$, $\sigma(X)$ denotes a $\sigma$–field
generated by the random vector $X$, $\psi_{1},\dots,\psi_{T}$ are values of
exogenous variables and they are known at time zero. We further suppose that
the transition probability matrix $\mathsf{P}$ is independent of the random
coefficient matrix $C_{s}$ and covariance matrix $\Gamma_{s}$ given initial
information $\mathcal{F}_{0}$ and regime–switching process $s_{t}$.

To ease of notations, for a generic vector $o=(o_{1},\dots,o_{T})^{\prime}$,
we denote its first $t$ and last $T-t$ sub vectors by $\bar{o}_{t}$ and
$\bar{o}_{t}^{c}$, respectively, that is,
$\bar{o}_{t}:=(o_{1},\dots,o_{t})^{\prime}$ and
$\bar{o}_{t}^{c}:=(o_{t+1},\dots,o_{T})^{\prime}$. We define $\sigma$–fields:
for $t=0,\dots,T$, $\mathcal{F}_{t}:=\mathcal{F}_{0}\vee\sigma(\bar{x}_{t})$
and
$\mathcal{H}_{t}:=\mathcal{F}_{t}\vee\sigma(\hat{C})\vee\sigma(\hat{\Gamma})\vee\sigma(\mathsf{P})\vee\sigma(s)$,
where for generic sigma fields $\mathcal{O}_{1},\dots,\mathcal{O}_{k}$,
$\vee_{i=1}^{k}\mathcal{O}_{i}$ is the minimal $\sigma$–field containing the
$\sigma$–fields $\mathcal{O}_{i}$, $i=1,\dots,k$. Observe that
$\mathcal{F}_{t}\subset\mathcal{H}_{t}$ for $t=0,\dots,T$.

For the first–order Markov chain, a conditional probability that the regime at
time $t+1$, $s_{t+1}$ equals some particular value conditional on the past
regimes, $s_{t},s_{t-1},\dots,s_{1}$ depends only through the most recent
regime at time $t$, $s_{t}$, that is,

$p_{s_{t}s_{t+1}}:=\mathbb{P}[s_{t+1}=s_{t+1}|s_{t}=s_{t},\mathsf{P},\mathcal{F}_{0}]=\mathbb{P}\big{[}s_{t+1}=s_{t+1}|\bar{s}_{t}=\bar{s}_{t},\mathsf{P},\mathcal{F}_{0}\big{]}$
(13)

for $t=0,\dots,T-1$, where
$p_{s_{1}}:=p_{s_{0}s_{1}}=\mathbb{P}[s_{1}=s_{1}|\mathsf{P},\mathcal{F}_{0}]$
is an initial probability. A distribution of a white noise vector
$\xi:=(\xi_{1}^{\prime},\dots,\xi_{T}^{\prime})^{\prime}$ is given by

$\xi=(\xi_{1}^{\prime},\dots,\xi_{T}^{\prime})^{\prime}~{}|~{}\mathcal{H}_{0}\sim\mathcal{N}(0,\bar{\Sigma}),$
(14)

where $\bar{\Sigma}:=\text{diag}\\{\Sigma_{s_{1}},\dots,\Sigma_{s_{T}}\\}$ is
a block diagonal matrix.

To remove duplicates in the random coefficient matrix $(C_{s},\Gamma_{s})$,
for a generic regime–switching vector with length $k$,
$o=(o_{1},\dots,o_{k})^{\prime}$, we define sets

$\mathcal{A}_{\bar{o}_{t}}:=\mathcal{A}_{\bar{o}_{t-1}}\cup\big{\\{}o_{t}\in\\{o_{1},\dots,o_{k}\\}\big{|}o_{t}\not\in\mathcal{A}_{\bar{o}_{t-1}}\big{\\}},~{}~{}~{}t=1,\dots,k,$
(15)

where for $t=1,\dots,k$, $o_{t}\in\\{1,\dots,N\\}$ and an initial set is the
empty set, i.e., $\mathcal{A}_{\bar{o}_{0}}=\O$. The final set
$\mathcal{A}_{o}=\mathcal{A}_{\bar{o}_{k}}$ consists of different regimes in
regime vector $o=\bar{o}_{k}$ and $|\mathcal{A}_{o}|$ represents a number of
different regimes in the regime vector $o$. Let us assume that elements of
sets $\mathcal{A}_{s}$, $\mathcal{A}_{\bar{s}_{t}}$, and difference sets
between the sets $\mathcal{A}_{\bar{s}_{t}^{c}}$ and
$\mathcal{A}_{\bar{s}_{t}}$ are given by
$\mathcal{A}_{s}=\\{\hat{s}_{1},\dots,\hat{s}_{r_{\hat{s}}}\\}$,
$\mathcal{A}_{\bar{s}_{t}}=\\{\alpha_{1},\dots,\alpha_{r_{\alpha}}\\}$, and
$\mathcal{A}_{\bar{s}_{t}^{c}}\backslash\mathcal{A}_{\bar{s}_{t}}=\\{\delta_{1},\dots,\delta_{r_{\delta}}\\}$,
respectively, where $r_{\hat{s}}:=|\mathcal{A}_{s}|$,
$r_{\alpha}:=|\mathcal{A}_{\bar{s}_{t}}|$, and
$r_{\delta}:=|\mathcal{A}_{\bar{s}_{t}^{c}}\backslash\mathcal{A}_{\bar{s}_{t}}|$
are numbers of elements of the sets, respectively. We introduce the following
regime vectors: $\hat{s}:=(\hat{s}_{1},\dots,\hat{s}_{r_{\hat{s}}})^{\prime}$
is an $(r_{\hat{s}}\times 1)$ vector,
$\alpha:=(\alpha_{1},\dots,\alpha_{r_{\alpha}})^{\prime}$ is an
$(r_{\alpha}\times 1)$ vector, and
$\delta=(\delta_{1},\dots,\delta_{r_{\delta}})^{\prime}$ is an
$(r_{\delta}\times 1)$ vector. For the regime vector
$a=(a_{1},\dots,a_{r_{a}})^{\prime}\in\\{\hat{s},\alpha,\delta\\}$, we also
introduce duplication removed random coefficient matrices, whose block
matrices are different: $C_{a}=[C_{a_{1}}:\dots:C_{a_{r_{a}}}]$ is an
$(\tilde{n}\times[lr_{a}])$ matrix,
$\Gamma_{a}=[\Gamma_{a_{1}}:\dots:\Gamma_{a_{r_{a}}}]$ is an
$(\tilde{n}\times[\tilde{n}r_{a}])$ matrix, and $(C_{a},\Gamma_{a})$.

We assume that for given duplication removed regime vector $\hat{s}$ and
initial information $\mathcal{F}_{0}$, the coefficient matrices
$(C_{\hat{s}_{1}},\Gamma_{\hat{s}_{1}}),\dots,(C_{\hat{s}_{r_{\hat{s}}}},\Gamma_{\hat{s}_{r_{\hat{s}}}})$
are independent under the real probability measure $\mathbb{P}$. Under the
assumption, conditional on $\hat{s}$ and $\mathcal{F}_{0}$, a joint density
function of the random coefficient random matrix
$(C_{\hat{s}},\Gamma_{\hat{s}})$ is represented by

$f\big{(}C_{\hat{s}},\Gamma_{\hat{s}}\big{|}\hat{s},\mathcal{F}_{0}\big{)}=\prod_{t=1}^{r_{\hat{s}}}f\big{(}C_{\hat{s}_{t}},\Gamma_{\hat{s}_{t}}\big{|}\hat{s}_{t},\mathcal{F}_{0}\big{)}$
(16)

under the real probability measure $\mathbb{P}$, where for a generic random
vector $X$, we denote its density function by $f(X)$ under the real
probability measure $\mathbb{P}$. Using the regime vectors $\alpha$ and
$\delta$, the above joint density function can be written by

$f\big{(}C_{\hat{s}},\Gamma_{\hat{s}}\big{|}\hat{s},\mathcal{F}_{0}\big{)}=f\big{(}C_{\alpha},\Gamma_{\alpha}\big{|}\alpha,\mathcal{F}_{0}\big{)}f_{*}\big{(}C_{\delta},\Gamma_{\delta}\big{|}\delta,\mathcal{F}_{0}\big{)}$
(17)

where the density function
$f_{*}\big{(}C_{\delta},\Gamma_{\delta}\big{|}\delta,\mathcal{F}_{0}\big{)}$
equals

$f_{*}\big{(}C_{\delta},\Gamma_{\delta}\big{|}\delta,\mathcal{F}_{0}\big{)}:=\begin{cases}f\big{(}C_{\delta},\Gamma_{\delta}\big{|}\delta,\mathcal{F}_{0}\big{)},&\text{if}~{}~{}~{}r_{\delta}\neq
0,\\\ 1,&\text{if}~{}~{}~{}r_{\delta}=0.\end{cases}$ (18)

###  3.1 Risk–Neutral Probability Measure

To price the European call and put options, we need to change from the real
probability measure to some risk–neutral measure. Let
$D_{t}:=\exp\\{-\tilde{r}_{1}-\dots-\tilde{r}_{t}\\}=1\big{/}\prod_{s=1}^{t}(1+r_{s})$
be a predictable discount process, where $\tilde{r}_{t-1}$ is the log spot
interest rate at time $t$. According to Pliska97 (see also Bjork20), for all
companies, a conditional expectation of the return processes
$k_{i,t}^{e}=(V_{i,t}^{e}+p_{i,t}^{e})/V_{i,t-1}^{e}-1$ and
$k_{i,t}^{\ell}=(V_{i,t}^{\ell}+p_{i,t}^{\ell})/V_{i,t-1}^{\ell}-1$ for
$i=1,\dots,n$ must equal the spot interest rate $r_{t}$ under some
risk–neutral probability measure $\tilde{\mathbb{P}}$ and a filtration
$\\{\mathcal{H}_{t}\\}_{t=0}^{T}$. Thus, it must hold

$\tilde{\mathbb{E}}\big{[}(V_{t}+p_{t})\oslash
V_{t-1}\big{|}\mathcal{H}_{t-1}\big{]}=\exp\big{\\{}\tilde{r}_{t}i_{2n}\big{\\}}$
(19)

for $t=1,\dots,T$, where $\tilde{\mathbb{E}}$ denotes an expectation under the
risk–neutral probability measure $\mathbb{\tilde{P}}$. According to equation
(4), condition (19) is equivalent to the following condition

$\tilde{\mathbb{E}}\big{[}\exp\big{\\{}u_{t}-\big{(}(i_{2n}-\delta)\tilde{r}_{t}-C_{k,s_{t}}\psi_{t}\big{)}\big{\\}}\big{|}\mathcal{H}_{t-1}\big{]}=i_{2n}.$
(20)

It should be noted that condition (20) corresponds only to the white noise
random process $u_{t}$. Thus, we need to impose a condition on the white noise
process $v_{t}$ under the risk–neutral probability measure. This condition is
fulfilled by
$\tilde{\mathbb{E}}[\exp\\{v_{t}\\}|\mathcal{H}_{t-1}]=\tilde{\theta}_{t}$ for
$\mathcal{H}_{t-1}$ measurable any random variable $\tilde{\theta}_{t}$.
Because for any admissible choices of $\tilde{\theta}_{t}$, condition (20)
holds, the market is incomplete. But prices of the options are still
consistent with the absence of arbitrage. For this reason, to price the
options, in this paper, we will use a unique optimal Girsanov kernel process
$\theta_{t}$, which minimizes the variance of a state price density process
and relative entropy. According to Battulga24a, the optimal kernel process
$\theta_{t}$ is obtained by

$\theta_{t}=\Theta_{t}\bigg{(}(i_{2n}-\delta)\tilde{r}_{t}-C_{k,s_{t}}\psi_{t}-\frac{1}{2}\mathcal{D}[\Sigma_{uu,s_{t}}]\bigg{)},$
(21)

where
$\Theta_{t}=\big{[}G_{t}:(\Sigma_{vu,s_{t}}\Sigma_{uu,s_{t}}^{-1})^{\prime}\big{]}^{\prime}$
and for a generic square matrix $O$, $\mathcal{D}[O]$ denotes a vector,
consisting of diagonal elements of the matrix $O$. Consequently, system (9)
can be written by

$\begin{cases}\tilde{V}_{t}=\tilde{\nu}_{V,t}+G_{t}\tilde{V}_{t-1}+G_{t}i_{2n}r_{t-1}+G_{t}\tilde{u}_{t}\\\
F_{t}\tilde{r}_{t}=\tilde{\nu}_{r,t}+\tilde{r}_{t-1}+\tilde{v}_{t},\end{cases}~{}~{}~{}\text{for}~{}t=1,\dots,T$
(22)

under the risk–neutral probability measure $\tilde{\mathbb{P}}$, where
$\tilde{\nu}_{V,t}:=-(G_{t}-I_{n})\tilde{p}_{t}-\frac{1}{2}G_{t}\mathcal{D}[\Sigma_{uu,s_{t}}]-h_{t}$
is an $(2n\times 1)$ intercept process of the log market value process
$\tilde{V}_{t}$,
$F_{t}:=1-\Sigma_{vu,s_{t}}\Sigma_{uu,s_{t}}^{-1}(i_{2n}-\delta)$ is a
$(1\times 1)$ process, and
$\tilde{\nu}_{r,t}:=c_{r,s_{t}}^{\prime}\psi_{t}-\Sigma_{vu,s_{t}}\Sigma_{uu,s_{t}}^{-1}\big{(}C_{k,s_{t}}\psi_{t}+\frac{1}{2}\mathcal{D}[\Sigma_{uu,s_{t}}]\big{)}$
is a $(1\times 1)$ intercept process of the log spot rate process
$\tilde{r}_{t}$. It is worth mentioning that a joint distribution of a random
vector
$\tilde{\xi}:=(\tilde{\xi}_{1}^{\prime},\dots,\tilde{\xi}_{T}^{\prime})^{\prime}$
with $\tilde{\xi}_{t}:=(\tilde{u}_{t}^{\prime},\tilde{v}_{t})^{\prime}$ equals
the joint distribution of the random vector
$\xi=(\xi_{1}^{\prime},\dots,\xi_{T}^{\prime})^{\prime}$, that is,

$\tilde{\xi}\sim\mathcal{N}\big{(}0,\bar{\Sigma}\big{)}$ (23)

under the risk–neutral probability measure $\mathbb{\tilde{P}}$, see
Battulga24a.

System (22) can be written in VAR(1) form, namely

$\tilde{Q}_{0,t}x_{t}=\tilde{\nu}_{t}+\tilde{Q}_{t}x_{t-1}+\mathsf{G}_{t}\tilde{\xi}_{t}$
(24)

under the risk–neutral probability measure $\mathbb{\tilde{P}}$, where
$\tilde{\nu}_{t}:=(\tilde{\nu}_{V,t}^{\prime},\tilde{\nu}_{r,t})^{\prime}$,
and
$\tilde{\xi}_{t}:=\big{(}\tilde{u}_{t}^{\prime},\tilde{v}_{t}\big{)}^{\prime}$
are intercept process and white noise processes of the VAR(1) process $x_{t}$,
respectively, and

$\tilde{Q}_{0,t}:=\begin{bmatrix}I_{2n}&0\\\
0&F_{t}\end{bmatrix},~{}~{}~{}\tilde{Q}_{1,t}:=\begin{bmatrix}G_{t}&G_{t}i_{2n}\\\
0&E_{t}\end{bmatrix},~{}~{}~{}\text{and}~{}~{}~{}\mathsf{G}_{t}=\begin{bmatrix}G_{t}&0\\\
0&1\end{bmatrix}$ (25)

are $(\tilde{n}\times\tilde{n})$ coefficient matrices with
$F_{t}:=1-\Sigma_{vu,t}\Sigma_{uu,t}^{-1}(i_{2n}-\delta)$ and
$E_{t}:=1+\Sigma_{vu,t}\Sigma_{uu,t}^{-1}(i_{2n}-\delta)$. By repeating
equation (24), one gets that for $i=t+1,\dots,T$,

$x_{i}=\tilde{\Pi}_{t,i}x_{t}+\sum_{\beta=t+1}^{i}\tilde{\Pi}_{\beta,i}\tilde{\nu}_{\beta}+\sum_{\beta=t+1}^{i}\tilde{\Pi}_{\beta,i}\mathsf{G}_{\beta}\tilde{\xi}_{\beta},$
(26)

where the coefficient matrices are

$\tilde{\Pi}_{\beta,i}:=\prod_{\alpha=\beta+1}^{i}\tilde{Q}_{0,\alpha}^{-1}\tilde{Q}_{1,\alpha}=\begin{bmatrix}\displaystyle\prod_{\alpha=\beta+1}^{i}G_{\alpha}&\displaystyle\sum_{\alpha=\beta+1}^{i}G_{i}\Bigg{(}\prod_{j_{1}=\alpha}^{i-1}G_{j_{1}}\Bigg{)}i_{2n}\Bigg{(}\prod_{j_{2}=\beta+1}^{\alpha-1}F_{j_{2}}^{-1}E_{j_{2}}\Bigg{)}\\\
\displaystyle
0&\displaystyle\prod_{\alpha=\beta+1}^{i}F_{\alpha}^{-1}E_{\alpha}\end{bmatrix}$
(27)

for $\beta=0,\dots,i-1$ and

$\tilde{\Pi}_{i,i}:=\tilde{Q}_{0,i}^{-1}=\begin{bmatrix}I_{2n}&0\\\
0&F_{i}^{-1}\end{bmatrix}.$ (28)

Here for a sequence of generic $(k\times k)$ square matrices
$O_{1},O_{2},\dots$, the products mean that for $v\leq u$,
$\prod_{j=v}^{u}O_{j}=O_{u}\dots O_{v}$ and for $v>u$,
$\prod_{j=v}^{u}O_{j}=I_{k}$.

Therefore, conditional on the information $\mathcal{H}_{t}$, for
$i=t+1,\dots,T$, a expectation at time $i$ and a conditional covariance matrix
at time $i_{1}$ and $i_{2}$ of the process $x_{t}$ is given by the following
equations

$\tilde{\mu}_{i|t}(\mathcal{H}_{t}):=\mathbb{\tilde{E}}\big{[}x_{i}\big{|}\mathcal{H}_{t}\big{]}=\mathsf{J}_{x}\tilde{\Pi}_{t,i}x_{t}+\sum_{\beta=t+1}^{i}\tilde{\Pi}_{\beta,i}\tilde{\nu}_{\beta}$
(29)

and

$\tilde{\Sigma}_{i_{1},i_{2}|t}(\mathcal{H}_{t}):=\widetilde{\text{Cov}}\big{[}x_{i_{1}},x_{i_{2}}\big{|}\mathcal{H}_{t}\big{]}=\sum_{\beta=t+1}^{i_{1}\wedge
i_{2}}\tilde{\Pi}_{\beta,i_{1}}\mathsf{G}_{\beta}\Sigma_{s_{\beta}}\mathsf{G}_{\beta}\tilde{\Pi}_{\beta,i_{2}}^{\prime},$
(30)

where $i_{1}\wedge i_{2}$ is a minimum of $i_{1}$ and $i_{2}$. Consequently,
due to equation (26), conditional on the information $\mathcal{H}_{t}$, joint
distribution of the random vector $\bar{x}_{t}^{c}$ is

$\bar{x}_{t}^{c}~{}|~{}\mathcal{H}_{t}\sim\mathcal{N}\big{(}\tilde{\mu}_{t}^{c}(\mathcal{H}_{t}),\tilde{\Sigma}_{t}^{c}(\mathcal{H}_{t})\big{)},~{}~{}~{}t=0,\dots,T-1$
(31)

under the risk–neutral probability measure $\tilde{\mathbb{P}}$, where
$\tilde{\mu}_{t}^{c}(\mathcal{H}_{t}):=\big{(}\tilde{\mu}_{t+1|t}(\mathcal{H}_{t}),\dots,\tilde{\mu}_{T|t}(\mathcal{H}_{t})\big{)}^{\prime}$
is a conditional expectation and
$\tilde{\Sigma}_{t}^{c}(\mathcal{H}_{t}):=\big{(}\tilde{\Sigma}_{i_{1},i_{2}|t}(\mathcal{H}_{t})\big{)}_{i_{1},i_{2}=t+1}^{T}$
is a conditional covariance matrix of the random vector $\bar{x}_{t}^{c}$ and
are calculated by equations (29) and (30), respectively.

###  3.2 Forward Probability Measure

According to Geman95, cleaver change of probability measure leads to a
significant reduction in the computational burden of derivative pricing. The
frequently used probability measure that reduces the computational burden is
the forward probability measure and to price the zero–coupon bond, the
European options, and the Margrabe exchange options we will apply it. To
define the forward probability measure, we need to zero–coupon bond. It is the
well–known fact that conditional on $\mathcal{F}_{t}$, price of zero–coupon
bond paying face value 1 at time $t$ is
$B_{t}(\mathcal{H}_{t}):=\frac{1}{D_{t}}\mathbb{\tilde{E}}\big{[}D_{T}\big{|}\mathcal{H}_{t}\big{]}$.
The $t$–forward probability measure is defined by

$\mathbb{\hat{P}}_{t}\big{[}A\big{|}\mathcal{H}_{t}\big{]}:=\frac{1}{D_{t}B_{t}(\mathcal{H}_{t})}\int_{A}D_{T}\mathbb{\tilde{P}}\big{[}\omega|\mathcal{H}_{t}\big{]}~{}~{}~{}\text{for
all}~{}A\in\mathcal{H}_{T}.$ (32)

Therefore, a negative exponent of $D_{T}/D_{t}$ in the zero–coupon bond
formula is represented by

$\sum_{\beta=t+1}^{T}\tilde{r}_{\beta}=\tilde{r}_{t+1}+j_{r}^{\prime}\Bigg{[}\sum_{\beta=t+1}^{T-1}J_{\beta|t}\Bigg{]}\bar{x}_{t}^{c}=\tilde{r}_{t+1}+\gamma_{t}^{\prime}\bar{x}_{t}^{c}$
(33)

where $j_{r}:=(0,1)^{\prime}$ is $(1\times\tilde{n})$ vector and it can be
used to extract the log spot rate process $\tilde{r}_{s}$ from the random
process $x_{s}$, $J_{\beta|t}:=[0:I_{\tilde{n}}:0]$ is
$(\tilde{n}\times\tilde{n}(T-t))$ matrix, whose $(\beta-t)$–th block matrix
equals $I_{\tilde{n}}$ and others are zero and it is used to extract the
random vector $x_{\beta}$ from the random vector $\bar{x}_{t}^{c}$, and
$\gamma_{t}^{\prime}:=j_{r}^{\prime}\sum_{\beta=t+1}^{T-1}J_{\beta|t}$.
Therefore, two times of negative exponent of the price at time $t$ of the
zero–coupon $B_{t}(\mathcal{H}_{t})$ is represented by

$\displaystyle
2\sum_{s=t+1}^{T}\tilde{r}_{s}+\big{(}\bar{x}_{t}^{c}-\tilde{\mu}_{t}^{c}(\mathcal{H}_{t})\big{)}^{\prime}\big{(}\tilde{\Sigma}_{t}^{c}(\mathcal{H}_{t})\big{)}^{-1}\big{(}\bar{x}_{t}^{c}-\tilde{\mu}_{t}^{c}(\mathcal{H}_{t})\big{)}$
$\displaystyle=\Big{(}\bar{x}_{t}^{c}-\tilde{\mu}_{t}^{c}(\mathcal{H}_{t})+\tilde{\Sigma}_{t}^{c}(\mathcal{H}_{t})\gamma_{t}\Big{)}^{\prime}\big{(}\tilde{\Sigma}_{t}^{c}(\mathcal{H}_{t})\big{)}^{-1}\Big{(}\bar{x}_{t}^{c}-\tilde{\mu}_{t}^{c}(\mathcal{H}_{t})+\tilde{\Sigma}_{t}^{c}(\mathcal{H}_{t})\gamma_{t}\Big{)}$
(34)
$\displaystyle+2\big{(}\tilde{r}_{t+1}+\gamma_{t}^{\prime}\tilde{\mu}_{t}^{c}(\mathcal{H}_{t})\big{)}-\gamma_{t}^{\prime}\tilde{\Sigma}_{t}^{c}(\mathcal{H}_{t})\gamma_{t}.$

As a result, for given $\mathcal{H}_{t}$, price at time $t$ of the zero–coupon
$B_{t}(\mathcal{H}_{t})$ is

$B_{t}(\mathcal{H}_{t})=\exp\bigg{\\{}-\tilde{r}_{t+1}-\gamma_{t}^{\prime}\tilde{\mu}_{t}^{c}(\mathcal{H}_{t})+\frac{1}{2}\gamma_{t}^{\prime}\tilde{\Sigma}_{t}^{c}(\mathcal{H}_{t})\gamma_{t}\bigg{\\}}.$
(35)

Consequently, conditional on the information $\mathcal{H}_{t}$, a joint
distribution of the random vector $\bar{x}_{t}^{c}$ is given by

$\bar{x}_{t}^{c}~{}|~{}\mathcal{H}_{t}\sim\mathcal{N}\big{(}\hat{\mu}_{t}^{c}(\mathcal{H}_{t}),\tilde{\Sigma}_{t}^{c}(\mathcal{H}_{t})\big{)},~{}~{}~{}t=0,\dots,T-1$
(36)

under the $t$–forward probability measure $\hat{\mathbb{P}}_{t}$, where
$\hat{\mu}_{t}^{c}(\mathcal{H}_{t}):=\tilde{\mu}_{t}^{c}(\mathcal{H}_{t})-\tilde{\Sigma}_{t}^{c}(\mathcal{H}_{t})\gamma_{t}$
and $\tilde{\Sigma}_{t}^{c}(\mathcal{H}_{t})$ are conditional expectation and
conditional covariance matrix, respectively, of the random vector
$\bar{x}_{t}^{c}$. Also, as
$J_{s_{1}|t}\tilde{\Sigma}_{t}^{c}(\mathcal{H}_{t})J_{s_{2}|t}^{\prime}=\tilde{\Sigma}_{s_{1},s_{2}|t}(\mathcal{H}_{t})$,
we have

$J_{s|t}\tilde{\Sigma}_{t}^{c}(\mathcal{H}_{t})\Bigg{(}\sum_{\beta=t+1}^{T-1}J_{\beta|t}\Bigg{)}=\sum_{\beta=t+1}^{T-1}\tilde{\Sigma}_{s,\beta|t}(\mathcal{H}_{t}),$
(37)

where $\tilde{\Sigma}_{s,\beta|t}(\mathcal{H}_{t})$ is calculated by equation
(30). Therefore, $(s-t)$–th block vector of the conditional expectation
$\hat{\mu}_{t}^{c}(\mathcal{H}_{t})$ is given by

$\hat{\mu}_{s|t}(\mathcal{H}_{t}):=J_{s|t}\hat{\mu}_{t}^{c}(\mathcal{H}_{t})=\tilde{\mu}_{s|t}(\mathcal{H}_{t})-\sum_{\beta=t+1}^{T-1}\big{(}\tilde{\Sigma}_{s,\beta|t}(\mathcal{H}_{t})\big{)}_{\tilde{n}},$
(38)

where for a generic matrix $O$, we denote its $j$–th column by $(O)_{j}$.
Similarly, it is clear that price at time $t$ of the zero–coupon bond is given
by

$B_{t}(\mathcal{H}_{t})=\exp\bigg{\\{}-\tilde{r}_{t+1}-\sum_{\beta=t+1}^{T-1}\big{(}\tilde{\mu}_{\beta|t}(\mathcal{H}_{t})\big{)}_{\tilde{n}}+\frac{1}{2}\sum_{\alpha=t+1}^{T-1}\sum_{\beta=t+1}^{T-1}\big{(}\tilde{\Sigma}_{\alpha,\beta|t}(\mathcal{H}_{t})\big{)}_{\tilde{n},\tilde{n}}\bigg{\\}}.$
(39)

where for a generic vector $o$, we denote its $j$–th element by $(o)_{j}$, and
for a generic square matrix $O$, we denote its $(i,j)$–th element by
$(O)_{i,j}$.

To price the European call and put options for asset value, we need a
distribution of the log market value process at time $T$. For this reason, it
follows from equation (36) that the distribution of the log market value
process at time $T$ is given by

$\tilde{V}_{T}~{}|~{}\mathcal{H}_{t}\sim\mathcal{N}\big{(}\hat{\mu}_{T|t}^{\tilde{V}}(\mathcal{H}_{t}),\tilde{\Sigma}_{T|t}^{\tilde{V}}(\mathcal{H}_{t})\big{)},$
(40)

under the $t$–forward probability measure $\mathbb{\hat{P}}$, where
$\hat{\mu}_{T|t}^{\tilde{V}}(\mathcal{H}_{t}):=J_{V}\hat{\mu}_{T|t}(\mathcal{H}_{t})$
is a conditional expectation, which is calculated from equation (38) and
$\tilde{\Sigma}_{T|t}^{\tilde{V}}(\mathcal{H}_{t}):=J_{V}\tilde{\Sigma}_{T,T|t}^{\tilde{V}}(\mathcal{H}_{t})J_{V}^{\prime}$
is a conditional covariance matrix, which is calculated from equation (30) of
the log market value at time $T$ given the information $\mathcal{H}_{t}$ and
$J_{V}:=[I_{2n}:0]$ is a ($2n\times\tilde{n}$) matrix, which is used to
extract the log market value process $\tilde{V}_{t}$ from the process $x_{t}$.

###  3.3 The European Call and Put Options

Let us assume that the recovery rates, corresponding to the market values of
assets of the companies are zero when they default. Then, because market
values at time $T$ of the equities and liabilities are given by the following
equations

$V_{T}^{e}=\max(V_{T}^{a}-L,0)=(V_{T}^{a}-L)^{+}~{}~{}~{}\text{and}~{}~{}~{}L_{T}=\min(V_{T}^{a},L)=L-(L-V_{T}^{a})^{+},$

respectively, where $L$ is a nominal value vector of the liabilities at
maturity $T$ of the companies. Therefore, a risk–neutral equity value at time
$t$ of a public company equals the European call option on its asset and
liabilities at time $t$ of the company is represented in terms of the European
put option on its asset. This Subsection is devoted to price the call and put
options.

According to equation (10) and (40), conditional on the information
$\mathcal{H}_{t}$, its distribution is given by

$\tilde{V}_{T}^{a}~{}|~{}\mathcal{H}_{t}\sim\mathcal{N}\big{(}\hat{\mu}_{T|t}^{a}(\mathcal{H}_{t}),\tilde{\Sigma}_{T|t}^{a}(\mathcal{H}_{t})\big{)}$
(41)

under the $t$–forward probability measure $\hat{\mathbb{P}}_{t}$, where
$\hat{\mu}_{T|t}^{a}(\mathcal{H}_{t}):=W_{T}^{a}\hat{\mu}_{T|t}^{\tilde{V}}(\mathcal{H}_{t})+(G_{T}^{a})^{-1}h_{T}^{a}$
and
$\tilde{\Sigma}_{T|t}^{a}(\mathcal{H}_{t}):=W_{T}^{a}\tilde{\Sigma}_{T|t}^{\tilde{V}}(\mathcal{H}_{t})(W_{T}^{a})^{\prime}$
are conditional mean and variance of the log asset value $\tilde{V}_{T}^{a}$,
respectively, given the information $\mathcal{H}_{t}$. Therefore, due to
equation (41) and Lemma 1, see Technical Annex, price vectors at time $t$ of
the Black–Scholes call and put options with strike price vector $L$ and
maturity $T$ are given by

$\displaystyle C_{T|t}(\mathcal{H}_{t})$ $\displaystyle=$
$\displaystyle\mathbb{\tilde{E}}\bigg{[}\frac{D_{T}}{D_{t}}\Big{(}V_{T}^{a}-L\Big{)}^{+}\bigg{|}\mathcal{H}_{t}\bigg{]}=B_{t}(\mathcal{H}_{t})\mathbb{\hat{E}}\big{[}\big{(}V_{T}^{a}-L\big{)}^{+}\big{|}\mathcal{H}_{t}\big{]}$
$\displaystyle=$ $\displaystyle
B_{t}(\mathcal{H}_{t})\bigg{(}\exp\bigg{\\{}\hat{\mu}_{T|t}^{a}(\mathcal{H}_{t})+\frac{1}{2}\mathcal{D}\big{[}\tilde{\Sigma}_{T|t}^{a}(\mathcal{H}_{t})\big{]}\bigg{\\}}\odot\Phi\big{(}d_{T|t}^{1}(\mathcal{H}_{t})\big{)}-L\odot\Phi\big{(}d_{T|t}^{2}(\mathcal{H}_{t})\big{)}\bigg{)}$

and

$\displaystyle P_{T|t}(\mathcal{H}_{t})$ $\displaystyle=$
$\displaystyle\mathbb{\tilde{E}}\bigg{[}\frac{D_{T}}{D_{t}}\Big{(}L-V_{T}^{a}\Big{)}^{+}\bigg{|}\mathcal{H}_{t}\bigg{]}=B_{t}(\mathcal{H}_{t})\mathbb{\hat{E}}\big{[}\big{(}L-V_{T}^{a}\big{)}^{+}\big{|}\mathcal{H}_{t}\big{]}$
$\displaystyle=$ $\displaystyle
B_{t}(\mathcal{H}_{t})\bigg{(}L\odot\Phi\big{(}-d_{T|t}^{2}(\mathcal{H}_{t})\big{)}-\exp\bigg{\\{}\hat{\mu}_{T|t}^{a}(\mathcal{H}_{t})+\frac{1}{2}\mathcal{D}\big{[}\tilde{\Sigma}_{T|t}^{a}(\mathcal{H}_{t})\big{]}\bigg{\\}}\odot\Phi\big{(}-d_{T|t}^{1}(\mathcal{H}_{t})\big{)}\bigg{)},$

where
$d_{T|t}^{1}(\mathcal{H}_{t}):=\Big{(}\hat{\mu}_{T|t}^{a}(\mathcal{H}_{t})+\mathcal{D}\big{[}\tilde{\Sigma}_{T|t}^{a}(\mathcal{H}_{t})\big{]}-\ln(L)\Big{)}\oslash\sqrt{\mathcal{D}\big{[}\tilde{\Sigma}_{T|t}^{a}(\mathcal{H}_{t})\big{]}}$
and
$d_{T|t}^{2}(\mathcal{H}_{t}):=d_{T|t}^{1}(\mathcal{H}_{t})-\sqrt{\mathcal{D}\big{[}\tilde{\Sigma}_{T|t}^{a}(\mathcal{H}_{t})\big{]}}$.

Therefore, due to Lemma 2 and the tower property of conditional expectation,
price vectors at time $t$ ($t=0,\dots,T-1$) of the Black–Scholes call and put
options on asset values with strike price vector $L$ and maturity $T$ are
obtained as

$C_{T|t}(\mathcal{F}_{t})=\tilde{\mathbb{E}}\bigg{[}\frac{D_{T}}{D_{t}}\Big{(}V_{T}^{a}-L\Big{)}^{+}\bigg{|}\mathcal{F}_{t}\bigg{]}=\sum_{s}\int_{C_{\hat{s}},\Gamma_{\hat{s}}}C_{T|t}(\mathcal{H}_{t})\tilde{f}(C_{\hat{s}},\Gamma_{\hat{s}},s|\mathcal{F}_{t})dC_{\hat{s}}d\Gamma_{\hat{s}}$
(44)

and

$P_{T|t}(\mathcal{F}_{t})=\tilde{\mathbb{E}}\bigg{[}\frac{D_{T}}{D_{t}}\Big{(}L-V_{T}^{a}\Big{)}^{+}\bigg{|}\mathcal{F}_{t}\bigg{]}=\sum_{s}\int_{C_{\hat{s}},\Gamma_{\hat{s}}}P_{T|t}(\mathcal{H}_{t})\tilde{f}(C_{\hat{s}},\Gamma_{\hat{s}},s|\mathcal{F}_{t})dC_{\hat{s}}d\Gamma_{\hat{s}}$
(45)

respectively.

As a result, according to formulas of the call and put options given in
equations (44) and (45), risk–neutral market values of the equities and
liabilities at time $t$ of the companies are given by

$\hat{V}_{t}^{e}=C_{T|t}(\mathcal{F}_{t})~{}~{}~{}\text{and}~{}~{}~{}\hat{L}_{t}=LB_{t}(\mathcal{F}_{t})-P_{T|t}(\mathcal{F}_{t}).$
(46)

###  3.4 Default Probability

Now, we move to default probabilities of a companies. In order to obtain the
default probabilities of the companies, for given the information
$\mathcal{H}_{t}$, we need a distribution of log asset value at time $T$ under
the real probability measure $\mathbb{P}$. For this reason, let us write
system (9) in VAR(1) form

$x_{t}=\nu_{t}+Q_{t}x_{t-1}+\mathsf{G}_{t}\xi_{t}$ (47)

under the real probability measure $\mathbb{P}$, where
$\nu_{t}:=(\nu_{V,t}^{\prime},\nu_{r,t})^{\prime}$ is intercept process of the
VAR(1) process $x_{t}$ and

$Q_{t}:=\begin{bmatrix}G_{t}&G_{t}\delta\\\ 0&1\end{bmatrix}$ (48)

is $(\tilde{n}\times\tilde{n})$ coefficient matrix. By repeating equation
(47), one gets that for $i=t+1,\dots,T$,

$x_{i}=\Pi_{t,i}x_{t}+\sum_{\beta=t+1}^{i}\Pi_{\beta,i}\nu_{\beta}+\sum_{\beta=t+1}^{i}\Pi_{\beta,i}\mathsf{G}_{\beta}\xi_{\beta},$
(49)

where the coefficient matrices are

$\Pi_{\beta,i}:=\prod_{\alpha=\beta+1}^{i}Q_{\alpha}=\begin{bmatrix}\displaystyle\prod_{\alpha=\beta+1}^{i}G_{\alpha}&\displaystyle\sum_{\alpha=\beta+1}^{i}G_{i}\Bigg{(}\prod_{j_{1}=\alpha}^{i-1}G_{j_{1}}\Bigg{)}\delta\\\
\displaystyle 0&1\end{bmatrix}$ (50)

for $\beta=0,\dots,i-1$ and $\Pi_{i,i}:=I_{\tilde{n}}$. Thus, conditional on
the information $\mathcal{H}_{t}$, for $i=t+1,\dots,T$, a expectation at time
$i$ and a conditional covariance matrix at time $i_{1}$ and $i_{2}$ of the
process $x_{t}$ is given by the following equations

$\mu_{i|t}(\mathcal{H}_{t}):=\mathbb{E}\big{[}x_{i}\big{|}\mathcal{H}_{t}\big{]}=\Pi_{t,i}x_{t}+\sum_{\beta=t+1}^{i}\Pi_{\beta,i}\nu_{\beta}$
(51)

and

$\Sigma_{i_{1},i_{2}|t}(\mathcal{H}_{t}):=\text{Cov}\big{[}x_{i_{1}},x_{i_{2}}\big{|}\mathcal{H}_{t}\big{]}=\sum_{\beta=t+1}^{i_{1}\wedge
i_{2}}\Pi_{\beta,i_{1}}\mathsf{G}_{\beta}\Sigma_{s_{\beta}}\mathsf{G}_{\beta}\Pi_{\beta,i_{2}}^{\prime}.$
(52)

Therefore, it follows from equations (51) and (52) that conditional on the
information $\mathcal{H}_{t}$, an expectation and covariance matrix of log
market value process at time $T$ under the real probability measure
$\mathbb{P}$ are given by the following equations

$\mu_{T|t}^{\tilde{V}}(\mathcal{H}_{t}):=\mathbb{E}\big{[}\tilde{V}_{T}\big{|}\mathcal{H}_{t}\big{]}=J_{V}\mu_{T|t}(\mathcal{H}_{t})$
(53)

and

$\displaystyle\Sigma_{T|t}^{\tilde{V}}(\mathcal{H}_{t})$ $\displaystyle:=$
$\displaystyle\text{Var}\big{[}\tilde{V}_{T}\big{|}\mathcal{H}_{t}\big{]}=J_{V}\Sigma_{T,T|t}(\mathcal{H}_{t})J_{V}^{\prime}.$
(54)

Consequently, due to equation (10), conditional on $\mathcal{H}_{t}$, a
distribution of the log asset value process at time $T$ is given by

$\tilde{V}_{T}^{a}~{}|~{}\mathcal{H}_{t}\sim\mathcal{N}\Big{(}\mu_{T|t}^{a}(\mathcal{H}_{t}),\Sigma_{T|t}^{a}(\mathcal{H}_{t})\Big{)}$
(55)

under the real probability measure $\mathbb{P}$, where
$\mu_{T|t}^{a}(\mathcal{H}_{t}):=W_{T}^{a}\mu_{T|t}^{\tilde{V}}(\mathcal{H}_{t})+(G_{T}^{a})^{-1}h_{T}^{a}$
and
$\Sigma_{T|t}^{a}(\mathcal{H}_{t}):=W_{T}^{a}\Sigma_{T|t}^{\tilde{V}}(\mathcal{H}_{t})(W_{T}^{a})^{\prime}$
are conditional mean and covariance matrix of the log asset value
$\tilde{V}_{T}^{a}$, respectively, given the information $\mathcal{H}_{t}$.
According to the structural model of default risk, if the asset value of a
company falls below the default threshold, representing liabilities, then
default occurs. Therefore, due to equation (55), conditional on the
information $\mathcal{H}_{t}$, the default probability at time $t$ of the
company is given by the following equation

$\mathbb{P}\big{[}V_{T}^{a}\leq\bar{L}|\mathcal{H}_{t}\big{]}=\mathbb{P}\big{[}\tilde{V}_{T}^{a}\leq\ln(\bar{L})|\mathcal{H}_{t}\big{]}=\Phi_{n}\left(\big{(}\Sigma_{T|t}^{a}(\mathcal{H}_{t})\big{)}^{-1}\big{(}\ln(\bar{L})-\mu_{T|t}^{a}(\mathcal{H}_{t})\big{)}\right),$
(56)

where $\bar{L}$ is the default threshold vector at maturity $T$ and for a
random vector $Z\sim\mathcal{N}(0,I_{n})$, $\Phi_{n}(z):=\mathbb{P}(Z\leq z)$
is a joint distribution function of the random vector $Z$. As a result, by the
tower property of conditional expectation formula and Lemma 3, we get that

$\mathbb{P}\big{[}V_{T}^{a}\leq\bar{L}|\mathcal{F}_{t}\big{]}=\sum_{s}\int_{C_{\hat{s}},\Gamma_{\hat{s}}}\mathbb{P}\big{[}V_{T}^{a}\leq\bar{L}|\mathcal{H}_{t}\big{]}f(C_{\hat{s}},\Gamma_{\hat{s}},s|\mathcal{F}_{t})dC_{\hat{s}}d\Gamma_{\hat{s}}.$
(57)

##  4 Parameter Estimation

To estimate parameters of the required rate of return $\tilde{k}_{t}$,
Battulga23b used the maximum likelihood method and Kalman filtering. For
Bayesian method, which removes duplication in regime vector, we refer to
Battulga24g. In this section, we assume that coefficient matrices
$C_{1},\dots,C_{N}$ and covariance matrices $\Sigma_{1},\dots,\Sigma_{N}$ are
deterministic. Here we apply the EM algorithm to estimate parameters of the
model. If we combine the equations (6) and (8), then we have that

$B_{0}y_{t}=C_{s_{t}}\psi_{t}+B_{1}y_{t-1}+\xi_{t},$ (58)

where $y_{t}:=(\tilde{k}_{t}^{\prime},\tilde{r}_{t})^{\prime}$ is an
$(\tilde{n}\times 1)$ vector of endogenous variables, $C_{s_{t}}$ is the
$(\tilde{n}\times l)$ matrix, which depends on the regime $s_{t}$,
$d:=(\delta^{\prime},1)^{\prime}$ is an $(\tilde{n}\times 1)$ vector, the
$(\tilde{n}\times\tilde{n})$ matrices are given by

$B_{0}:=\begin{bmatrix}I_{2n}&\delta\\\
0&1\end{bmatrix}~{}~{}~{}\text{and}~{}~{}~{}B_{1}:=\begin{bmatrix}0&1\\\
0&0\end{bmatrix}.$ (59)

For $t=0,\dots,T$, let $\mathcal{Y}_{t}$ be the available data at time $t$,
which is used to estimate parameters of the model, that is,
$\mathcal{Y}_{t}:=\sigma(\tilde{d}_{0},\tilde{r}_{0},y_{1},\dots,y_{t}).$
Then, it is clear that the log–likelihood function of our model is given by
the following equation

$\mathcal{L}(\theta)=\sum_{t=1}^{T}\ln\big{(}f(y_{t}|\mathcal{Y}_{t-1};\theta)\big{)}$
(60)

where
$\theta:=\big{(}\text{vec}(C_{1})^{\prime},\dots,\text{vec}(C_{N})^{\prime},\text{vec}(\Sigma_{1})^{\prime},\dots,\text{vec}(\Sigma_{N})^{\prime},\text{vec}(\mathsf{P})^{\prime}\big{)}^{\prime}$
is a vector, which consists of all population parameters of the model and
$f(y_{t}|\mathcal{Y}_{t-1};\theta)$ is a conditional density function of the
random vector $y_{t}$ given the information $\mathcal{Y}_{t-1}$. The
log–likelihood function is used to obtain the maximum likelihood estimator of
the parameter vector $\theta$. Note that the log–likelihood function depends
on all observations, which are collected in $\mathcal{Y}_{T}$, but does not
depend on regime–switching process $s_{t}$, whose values are unobserved. If we
assume that the regime–switching process in regime $j$ at time $t$, then
because conditional on the information $\mathcal{Y}_{t-1}$, $\xi_{t}$ follows
a multivariate normal distribution with mean zero and covariance matrix
$\Sigma_{j}$, the conditional density function of the random vector $y_{t}$ is
given by the following equation

$\displaystyle\eta_{t,j}$ $\displaystyle:=$ $\displaystyle
f(y_{t}|s_{t}=j,\mathcal{Y}_{t-1};\alpha)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{\tilde{n}/2}|\Sigma_{j}|^{1/2}}\exp\bigg{\\{}-\frac{1}{2}\Big{(}B_{0}y_{t}-C_{j}\psi_{t}-B_{1}y_{t-1}\Big{)}^{\prime}\Sigma_{j}^{-1}\Big{(}B_{0}y_{t}-C_{j}\psi_{t}-B_{1}y_{t-1}\Big{)}\bigg{\\}}$

for $t=1,\dots,T$ and $j=1,\dots,N$, where
$\alpha:=\big{(}\text{vec}(C_{1})^{\prime},\dots,\text{vec}(C_{N})^{\prime},\text{vec}(\Sigma_{1})^{\prime},\dots,\text{vec}(\Sigma_{N})^{\prime}\big{)}^{\prime}$
is a parameter vector, which differs from the vector of all parameters
$\theta$ by the transition probability matrix $\mathsf{P}$. For all
$t=1,\dots,T$, we collect the conditional density functions of the price at
time $t$ into an $(N\times 1)$ vector $\eta_{t}$, that is,
$\eta_{t}:=(\eta_{t,1},\dots,\eta_{t,N})^{\prime}$.

Let us denote a probabilistic inference about the value of the
regime–switching process $s_{t}$ equals to $j$, based on the information
$\mathcal{Y}_{t}$ and the parameter vector $\theta$ by
$\mathbb{P}(s_{t}=j|\mathcal{Y}_{t},\theta)$. Collect these conditional
probabilities $\mathbb{P}(s_{t}=j|\mathcal{Y}_{t},\theta)$ for $j=1,\dots,N$
into an $(N\times 1)$ vector $z_{t|t}$, that is,
$z_{t|t}:=\big{(}\mathbb{P}(s_{t}=1|\mathcal{Y}_{t};\theta),\dots,\mathbb{P}(s_{t}=N|\mathcal{Y}_{t};\theta)\big{)}^{\prime}$.
Also, we need a probabilistic forecast about the value of the regime–switching
process at time $t+1$ equals $j$ conditional on data up to and including time
$t$. Collect these forecasts into an $(N\times 1)$ vector $z_{t+1|t}$, that
is,
$z_{t+1|t}:=\big{(}\mathbb{P}(s_{t+1}=1|\mathcal{Y}_{t};\theta),\dots,\mathbb{P}(s_{t+1}=N|\mathcal{Y}_{t};\theta)\big{)}^{\prime}$.

The probabilistic inference and forecast for each time $t=1,\dots,T$ can be
found by iterating on the following pair of equations:

$z_{t|t}=\frac{(z_{t|t-1}\odot\eta_{t})}{i_{N}^{\prime}(z_{t|t-1}\odot\eta_{t})}~{}~{}~{}\text{and}~{}~{}~{}z_{t+1|t}=\hat{\mathsf{P}}^{\prime}z_{t|t},~{}~{}~{}t=1,\dots,T,$
(62)

see book of Hamilton94, where $\eta_{t}$ is the $(N\times 1)$ vector, whose
$j$-th element is given by equation (4), $\hat{\mathsf{P}}$ is the $(N\times
N)$ transition probability matrix, which is defined by omitting the first row
of the matrix $\mathsf{P}$, and $i_{N}$ is an $(N\times 1)$ vector, whose
elements equal 1. Given a starting value $z_{1|0}$ and an assumed value for
the population parameter vector $\theta$, one can iterate on (62) for
$t=1,\dots,T$ to calculate the values of $z_{t|t}$ and $z_{t+1|t}$.

To obtain MLE of the population parameters, in addition to the inferences and
forecasts we need a smoothed inference about the regime–switching process is
in at time $t$ based on full information $\mathcal{Y}_{T}$. Collect these
smoothed inferences into an $(N\times 1)$ vector $z_{t|T}$, that is,
$z_{t|T}:=\big{(}\mathbb{P}(s_{t}=1|\mathcal{Y}_{T};\theta),\dots,\mathbb{P}(s_{t}=N|\mathcal{Y}_{T};\theta)\big{)}^{\prime}$.
The smoothed inferences can be obtained by using the Battulga24g’s exact
smoothing algorithm:

$z_{T-1|T}=\frac{1}{i_{N}^{\prime}(z_{T|T-1}\odot\eta_{t})}\big{(}\hat{\mathsf{P}}\mathsf{H}_{T}i_{N}\big{)}\odot
z_{T-1|T-1}$ (63)

and for $t=T-2,\dots,1$,

$z_{t|T}=\frac{1}{i_{N}^{\prime}(z_{t+1|t}\odot\eta_{t+1})}\Big{(}\hat{\mathsf{P}}\mathsf{H}_{t+1}\big{(}z_{t+1|T}\oslash
z_{t+1|t+1}\big{)}\Big{)}\odot z_{t|t},$ (64)

where $\oslash$ is an element–wise division of two vectors and
$\mathsf{H}_{t+1}:=\mathrm{diag}\\{\eta_{t+1,1},\dots,\eta_{t+1,N}\\}$ is an
$(N\times N)$ diagonal matrix. For $t=2,\dots,T$, joint probability of the
regimes $s_{t-1}$ and $s_{t}$ is

$\mathbb{P}(s_{t-1}=i,s_{t}=j|\mathcal{F}_{t};\theta)=\frac{(z_{t|T})_{j}\eta_{t,j}p_{s_{t-1}s_{t}}(z_{t-1|t-1})_{i}}{(z_{t|t})_{j}i_{N}^{\prime}(z_{t|t-1}\odot\eta_{t})},$
(65)

where for a generic vector $o$, $(o)_{j}$ denotes $j$–th element of the vector
$o$.

The EM algorithm is an iterative method to obtain (local) maximum likelihood
estimate of parameters of distribution functions, which depend on unobserved
(latent) variables. The EM algorithm alternates an expectation (E) step and a
maximization (M) step. In E–Step, we consider that conditional on the full
information $\mathcal{Y}_{T}$ and parameter at iteration $k$, $\theta^{[k]}$,
expectation of augmented log–likelihood of the data $\mathcal{Y}_{T}$ and
unobserved (latent) transition probability matrix $\mathsf{P}$. The E–Step
defines a objective function $\mathcal{L}$, namely,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathbb{E}\bigg{[}-\frac{T\tilde{n}}{2}\ln(2\pi)-\frac{1}{2}\sum_{t=1}^{T}\sum_{j=1}^{N}\ln(\Sigma_{j})1_{\\{s_{t}=j\\}}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\sum_{t=1}^{T}\sum_{j=1}^{N}\Big{(}B_{0}y_{t}-C_{j}\psi_{t}-B_{1}y_{t-1}\Big{)}^{\prime}\Sigma_{j}^{-1}\Big{(}B_{0}y_{t}-C_{j}\psi_{t}-B_{1}y_{t-1}\Big{)}$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{N}p_{0j}1_{\\{s_{1}=j\\}}+\sum_{t=2}^{T}\sum_{i=1}^{N}\sum_{j=1}^{N}\ln(p_{ij})1_{\\{s_{t-1}=i,s_{t}=j\\}}-\sum_{i=0}^{N}\mu_{i}\bigg{(}\sum_{j=1}^{N}p_{ij}-1\bigg{)}\bigg{|}\mathcal{Y}_{T};\theta^{[k]}\bigg{]}$

In M–Step, to obtain parameter estimate of next iteration $\theta^{[k+1]}$,
one maximizes the objective function with respect to the parameter $\theta$.
First, let us consider partial derivative from the objective function with
respect to the parameter $C_{j}$ for $j=1,\dots,N$. Let $c_{j}$ is a
vectorization of the matrix $C_{j}$, i.e., $c_{j}=\text{vec}(C_{j})$. Since
$C_{j}\psi_{t}=(\psi_{t}^{\prime}\otimes I_{2n+1})c_{j}$, we have that

$\frac{\partial\mathcal{L}}{\partial
c_{j}^{\prime}}=\sum_{t=1}^{T}\Big{(}B_{0}y_{t}-\big{(}\psi_{t}^{\prime}\otimes
I_{2n+1}\big{)}c_{j}-B_{1}y_{t-1}\Big{)}^{\prime}\big{(}\Sigma_{j}^{[k]}\big{)}^{-1}\big{(}\psi_{t}^{\prime}\otimes
I_{2n+1}\big{)}\big{(}z_{t|T}^{[k]}\big{)}_{j}.$ (67)

Consequently, an estimator at iteration $(k+1)$ of the parameter $c_{j}$ is
given by

$\displaystyle c_{j}^{[k+1]}$ $\displaystyle=$
$\displaystyle\bigg{(}\sum_{t=1}^{T}\big{(}\psi_{t}\otimes
I_{2n+1}\big{)}\big{(}\Sigma_{j}^{[k]}\big{)}^{-1}\big{(}\psi_{t}\otimes
I_{2n+1}\big{)}\big{(}z_{t|T}^{[k]}\big{)}_{j}\bigg{)}^{-1}$ (68)
$\displaystyle\times$ $\displaystyle\sum_{t=1}^{T}\big{(}\psi_{t}\otimes
I_{2n+1}\big{)}\big{(}\Sigma_{j}^{[k]}\big{)}^{-1}\big{(}B_{0}y_{t}-B_{1}y_{t-1}\big{)}\big{(}z_{t|T}^{[k]}\big{)}_{j}.$

As a result, an estimator at iteration $(k+1)$ of the parameter $C_{j}$ is
given by

$C_{j}^{[k+1]}=\big{(}B_{0}\bar{y}_{j}^{[k]}-B_{1}\bar{y}_{j,-1}^{[k]}\big{)}\big{(}\bar{\psi}_{j}^{[k]}\big{)}^{\prime}\big{(}\bar{\psi}_{j}^{[k]}\big{(}\bar{\psi}_{j}^{[k]}\big{)}^{\prime}\big{)}^{-1},$
(69)

where
$\bar{y}_{j}^{[k]}:=\Big{[}y_{1}\sqrt{\big{(}z_{1|T}^{[k]}\big{)}_{j}}:\dots:y_{T}\sqrt{\big{(}z_{T|T}^{[k]}\big{)}_{j}}\Big{]}$
is a $(\tilde{n}\times T)$ matrix,
$\bar{y}_{j,-1}^{[k]}:=\Big{[}y_{0}\sqrt{\big{(}z_{1|T}^{[k]}\big{)}_{j}}:\dots:y_{T-1}\sqrt{\big{(}z_{T|T}^{[k]}\big{)}_{j}}\Big{]}$
is a $(\tilde{n}\times T)$ matrix, and
$\bar{\psi}_{j}^{[k]}:=\Big{[}\psi_{1}\sqrt{\big{(}z_{1|T}^{[k]}\big{)}_{j}}:\dots:\psi_{T}\sqrt{\big{(}z_{T|T}^{[k]}\big{)}_{j}}\Big{]}$
is an $(l\times T)$ matrix. Second, a partial derivative from the objective
function with respect to the parameter $\Sigma_{j}$ for $j=1,\dots,N$ is given
by

$\displaystyle\frac{\partial\mathcal{L}}{\partial\Sigma_{j}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\Sigma_{j}^{-1}\sum_{t=1}^{T}\big{(}z_{t|T}^{[k]}\big{)}_{j}$
(70) $\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{t=1}^{T}\Sigma_{j}^{-1}\big{(}y_{t}-C_{j}^{[k]}\psi_{t}-Dy_{t-1}\big{)}\big{(}y_{t}-C_{j}^{[k]}\psi_{t}-Dy_{t-1}\big{)}^{\prime}\Sigma_{j}^{-1}\big{(}z_{t|T}^{[k]}\big{)}_{j}.$

Consequently, an estimator at iteration $(k+1)$ of the parameter $\Sigma_{j}$
is given by

$\Sigma_{j}^{[k+1]}=\frac{1}{\sum_{t=1}^{T}\big{(}z_{t|T}^{[k]}\big{)}_{j}}\sum_{t=1}^{T}\big{(}B_{0}y_{t}-C_{j}^{[k]}\psi_{t}-B_{1}y_{t-1}\big{)}\big{(}B_{0}y_{t}-C_{j}^{[k]}\psi_{t}-B_{1}y_{t-1}\big{)}^{\prime}\big{(}z_{t|T}^{[k]}\big{)}_{j}.$
(71)

Third, a partial derivative from the objective function with respect to the
parameter $p_{ij}$ for $i,j=1,\dots,N$ is given by

$\displaystyle\frac{\partial\mathcal{L}}{\partial
p_{ij}}=\frac{1}{p_{ij}}\sum_{t=2}^{T}\mathbb{P}\big{(}s_{t-1}=i,s_{t}=j|\mathcal{F}_{T};\theta^{[k]}\big{)}-\mu_{i}.$
(72)

Consequently, an estimator at iteration $(k+1)$ of the parameter $p_{ij}$ is
given by

$p_{ij}^{[k+1]}=\frac{1}{\sum_{t=2}^{T}\big{(}z_{t|T}^{[k]}\big{)}_{i}}\sum_{t=2}^{T}\mathbb{P}\big{(}s_{t-1}=i,s_{t}=j|\mathcal{F}_{T};\theta^{[k]}\big{)}$
(73)

where the joint probability
$\mathbb{P}\big{(}s_{t-1}=i,s_{t}=j|\mathcal{F}_{T};\theta^{[k]}\big{)}$ is
calculated by equation (65). Fourth, a partial derivative from the objective
function with respect to the parameter $p_{0j}$ for $j=1,\dots,N$ is given by

$\displaystyle\frac{\partial\mathcal{L}}{\partial
p_{0j}}=\frac{1}{p_{0j}}\mathbb{P}\big{(}s_{1}=j|\mathcal{F}_{T};\theta^{[k]}\big{)}-\mu_{0}.$
(74)

Consequently, an estimator at iteration $(k+1)$ of the parameter $p_{0j}$ is
given by

$p_{0j}^{[k+1]}=\big{(}z_{1|T}^{[k]}\big{)}_{j}.$ (75)

Alternating between these steps, the EM algorithm produces improved parameter
estimates at each step (in the sense that the value of the original
log–likelihood is continually increased) and it converges to the maximum
likelihood (ML) estimates of the parameters.

##  5 Conclusion

In this paper, we developed the Merton’s structural model for public companies
under an assumption that liabilities of the companies are observed. By
modeling the market values of equities, liabilities and assets of companies
using the Campbell Shiller’s 1988 approximation method, we obtain formulas for
risk–neutral equity and liability values and default probabilities of the
companies. Finally, we study ML estimators of suggested model’s parameters. It
is worth mentioning that following the ideas in Battulga24e one can develop
option pricing formulas with default risk and portfolio selection theory with
default risk for public companies.

##  6 Technical Annex

Here we give the Propositions, Corollary, and Lemmas, which are used in the
paper and their proofs.

######  Lemma 1.

Let $X\sim\mathcal{N}(\mu,\sigma^{2})$. Then for all $K>0$,

$\mathbb{E}\big{[}\big{(}e^{X}-K\big{)}^{+}\big{]}=\exp\bigg{\\{}\mu+\frac{\sigma^{2}}{2}\bigg{\\}}\Phi(d_{1})-K\Phi(d_{2})$

and

$\mathbb{E}\big{[}\big{(}K-e^{X}\big{)}^{+}\big{]}=K\Phi(-d_{2})-\exp\bigg{\\{}\mu+\frac{\sigma^{2}}{2}\bigg{\\}}\Phi(-d_{1}),$

where $d_{1}:=\big{(}\mu+\sigma^{2}-\ln(K)\big{)}/\sigma$,
$d_{2}:=d_{1}-\sigma$, and
$\Phi(x)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{-u^{2}/2}du$ is the
cumulative standard normal distribution function.

###### Proof.

See, e.g., Battulga24a and Battulga24e. ∎

Let us denote conditional on a generic $\sigma$-field $\mathcal{O}$, a joint
density functions of a generic random vector $X$ by $f(X|\mathcal{O})$ and
$\tilde{f}(X|\mathcal{O})$ under $\mathbb{P}$ and $\mathbb{\tilde{P}}$,
respectively, and let
$\mathcal{J}_{t}:=\sigma(\bar{C}_{t})\vee\sigma(\bar{\Gamma}_{t})\vee\sigma(\bar{s}_{t})\vee\mathcal{F}_{0}$.
Then, the following Lemmas hold.

######  Lemma 2.

Conditional on $\mathcal{F}_{t}$, a joint density of
$\big{(}\Pi_{\hat{s}},\Gamma_{\hat{s}},s,\mathsf{P}\big{)}$ is given by

$\tilde{f}\big{(}C_{\hat{s}},\Gamma_{\hat{s}},s,\mathsf{P}|\mathcal{F}_{t}\big{)}=\frac{\tilde{f}(\bar{y}_{t}|C_{\alpha},\Gamma_{\alpha},\bar{s}_{t},\mathcal{F}_{0})f(C_{\hat{s}},\Gamma_{\hat{s}}|\hat{s},\mathcal{F}_{0})f(s,\mathsf{P}|\mathcal{F}_{0})}{\displaystyle\sum_{\bar{s}_{t}}\bigg{(}\int_{C_{\alpha},\Gamma_{\alpha}}\tilde{f}(\bar{y}_{t}|C_{\alpha},\Gamma_{\alpha},\bar{s}_{t},\mathcal{F}_{0})f(C_{\alpha},\Gamma_{\alpha}|\alpha,\mathcal{F}_{0})dC_{\alpha}d\Gamma_{\alpha}\bigg{)}f(\bar{s}_{t}|\mathcal{F}_{0})}$
(76)

for $t=1,\dots,T$, where for $t=1,\dots,T$,

$\tilde{f}(\bar{y}_{t}|C_{\alpha},\Gamma_{\alpha},\bar{s}_{t},\mathcal{F}_{0})=\frac{1}{(2\pi)^{nt/2}|\Sigma_{11}|^{1/2}}\exp\Big{\\{}-\frac{1}{2}\big{(}\bar{y}_{t}-\tilde{\mu}_{1}\big{)}^{\prime}\tilde{\Sigma}_{11}^{-1}\big{(}\bar{y}_{t}-\tilde{\mu}_{1}\big{)}\Big{\\}}$
(77)

with
$\tilde{\mu}_{1}:=\big{(}\tilde{\mu}_{1|0}(\mathcal{H}_{0}),\dots,\tilde{\mu}_{t|0}(\mathcal{H}_{0})\big{)}^{\prime}$
and
$\tilde{\Sigma}_{11}:=\big{(}\tilde{\Sigma}_{i_{1},i_{2}|0}(\mathcal{H}_{0})\big{)}_{i_{1},i_{2}=1}^{t}$.
In particular, we have that

$\tilde{f}\big{(}C_{\hat{s}},\Gamma_{\hat{s}},s|\mathcal{F}_{t}\big{)}=\frac{\tilde{f}(\bar{y}_{t}|C_{\alpha},\Gamma_{\alpha},\bar{s}_{t},\mathcal{F}_{0})f(C_{\hat{s}},\Gamma_{\hat{s}}|\hat{s},\mathcal{F}_{0})f(s|\mathcal{F}_{0})}{\displaystyle\sum_{\bar{s}_{t}}\bigg{(}\int_{C_{\alpha},\Gamma_{\alpha}}\tilde{f}(\bar{y}_{t}|C_{\alpha},\Gamma_{\alpha},\bar{s}_{t},\mathcal{F}_{0})f(C_{\alpha},\Gamma_{\alpha}|\alpha,\mathcal{F}_{0})dC_{\alpha}d\Gamma_{\alpha}\bigg{)}f(\bar{s}_{t}|\mathcal{F}_{0})}$
(78)

for $t=1,\dots,T$.

###### Proof.

See, e.g., Battulga24a. ∎

######  Lemma 3.

Let

$f(\bar{y}_{t}|C_{\alpha},\Gamma_{\alpha},\bar{s}_{t},\mathcal{F}_{0})=\frac{1}{(2\pi)^{nt/2}|\Sigma_{11}|^{1/2}}\exp\Big{\\{}-\frac{1}{2}\big{(}\bar{y}_{t}-\mu_{1}\big{)}^{\prime}\Sigma_{11}^{-1}\big{(}\bar{y}_{t}-\mu_{1}\big{)}\Big{\\}},$
(79)

where
$\mu_{1}:=\big{(}\mu_{1|0}(\mathcal{H}_{0}),\dots,\mu_{t|0}(\mathcal{H}_{0})\big{)}^{\prime}$
and
$\Sigma_{11}:=\big{(}\Sigma_{i_{1},i_{2}|0}(\mathcal{H}_{0})\big{)}_{i_{1},i_{2}=1}^{t}$.
Then, we have that

$f\big{(}C_{\hat{s}},\Gamma_{\hat{s}},s|\mathcal{F}_{t}\big{)}=\frac{f(\bar{y}_{t}|C_{\alpha},\Gamma_{\alpha},\bar{s}_{t},\mathcal{F}_{0})f(C_{\hat{s}},\Gamma_{\hat{s}}|\hat{s},\mathcal{F}_{0})f(s|\mathcal{F}_{0})}{\displaystyle\sum_{\bar{s}_{t}}\bigg{(}\int_{C_{\alpha},\Gamma_{\alpha}}f(\bar{y}_{t}|C_{\alpha},\Gamma_{\alpha},\bar{s}_{t},\mathcal{F}_{0})f(C_{\alpha},\Gamma_{\alpha}|\alpha,\mathcal{F}_{0})dC_{\alpha}d\Gamma_{\alpha}\bigg{)}f(\bar{s}_{t}|\mathcal{F}_{0})}$
(80)

for $t=1,\dots,T$.

###### Proof.

By following Battulga24a, one can prove the Lemma 3. ∎

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