# Augmented Dynamic Gordon Growth Model

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

In this paper, we introduce a dynamic Gordon growth model, which is augmented
by a time–varying spot interest rate and the Gordon growth model for
dividends. Using the risk–neutral valuation method and locally risk–minimizing
strategy, we obtain pricing and hedging formulas for the dividend–paying
European call and put options, Margrabe exchange options, and equity–linked
life insurance products. Also, we provide ML estimator of the model.

Keywords: Options, equity–linked life insurance, dynamic Gordon growth model,
locally–risk minimizing strategy, 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 the
equity of a firm is equal to the present value of a sum of a dividend paid by
the firm and the price of the firm, which correspond to the next period. The
same idea can be used to value the liabilities of the firm. 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. An interesting review of some existing DDMs that include
deterministic and stochastic models can be found in dAmico20a.

Existing stochastic DDMs have one common disadvantage: If dividend and debt
payments have chances to take negative values, then the market values the
firm’s equity and liability 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 price and dividend get positive values. For private
companies, using the log private company valuation model, based on the dynamic
Gordon growth model, Battulga23c developed closed–form pricing and hedging
formulas for the European options and equity–linked life insurance products
and valuation formula. In this paper, to obtain pricing and hedging formulas
of some options and equity–linked life insurance products for public
companies, by modeling dividend payments and spot interest rates, we will
augment the dynamic Gordon growth model.

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 the 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, Battulga23a
developed option pricing formulas for some frequently used options by using
Markov–Switching Vector Autoregressive process.

In Section 2 of the paper, we develop stochastic DDM, which is known as the
dynamic Gordon growth model using the Campbell Shiller’s 1988 approximation
method. Also, we obtain closed–form pricing formulas of the European call and
put options and Margrabe exchange options in Section 3. Section 4 provides
calculations of net single premiums of equity–linked life insurance products.
Section 5 is dedicated to hedging formulas for the options and equity–linked
life insurance products. In Section 6, we study ML estimators our model’s
parameters. In Section 6, we conclude the study. Finally, in Section 7 we
provide Lemmas, which will be used to the paper.

##  2 Dynamic Gordon Growth Model

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
$\mathsf{P}:=\\{p_{ij}\\}_{i,j=1}^{N}$ is a random transition probability
matrix.

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 price
of a stock at time $t-1$ of the firm equals the sum of the market price of the
stock at time $t$ and dividend payment at time $t$ discounted at risk–adjusted
rates (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

$P_{i,t}=(1+k_{i,t})P_{i,t-1}-d_{i,t},~{}~{}~{}t=1,\dots,T,$ (1)

where $k_{i,t}:=k_{i}(s_{t})$ is the required rate of return on stock at
regime $s_{t}$, $P_{i,t}$ is the market price of the stock, and $d_{i,t}$ is
the dividend payment for investors, respectively, at time $t$ of $i$–th
company.

To keep notations simple, let $P_{t}:=(P_{1,t},\dots,P_{n,t})^{\prime}$ be an
$(n\times 1)$ vector of market price of stock,
$k_{t}:=(k_{1,t},\dots,k_{n,t})^{\prime}$ be an $(n\times 1)$ vector of
required rate of return on stock, $d_{t}:=(d_{1,t},\dots,d_{n,t})^{\prime}$ be
an $(n\times 1)$ vector of dividend payment, 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.

As mentioned above, if payments of dividends have a chance to take negative
values, then the stock prices of a company can take negative values with a
positive probability, which is an undesirable property for the stock price.
That is why, we follow the idea in Campbell88. As a result, the stock price of
the company takes positive values. Following the idea in Campbell88, one can
obtain the following approximation

$\exp\\{\tilde{k}_{t}\\}=(P_{t}+d_{t})\oslash
P_{t-1}\approx\exp\Big{\\{}\tilde{P}_{t}-\tilde{P}_{t-1}+\ln(g_{t})+G_{t}^{-1}(G_{t}-I_{n})\big{(}\tilde{d}_{t}-\tilde{P}_{t}-\mu_{t}\big{)}\Big{\\}},$
(2)

where $\oslash$ is a component–wise division of two vectors,
$\tilde{k}_{t}:=\ln(i_{n}+k_{t})$ is an ($n\times 1$) log required rate of
return process, $\tilde{P}_{t}:=\ln(P_{t})$ is a $(n\times 1)$ log stock price
process, $\tilde{d}_{t}:=\ln(d_{t})$ is a $(n\times 1)$ log dividend process,
$\mu_{t}:=\mathbb{E}\big{[}\tilde{d}_{t}-\tilde{P}_{t}\big{|}\mathcal{F}_{0}\big{]}$
is a $(n\times 1)$ mean log dividend–to–price process, respectively, at time
$t$ of the companies and $\mathcal{F}_{0}$ is an initial information, which
will be defined below, $g_{t}:=i_{n}+\exp\\{\mu_{t}\\}$ is a $(n\times 1)$
linearization parameter, and $G_{t}:=\text{diag}\\{g_{t}\\}$ is a $(n\times
n)$ diagonal matrix, whose diagonal elements are $g_{t}$. As a result, for the
log stock price process at time $t$, the following approximation holds

$\tilde{P}_{t}\approx
G_{t}(\tilde{P}_{t-1}-\tilde{d}_{t}+\tilde{k}_{t})+\tilde{d}_{t}-h_{t}.$ (3)

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. For the quality of the approximation, we refer to
Campbell97. Henceforth, the notation of approximation $(\approx)$ will be
replaced by the notation of equality $(=)$. To estimate the parameters of the
dynamic Gordon growth model and to price the Black–Scholes call and put
options, Margrabe exchange options, and equity–linked life insurance products,
we suppose that the log required rate of return process at time $t$ is
represented by the following equation

$\tilde{k}_{t}=C_{k,t}\psi_{k,t}+u_{t},$ (4)

where $\psi_{k,t}:=(\psi_{1,t},\dots,\psi_{l_{k},t})^{\prime}$ is an
$(l_{k}\times 1)$ vector, which consists of exogenous variables,
$C_{k,t}:=C_{k}(s_{t})$ is an $(n\times l_{k})$ random matrix at regime
$s_{t}$, $u_{t}$ is an $(n\times 1)$ white noise process with random
covariance matrix $\Sigma_{uu,t}:=\Sigma_{uu}(s_{t})$ at regime $s_{t}$. In
this case, equation (3) becomes

$\tilde{P}_{t}=G_{t}(\tilde{P}_{t-1}-\tilde{d}_{t}+C_{k,t}\psi_{t})+\tilde{d}_{t}-h_{t}+G_{t}u_{t}.$
(5)

We model the dividend process $d_{t}$ by the Gordon growth model. Therefore,
successive log dividends are modeled by the following relationship

$\tilde{d}_{t}=C_{d,t}\psi_{y,t}+\tilde{d}_{t-1}+v_{t},$ (6)

where $\psi_{y,t}:=(\psi_{1,t},\dots,\psi_{l_{y},t})^{\prime}$ is an
$(l_{y}\times 1)$ vector, which consists of exogenous variables,
$C_{d,t}:=C_{d}(s_{t})$ is an $(l_{y}\times 1)$ random matrix at regime
$s_{t}$, and $v_{t}$ is a white noise process.

Finally, we model log spot interest rate $\tilde{r}_{t}$. Let $r_{t}$ be a
spot interest rate for borrowing and lending over a period $(t-1,t]$. Then,
the log spot interest rate is defined by $\tilde{r}_{t}:=\ln(1+r_{t})$. By
using the Dickey–Fuller test, it can be confirmed that the quarterly log spot
interest rate is the unit–root process with drift, see data IRX of Yahoo
Finance. Consequently, the log spot rate is modeled by the following equation

$\tilde{r}_{t}=c_{r,t}^{\prime}\psi_{y,t}+\tilde{r}_{t-1}+w_{t},$ (7)

where $c_{r,t}:=c_{r}(s_{t})$ is an $(l_{y}\times 1)$ random vector at regime
$s_{t}$ and $w_{t}$ is a white noise process.

As a result, by combining equations (5)–(7), we arrive the following system

$\begin{cases}\tilde{P}_{t}=\nu_{P,t}-(G_{t}-I_{n})\tilde{d}_{t}+G_{t}\tilde{P}_{t-1}+G_{t}u_{t}\\\
y_{t}=\nu_{y,t}+y_{t-1}+\eta_{t}\end{cases}~{}~{}~{}\text{for}~{}t=1,\dots,T$
(8)

under the real probability measure $\mathbb{P}$, where
$y_{t}:=(\tilde{d}_{t}^{\prime},\tilde{r}_{t})^{\prime}$ is $([n+1]\times 1)$
process, which consists of the log dividend process and log spot rate process,
$\eta_{t}:=(v_{t}^{\prime},w_{t})$ is an ($[n+1]\times 1$) white noise process
with random covariance matrix $\Sigma_{\eta\eta,t}:=\Sigma_{\eta\eta}(s_{t})$,
$\nu_{P,t}:=G_{t}C_{k,t}\psi_{k,t}-h_{t}$ is an $(n\times 1)$ intercept
process of log stock price process $\tilde{P}_{t}$ and
$\nu_{y,t}:=C_{y,t}\psi_{y,t}$ is an $([n+1]\times 1)$ intercept process of
the process $y_{t}$ with $C_{y,t}:=[C_{d,t}^{\prime}:c_{r,t}]^{\prime}$. Let
us denote a dimension of system (8) by $\tilde{n}$, that is,
$\tilde{n}:=2n+1$.

The stochastic properties of system (11) is governed by the random vectors
$\\{u_{1},\dots,u_{T},\eta_{1},\dots,\eta_{T}\\}.$ We assume that for
$t=1,\dots,T$, the white noise process
$\xi_{t}:=(u_{t}^{\prime},\eta_{t}^{\prime})^{\prime}$ follows normal
distribution, namely,

$\xi_{t}\sim\mathcal{N}(0,\Sigma_{t})$ (9)

under the real probability measure $\mathbb{P}$, where $\xi_{t}$ is an
$(\tilde{n}\times 1)$ white noise process with random covariance matrix

$\Sigma_{t}:=\Sigma(s_{t})=\begin{bmatrix}\Sigma_{uu}(s_{t})&\Sigma_{u\eta}(s_{t})\\\
\Sigma_{\eta u}(s_{t})&\Sigma_{\eta\eta}(s_{t})\end{bmatrix}.$ (10)

##  3 Options Pricing

Black73 developed a closed–form formula for evaluating a European option. The
formula assumes that the underlying asset follows geometric Brownian motion,
but does not take dividends into account. Most stock options traded on the
options exchange pay dividends at least once before they expire. Therefore, it
is important to develop a formula for options on dividend–paying stocks from a
practical point of view. Merton Merton73 first time used continuous dividend
in the Black–Scholes framework and obtained a similar pricing formula with the
Black–Scholes formula. However, if the dividend process does not depend on the
stock level, the Black–Scholes framework with dividends will collapse. In this
paper, we develop an option pricing model where the dividend process is
discrete and correlated with the stock level.

Let $T$ be a time to maturity of the European call and put options,
$x_{t}:=(\tilde{P}_{t}^{\prime},y_{t})^{\prime}$ be $(\tilde{n}\times 1)$
process at time $t$ of endogenous variables,
$\psi_{t}:=(\psi_{k,t}^{\prime},\psi_{y,t}^{\prime})^{\prime}$ be a process of
exogenous variables, and
$c_{t}:=\big{(}\text{vec}(C_{k,t})^{\prime},\text{vec}(C_{y,t})^{\prime}\big{)}^{\prime}$
be random coefficient vector at regime $s_{t}$. We introduce stacked vectors
and matrices: $x:=(x_{1}^{\prime},\dots,x_{T}^{\prime})^{\prime}$,
$s:=(s_{1},\dots,s_{T})^{\prime}$, $C:=[c_{1}:\dots:c_{T}]$, and
$\Gamma:=[\Sigma_{1}:\dots:\Sigma_{T}]$. We suppose that the white noise
process $\\{\xi_{t}\\}_{t=1}^{T}$ is independent of the random coefficient
matrix $C$, random transition matrix $\mathsf{P}$, and regime–switching
process $\\{s_{t}\\}_{t=1}^{T}$ conditional on initial information
$\mathcal{F}_{0}:=\sigma(\mathcal{Y}_{0},\psi_{1},\dots,\psi_{T})$. 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, and for MS–VAR$(1)$ process,
$\mathcal{Y}_{0}:=x_{0}$ is an initial value vector of the process $x_{t}$ and
for Bayesian MS–VAR$(1)$ process,
$\mathcal{Y}_{0}:=(x_{-T_{*}}^{\prime},\dots,x_{0}^{\prime},\psi_{1-T_{*}}^{\prime},\dots,\psi_{0}^{\prime})^{\prime}$
is the data, covering the period $T_{*}$ before time 1 for the posterior
distribution. We further suppose that the transition probability matrix
$\mathsf{P}$ is independent of the random coefficient matrix $C$ given initial
information $\mathcal{F}_{0}$ and regime–switching process $s$.

To ease of notations, for a generic matrix $O=[O_{1}:\dots:O_{T}]$, we denote
its first $t$ and last $T-t$ block matrices by $\bar{O}_{t}$ and
$\bar{O}_{t}^{c}$, respectively, that is, $\bar{O}_{t}:=[O_{1}:\dots:O_{t}]$
and $\bar{O}_{t}^{c}:=[O_{t+1}:\dots:O_{T}]$. This notation also holds for
vectors. We define $\sigma$–fields: for $t=0,\dots,T$,
$\mathcal{F}_{t}:=\mathcal{F}_{0}\vee\sigma(\bar{x}_{t})$,
$\mathcal{G}_{t}:=\mathcal{F}_{t}\vee\sigma(C)\vee\sigma(\Gamma)\vee\sigma(\mathsf{P})\vee\sigma(\bar{s}_{t})$,
and
$\mathcal{H}_{t}:=\mathcal{F}_{t}\vee\sigma(C)\vee\sigma(\Gamma)\vee\sigma(\mathsf{P})\vee\sigma(s)$,
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
$\mathcal{F}_{t}\subset\mathcal{G}_{t}\subset\mathcal{H}_{t}$ 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,

$p_{s_{t}s_{t+1}}:=\mathbb{P}[s_{t+1}=s_{t+1}|s_{t}=s_{t},\mathsf{P},\mathcal{F}_{0}]=\mathbb{P}\big{[}s_{t+1}=s_{t+1}|\bar{s}_{t}=\bar{s}_{t},\mathsf{P},\mathcal{F}_{0}\big{]}$
(11)

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

$\xi=(\xi_{1}^{\prime},\dots,\xi_{T}^{\prime})^{\prime}~{}|~{}\mathcal{H}_{0}\sim\mathcal{N}(0,\bar{\Sigma}),$
(12)

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

###  3.1 Risk–Neutral Probability Measure

To price the European call and put options, Margrabe exchange options, and
equity–liked life insurance products, 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 $\tilde{r}_{t}$ 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
$k_{i,t}=(P_{i,t}+d_{i,t})/P_{i,t-1}-1$ 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

$\tilde{\mathbb{E}}\big{[}(P_{t}+d_{t})\oslash
P_{t-1}\big{|}\mathcal{H}_{t-1}\big{]}=\exp\big{\\{}\tilde{r}_{t}i_{n}\big{\\}}$
(13)

for $t=1,\dots,T$, where $\tilde{\mathbb{E}}$ denotes an expectation under the
risk–neutral probability measure $\mathbb{\tilde{P}}$. According to equation
(2), condition (13) is equivalent to the following condition

$\tilde{\mathbb{E}}\big{[}\exp\big{\\{}u_{t}-\big{(}\tilde{r}_{t}i_{n}-C_{k,t}\psi_{k,t}\big{)}\big{\\}}\big{|}\mathcal{H}_{t-1}\big{]}=i_{n}.$
(14)

It should be noted that condition (14) corresponds only to the white noise
random process $u_{t}$. Thus, we need to impose a condition on the white noise
random process $\eta_{t}$ under the risk–neutral probability measure. This
condition is fulfilled by
$\tilde{\mathbb{E}}[\exp\\{\eta_{t}\\}|\mathcal{H}_{t-1}]=\tilde{\theta}_{t}$
for $\mathcal{H}_{t-1}$ measurable any random vector $\tilde{\theta}_{t}$.
Because for any admissible choices of $\tilde{\theta}_{t}$, condition (14)
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 and life insurance products, 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 Battulga23a,
the optimal kernel process $\theta_{t}$ is obtained by

$\theta_{t}=\Theta_{t}\bigg{(}\tilde{r}_{t}i_{n}-C_{k,t}\psi_{k,t}-\frac{1}{2}\mathcal{D}[\Sigma_{uu,t}]\bigg{)},$
(15)

where $\Theta_{t}=\big{[}G_{t}:(\Sigma_{\eta
u,t}\Sigma_{uu,t}^{-1})^{\prime}\big{]}^{\prime}$ and for a generic square
matrix $O$, $\mathcal{D}[O]$ denotes a vector, consisting of diagonal elements
of the matrix $O$. Consequently, system (8) can be written by

$\begin{cases}\tilde{P}_{t}=\tilde{\nu}_{P,t}-(G_{t}-I_{n})\tilde{d}_{t}+G_{t}\tilde{P}_{t-1}+G_{t}i_{n}j_{r}^{\prime}y_{t-1}+G_{t}\tilde{u}_{t}\\\
y_{t}=\tilde{\nu}_{y,t}+\big{(}I_{n+1}+\Sigma_{\eta
u,t}\Sigma_{uu,t}^{-1}i_{n}j_{r}^{\prime}\big{)}y_{t-1}+\tilde{\eta}_{t},\end{cases}~{}~{}~{}\text{for}~{}t=1,\dots,T$
(16)

under the risk–neutral probability measure $\tilde{\mathbb{P}}$, where
$j_{r}:=(0,1)^{\prime}$ is an $([n+1]\times 1)$ vector, which is used to
extract the log spot rate process $\tilde{r}_{t}$ from the process $y_{t}$,
i.e., $\tilde{r}_{t}=j_{r}^{\prime}y_{t}$,
$\tilde{\nu}_{V,t}:=-\frac{1}{2}G_{t}\mathcal{D}[\Sigma_{uu}]-h_{t}$ is an
$(n\times 1)$ intercept process of the log stock price process $\tilde{P}_{t}$
and
$\tilde{\nu}_{y,t}:=C_{y,t}^{\prime}\psi_{y,t}-\Sigma_{vu,t}\Sigma_{uu,t}^{-1}\big{(}C_{k,t}\psi_{k,t}+\frac{1}{2}\mathcal{D}[\Sigma_{uu,t}]\big{)}$
is an $([n+1]\times 1)$ 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 $\tilde{\xi}_{t}:=(\tilde{u}_{t}^{\prime},\tilde{\eta}_{t})^{\prime}$
equals the joint distribution of the random vector
$\xi=(\xi_{1}^{\prime},\dots,\xi_{T}^{\prime})^{\prime}$, that is,

$\tilde{\xi}\sim\mathcal{N}\big{(}0,\bar{\Sigma}\big{)}$ (17)

under the risk–neutral probability measure $\mathbb{\tilde{P}}$, see
Battulga23a.

System (16) can be written in VAR(1) form, namely

$Q_{0,t}x_{t}=\tilde{\nu}_{t}+Q_{1,t}x_{t-1}+\mathsf{G}_{t}\tilde{\xi}_{t}$
(18)

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

$Q_{0,t}:=\begin{bmatrix}I_{n}&H_{t}\\\
0&I_{n+1}\end{bmatrix},~{}~{}~{}Q_{1,t}:=\begin{bmatrix}G_{t}&G_{t}i_{n}j_{r}^{\prime}\\\
0&E_{t}\end{bmatrix},~{}~{}~{}\text{and}~{}~{}~{}\mathsf{G}_{t}=\begin{bmatrix}G_{t}&0\\\
0&I_{n+1}\end{bmatrix}$ (19)

are $(\tilde{n}\times\tilde{n})$ coefficient matrices,
$H_{t}:=[G_{t}-I_{n}:0]$ is an $(n\times[n+1])$ matrix,
$E_{t}:=I_{n+1}+\Sigma_{\eta u,t}\Sigma_{uu,t}^{-1}i_{n}j_{r}^{\prime}$ is an
$([n+1]\times[n+1])$ matrix,. By repeating equation (18), one gets that for
$i=t+1,\dots,T$,

$x_{i}=\Pi_{t,i}x_{t}+\sum_{\beta=t+1}^{i}\Pi_{\beta,i}\tilde{\nu}_{\beta}+\sum_{\beta=t+1}^{i}\Pi_{\beta,i}\mathsf{G}_{\beta}\tilde{\xi}_{\beta},$
(20)

where the coefficient matrices are

$\Pi_{\beta,i}:=\prod_{\alpha=\beta+1}^{i}Q_{0,\alpha}^{-1}Q_{1,\alpha}=\begin{bmatrix}\displaystyle\prod_{\alpha=\beta+1}^{i}G_{\alpha}&\displaystyle\sum_{\alpha=\beta+1}^{i}\Bigg{(}\prod_{j_{1}=\alpha+1}^{i}G_{j_{1}}\Bigg{)}\Psi_{\alpha}\Bigg{(}\prod_{j_{2}=\beta+1}^{\alpha-1}E_{j_{2}}\Bigg{)}\\\
\displaystyle
0&\displaystyle\prod_{\alpha=\beta+1}^{i}E_{\alpha}\end{bmatrix}$ (21)

for $\beta=0,\dots,i-1$,
$\Psi_{\alpha}:=G_{\alpha}i_{n}j_{r}^{\prime}-H_{\alpha}E_{\alpha}$, and

$\Pi_{i,i}:=Q_{0,i}^{-1}=\begin{bmatrix}I_{n}&-H_{i}\\\
0&I_{n+1}\end{bmatrix}.$ (22)

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

$\tilde{\mu}_{i|t}:=\mathbb{\tilde{E}}\big{[}x_{i}\big{|}\mathcal{H}_{t}\big{]}=\Pi_{t,i}x_{t}+\sum_{\beta=t+1}^{i}\Pi_{\beta,i}\tilde{\nu}_{\beta}$
(23)

and

$\Sigma_{i_{1},i_{2}|t}:=\widetilde{\text{Cov}}\big{[}x_{i_{1}},x_{i_{2}}\big{|}\mathcal{H}_{t}\big{]}=\sum_{\beta=t+1}^{i_{1}\wedge
i_{2}}\Pi_{\beta,i_{1}}\mathsf{G}_{\beta}\Sigma_{\beta}\mathsf{G}_{\beta}\Pi_{\beta,i_{2}}^{\prime},$
(24)

where $i_{1}\wedge i_{2}$ is a minimum of $i_{1}$ and $i_{2}$. It should be
noted that the expectation $\tilde{\mu}_{i|t}$ and covariance matrix
$\Sigma_{i_{1},i_{2}|t}$ are depend on the information $\mathcal{H}_{t}$.
Consequently, due to equation (20), conditional on the information
$\mathcal{H}_{t}$, a joint distribution of the random vector $\bar{x}_{t}^{c}$
is

$\bar{x}_{t}^{c}~{}|~{}\mathcal{H}_{t}\sim\mathcal{N}\big{(}\tilde{\mu}_{t}^{c},\Sigma_{t}^{c}\big{)},~{}~{}~{}t=0,\dots,T-1$
(25)

under the risk–neutral probability measure $\tilde{\mathbb{P}}$, where
$\tilde{\mu}_{t}^{c}:=\big{(}\tilde{\mu}_{t+1|t},\dots,\tilde{\mu}_{T|t}\big{)}^{\prime}$
is a conditional expectation and
$\Sigma_{t}^{c}:=\big{(}\Sigma_{i_{1},i_{2}|t}\big{)}_{i_{1},i_{2}=t+1}^{T}$
is a conditional covariance matrix of the random vector $\bar{x}_{t}^{c}$ and
are calculated by equations (23) and (24), 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, the Margrabe exchange options, and the equity–linked life
insurance products, we will apply it. To define the forward probability
measure, we need zero–coupon bond. It is the well–known fact that conditional
on $\mathcal{F}_{t}$, price at time $t$ of zero–coupon bond paying face value
1 at time $u$ is
$B_{t,u}(\mathcal{H}_{t}):=\frac{1}{D_{t}}\mathbb{\tilde{E}}\big{[}D_{u}\big{|}\mathcal{H}_{t}\big{]}$.
The $(t,u)$–forward probability measure is defined by

$\mathbb{\hat{P}}_{t,u}\big{[}A\big{|}\mathcal{H}_{t}\big{]}:=\frac{1}{D_{t}B_{t,u}(\mathcal{H}_{t})}\int_{A}D_{u}\mathbb{\tilde{P}}\big{[}\omega|\mathcal{H}_{t}\big{]}~{}~{}~{}\text{for
all}~{}A\in\mathcal{H}_{T}.$ (26)

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

$\sum_{\beta=t+1}^{u}\tilde{r}_{\beta}=\tilde{r}_{t+1}+j_{r}^{\prime}J_{y}\Bigg{[}\sum_{\beta=t+1}^{u-1}J_{\beta|t}\Bigg{]}\bar{x}_{t}^{c}=\tilde{r}_{t+1}+\gamma_{t,u}^{\prime}\bar{x}_{t}^{c}$
(27)

where $J_{y}:=[0:I_{n+1}]$ is $([n+1]\times\tilde{n})$ matrix, whose second
block matrix equals $I_{n+1}$ and other block is zero and it can be used to
extract the random process $y_{s}$ 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
$\gamma_{t,u}^{\prime}:=j_{r}^{\prime}J_{y}\sum_{\beta=t+1}^{u-1}J_{\beta|t}$.
Therefore, two times of negative exponent of the price at time $t$ of the
zero–coupon $B_{t,u}$ is represented by

$\displaystyle
2\sum_{s=t+1}^{u}\tilde{r}_{s}+\big{(}\bar{x}_{t}^{c}-\tilde{\mu}_{t}^{c}\big{)}^{\prime}\big{(}\Sigma_{t}^{c}\big{)}^{-1}\big{(}\bar{x}_{t}^{c}-\tilde{\mu}_{t}^{c}\big{)}$
$\displaystyle=\Big{(}\bar{x}_{t}^{c}-\tilde{\mu}_{t}^{c}+\Sigma_{t}^{c}\gamma_{t,u}\Big{)}^{\prime}\big{(}\Sigma_{t}^{c}\big{)}^{-1}\Big{(}\bar{x}_{t}^{c}-\tilde{\mu}_{t}^{c}+\Sigma_{t}^{c}\gamma_{t,u}\Big{)}$
$\displaystyle+2\big{(}\tilde{r}_{t+1}+\gamma_{t,u}^{\prime}\tilde{\mu}_{t}^{c}\big{)}-\gamma_{t,u}^{\prime}\Sigma_{t}^{c}\gamma_{t,u}.$

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

$B_{t,u}(\mathcal{H}_{t})=\exp\bigg{\\{}-\tilde{r}_{t+1}-\gamma_{t,u}^{\prime}\tilde{\mu}_{t}^{c}+\frac{1}{2}\gamma_{t,u}^{\prime}\Sigma_{t}^{c}\gamma_{t,u}\bigg{\\}}.$
(29)

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

$\bar{x}_{t}^{c}~{}|~{}\mathcal{H}_{t}\sim\mathcal{N}\big{(}\hat{\mu}_{t,u}^{c},\Sigma_{t}^{c}\big{)},~{}~{}~{}t=0,\dots,T-1$
(30)

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

$J_{s|t}\Sigma_{t}^{c}\Bigg{(}\sum_{\beta=t+1}^{T-1}J_{\beta|t}\Bigg{)}=\sum_{\beta=t+1}^{T-1}\Sigma_{s,\beta|t},$
(31)

where $\Sigma_{s,\beta|t}$ is calculated by equation (24). Therefore, for
$s=t+1,\dots,T$, $(s-t)$–th block vector of the conditional expectation
$\hat{\mu}_{t,u}^{c}$ is given by

$\hat{\mu}_{s|t,u}:=J_{s|t}\hat{\mu}_{t,u}^{c}=\tilde{\mu}_{s|t}-\sum_{\beta=t+1}^{u-1}\big{(}\Sigma_{s,\beta|t}\big{)}_{\tilde{n}},$
(32)

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

$B_{t,u}=\exp\bigg{\\{}-\tilde{r}_{t+1}-\sum_{\beta=t+1}^{u-1}\big{(}\tilde{\mu}_{\beta|t}\big{)}_{\tilde{n}}+\frac{1}{2}\sum_{\alpha=t+1}^{u-1}\sum_{\beta=t+1}^{u-1}\big{(}\Sigma_{\alpha,\beta|t}\big{)}_{\tilde{n},\tilde{n}}\bigg{\\}}.$
(33)

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, Margrabe exchange options, and
equity–linked life insurance products, we need a distribution of the log stock
price process at time $k$ for $k=t+1,\dots,T$ under the $(t,u)$–forward
probability measure $\hat{\mathbb{P}}_{t,u}$. For this reason, it follows from
equation (30) that the distribution of the log stock price process at time $T$
is given by

$\tilde{P}_{k}~{}|~{}\mathcal{H}_{t}\sim\mathcal{N}\Big{(}\hat{\mu}_{k|t,u}^{\tilde{P}},\Sigma_{k|t}^{\tilde{P}}\Big{)}$
(34)

for $k=t+1,\dots,T$ under the $(t,u)$–forward probability measure
$\mathbb{\hat{P}}_{t,u}$, where
$\hat{\mu}_{k|t,u}^{\tilde{P}}:=J_{P}\hat{\mu}_{k|t,u}$ is a conditional
expectation, which is calculated from equation (32) and
$\Sigma_{k|t}^{\tilde{P}}:=J_{P}\Sigma_{k,k|t}J_{P}^{\prime}$ is a conditional
covariance matrix, which is calculated from equation (24) of the log stock
price at time $k$ given the information $\mathcal{H}_{t}$ and
$J_{P}:=[I_{n}:0]$ is a ($n\times\tilde{n}$) matrix, which is used to extract
the log stock price process $\tilde{P}_{t}$ from the process $x_{t}$.

Therefore, according to equation (34) and Lemma 1, see Technical Annex, for
$i=1,\dots,n$, conditional on the information $\mathcal{H}_{t}$, price vectors
at time $t$ of the Black–Sholes call and put options with strike price vector
$K$ and maturity $T$ is given by

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

and

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

where $\mathbb{\hat{E}}_{t,u}$ is an expectation under the ($t,u$)-forward
probability measure $\mathbb{\hat{P}}_{t,u}$,
$d_{T|t}^{1}(\mathcal{F}):=\Big{(}\hat{\mu}_{T|t,T}^{\tilde{P}}(\mathcal{F})+\mathcal{D}\big{[}\Sigma_{T|t}^{\tilde{P}}(\mathcal{F})\big{]}-\ln(K)\Big{)}\oslash\sqrt{\mathcal{D}\big{[}\Sigma_{T|t}^{\tilde{P}}(\mathcal{F})\big{]}}$
is an $(n\times 1)$ vector and same dimension holds for the vector
$d_{T|t}^{2}(\mathcal{F}):=d_{T|t}^{1}(\mathcal{F})-\sqrt{\mathcal{D}\big{[}\Sigma_{T|t}^{\tilde{P}}(\mathcal{F})\big{]}}$.
Consequently, by the tower property of conditional expectation, Lemmas 2 and
3, and equations (3.2) and (3.2), a price vector at time $t$ of the
Black–Sholes call and put options with strike price vector $K$ and maturity
$T$ is obtained by

$\displaystyle
C_{T|t}=\sum_{s}\int_{C,\Gamma,\mathsf{P}}C_{T|t}(\mathcal{H}_{t})f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P}$ (37)

and

$\displaystyle
P_{T|t}=\sum_{s}\int_{C,\Gamma,\mathsf{P}}P_{T|t}(\mathcal{H}_{t})f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P},$ (38)

see Battulga23a.

Now, we consider a Margrabe exchange option, which gives the owner a right to
exchange $w_{j}$ units of stock of $j$–th firm into $w_{i}$ units of stock of
$i$–th firm at time $T$. For given information $\mathcal{H}_{t}$, the price at
time $t$ of the Margrabe exchange option is given by the following equation

$\displaystyle O_{T|t}^{ij}(\mathcal{H}_{t})$ $\displaystyle:=$
$\displaystyle\mathbb{\tilde{E}}\bigg{[}\frac{D_{T}}{D_{t}}\Big{(}w_{i}P_{i,T}-w_{j}P_{j,T}\Big{)}^{+}\bigg{|}\mathcal{H}_{t}\bigg{]}$
$\displaystyle=$ $\displaystyle
B_{t,T}(\mathcal{H}_{t})\Big{(}w_{i}\mathbb{\hat{E}}_{t,T}\big{[}P_{i,T}1_{A}\big{|}\mathcal{H}_{t}\big{]}-w_{j}\mathbb{\hat{E}}_{t,T}\big{[}P_{j,T}1_{A}\big{|}\mathcal{H}_{t}\big{]}\Big{)}$

for $i,j=1,\dots,n$ $(i\neq j)$, where
$A:=\big{\\{}\tilde{P}_{i,T}-\tilde{P}_{j,T}\geq\ln(w_{j}/w_{i})\big{\\}}$ is
an event that represents the Margrabe exchange option expires in the money.
The first expectation of the above equation can be written by

$\mathbb{\hat{E}}_{t,T}\big{[}P_{i,T}1_{A}\big{|}\mathcal{H}_{t}\big{]}=\mathbb{\hat{E}}_{t,T}\Big{[}\exp\big{\\{}e_{i}^{\prime}\tilde{P}_{T}\big{\\}}1_{\\{(e_{i}-e_{j})^{\prime}\tilde{P}_{T}\geq\ln(w_{j}/w_{i})\\}}\Big{|}\mathcal{H}_{t}\Big{]},$
(40)

where $e_{i}$ is an ($n\times 1$) unit vector, whose $i$–th component equals
one and others equal zero. Consequently, according to Lemma 4 in Technical
Annex, we get that

$\displaystyle\mathbb{\hat{E}}_{t,T}\big{[}P_{i,T}1_{A}\big{|}\mathcal{H}_{t}\big{]}=\exp\bigg{\\{}e_{i}^{\prime}\hat{\mu}_{T|t,T}^{\tilde{P}}+\frac{1}{2}e_{i}^{\prime}\Sigma_{T|t}^{\tilde{P}}e_{i}\bigg{\\}}\Phi\Big{(}d_{T|t}^{ij,1}\Big{)}.$
(41)

where

$d_{T|t}^{ij,1}:=\frac{(e_{i}-e_{j})^{\prime}\hat{\mu}_{T|t,T}^{\tilde{P}}+(e_{i}-e_{j})^{\prime}\Sigma_{T|t}^{\tilde{P}}e_{i}-\ln(w_{j}/w_{i})}{\sqrt{(e_{i}-e_{j})^{\prime}\Sigma_{T|t}^{\tilde{P}}(e_{i}-e_{j})}}$
(42)

A similar formula holds for the second expectation of equation (3.2).
Consequently, the price at time $t$ of the Margrabe exchange option is given
by the following equation

$\displaystyle O_{T|t}^{ij}(\mathcal{H}_{t})$ $\displaystyle=$ $\displaystyle
B_{t,T}(\mathcal{H}_{t})\bigg{[}w_{i}\exp\bigg{\\{}e_{i}^{\prime}\hat{\mu}_{T|t}^{\tilde{P}}+\frac{1}{2}e_{i}^{\prime}\Sigma_{T|t}^{\tilde{P}}e_{i}\bigg{\\}}\Phi\Big{(}d_{T|t}^{ij,1}\Big{)}$
(43) $\displaystyle-$ $\displaystyle
w_{j}\exp\bigg{\\{}e_{j}^{\prime}\hat{\mu}_{T|t}^{\tilde{P}}+\frac{1}{2}e_{j}^{\prime}\Sigma_{T|t}^{\tilde{P}}e_{j}\bigg{\\}}\Phi\Big{(}d_{T|t}^{ij,2}\Big{)}\bigg{]}$

for $i,j=1,\dots,n$ $(i\neq j)$, where
$d_{T|t}^{ij,2}=d_{T|t}^{ij,1}-\sqrt{(e_{i}-e_{j})^{\prime}\Sigma_{T|t}^{\tilde{P}}(e_{i}-e_{j})}$.
As a result, the price at time $t$ of Margrabe exchange options, which gives
the owner a right to exchange $w_{j}$ units of stock of $j$–th firm into
$w_{i}$ units of stock of $i$–th firm at time $T$ is given by the following
equation

$\displaystyle
O_{T|t}^{ij}=\sum_{s}\int_{C,\Gamma,\mathsf{P}}O_{T|t}^{ij}(\mathcal{H}_{t})f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P}.$ (44)

##  4 Life Insurance Products

Now we consider the pricing of some equity–linked life insurance products
using the risk–neutral measure. Here we will price segregated funds contract
with guarantees, see Hardy01 and unit–linked life insurances with guarantees,
see Aase94 and Moller98. We suppose that the stocks represent some funds and
an insured receives dividends from the funds. Let $T_{x}$ be $x$ aged
insured’s future lifetime random variable,
$\mathcal{T}_{t}^{x}=\sigma(1_{\\{T_{x}>s\\}}:s\in[0,t])$ be $\sigma$–field,
which is generated by a death indicator process $1_{\\{T_{x}\leq t\\}}$,
$F_{t}^{*}$ be an $(n\times 1)$ vector of units of the funds, and $G_{t}^{*}$
be a $(n\times 1)$ vector of amounts of the guarantees, respectively, at time
$t$. We assume that the $\sigma$–fields $\mathcal{H}_{T}$ and
$\mathcal{T}_{T}^{x}$ are independent, and operational expenses, which are
deducted from the funds and withdrawals are omitted from the life insurance
products. A common life insurance product in practice is endowment insurance,
and combinations of term life insurance and pure endowment insurance lead to
interesting endowment insurances, see Aase94. Thus, it is sufficient to
consider only the term life insurance and the pure endowment insurance.

A $T$–year pure endowment insurance provides payment of a sum insured at the
end of the $T$ years only if the insured is alive at the end of $T$ years from
the time of policy issue. For the pure endowment insurance, we assume that the
sum insured is forming $f(P_{T})$ for some Borel function
$f:\mathbb{R}_{+}^{n}\to\mathbb{R}_{+}^{n}$, where
$\mathbb{R}_{+}^{n}:=\\{x\in\mathbb{R}^{n}|x>0\\}$ is the set of $(n\times 1)$
positive real vectors. In this case, the sum insured depends on the random
stock price at time $T$, and the form of the function $f$ depends on an
insurance contract. Choices of $f$ give us different types of life insurance
products. For example, for $x,K\in\mathbb{R}_{+}^{n}$, $f(x)=i_{n}$, $f(x)=x$,
$f(x)=\max\\{x,K\\}=[x-K]^{+}+K$, and $f(x)=[K-x]^{+}$ correspond to simple
life insurance, pure unit–linked, unit–linked with guarantee, and segregated
fund contract with guarantee, respectively, see Aase94, Bowers97, and Hardy01.
As a result, a discounted contingent claim of the $T$–year pure endowment
insurance can be represented by the following equation

$\overline{H}_{T}:=D_{T}f(P_{T})1_{\\{T_{x}>T\\}}.$ (45)

To price the contingent claim we define $\sigma$–fields: for each
$t=1,\dots,T$, $\mathcal{H}_{t}^{x}:=\mathcal{H}_{t}\vee\mathcal{T}_{t}^{x}$
is a minimal $\sigma$–field that contains the $\sigma$–fields
$\mathcal{H}_{t}$ and $\mathcal{T}_{t}^{x}$. Since the $\sigma$–fields
$\mathcal{H}_{T}$ and $\mathcal{T}_{T}^{x}$ are independent, one can obtain
that value at time $t$ of a contingent claim $f(P_{T})1_{\\{T_{x}>T\\}}$ is
given by

$V_{t}(\mathcal{H}_{t})=\frac{1}{D_{t}}\mathbb{\tilde{E}}[\overline{H}_{T}|\mathcal{H}_{t}^{x}]=\frac{1}{D_{t}}\mathbb{\tilde{E}}[D_{T}f(P_{T})|\mathcal{H}_{t}]{}_{T-t}p_{x+t},$
(46)

where ${}_{t}p_{x}:=\mathbb{P}[T_{x}>t]$ represents the probability that
$x$–aged insured will attain age $x+t$.

A $T$–year term life insurance is an insurance that provides payment of a sum
insured only if death occurs in $T$ years. In contrast to pure endowment
insurance, the term life insurance’s sum insured depends on time $t$, that is,
its sum insured form is $f(P_{t})$ because random death occurs at any time in
$T$ years. Therefore, a discounted contingent claim of the $T$–term life
insurance is given by

$\overline{H}_{T}:=D_{K_{x}+1}f(P_{K_{x}+1})1_{\\{K_{x}+1\leq
T\\}}=\sum_{k=0}^{T-1}D_{k+1}f(P_{t+k})1_{\\{K_{x}=k\\}},$ (47)

where $K_{x}:=[T_{x}]$ is the curtate future lifetime random variable of
life–aged–$x$. For the contingent claim of the term life insurance, providing
a benefit at the end of the year of death, it follows from the fact that
$\mathcal{H}_{T}$ and $\mathcal{T}_{T}^{x}$ are independent that a value
process at time $t$ of the term insurance is

$V_{t}(\mathcal{H}_{t})=\frac{1}{D_{t}}\mathbb{\tilde{E}}[\overline{H}_{T}|\mathcal{H}_{t}^{x}]=\sum_{k=t}^{T-1}\frac{1}{D_{t}}\mathbb{\tilde{E}}[D_{k+1}f(P_{k+1})|\mathcal{H}_{t}]{}_{k-t}p_{x+t}q_{x+k}.$
(48)

where ${}_{t}q_{x}:=\mathbb{P}[T_{x}\leq t]$ represents the probability that
$x$–aged insured will die within $t$ years.

For the $T$–year term life insurance and $T$–year pure endowment insurance
both of which correspond to the segregated fund contract, observe that the sum
insured forms are $f(P_{k})=F_{k}^{*}\odot\big{[}(G_{k}^{*}\oslash
F_{k}^{*})-P_{k}\big{]}^{+}$ for $k=1,\dots,T$. On the other hand, the sum
insured forms of the unit–linked life insurance are
$f(P_{k})=F_{k}^{*}\odot\big{[}P_{k}-(G_{k}^{*}\oslash
F_{k}^{*})\big{]}^{+}+G_{k}$ for $k=1,\dots,T$. Therefore, from the structure
of the sum insureds of the segregated funds and the unit–linked life
insurances, one can conclude that to price the life insurance products it is
sufficient to consider European call and put options with strike price
$(G_{k}^{*}\oslash F_{k}^{*})$ and maturity $k$ for $k=t+1,\dots,T$.

Similarly to equations (3.2) and (3.2), one can obtain that for
$k=t+1,\dots,T$,

$\displaystyle
C_{k|t}(\mathcal{H}_{t})=\mathbb{\tilde{E}}_{t,k}\bigg{[}\frac{D_{k}}{D_{t}}\Big{(}P_{k}-(G_{k}^{*}\oslash
F_{k}^{*})\Big{)}^{+}\bigg{|}\mathcal{H}_{t}\bigg{]}$ (49)
$\displaystyle=B_{t,k}(\mathcal{H}_{t})\bigg{(}\exp\bigg{\\{}\hat{\mu}_{k|t,k}^{\tilde{P}}+\frac{1}{2}\mathcal{D}\big{[}\Sigma_{k|t}^{\tilde{P}}\big{]}\bigg{\\}}\odot\Phi\big{(}d_{k|t}^{1}\big{)}-(G_{k}^{*}\oslash
F_{k}^{*})\odot\Phi\big{(}d_{k|t}^{2}\big{)}\bigg{)}$

and

$\displaystyle
P_{k|t}(\mathcal{H}_{t})=\mathbb{\tilde{E}}_{t,k}\bigg{[}\frac{D_{k}}{D_{t}}\Big{(}(G_{k}^{*}\oslash
F_{k}^{*})-P_{k}\Big{)}^{+}\bigg{|}\mathcal{H}_{t}\bigg{]}$ (50)
$\displaystyle=B_{t,k}(\mathcal{H}_{t})\bigg{(}(G_{k}^{*}\oslash
F_{k}^{*})\odot\Phi\big{(}-d_{k|t}^{2}\big{)}-\exp\bigg{\\{}\hat{\mu}_{k|t,k}^{\tilde{P}}+\frac{1}{2}\mathcal{D}\big{[}\Sigma_{k|t}^{\tilde{P}}\big{]}\bigg{\\}}\odot\Phi\big{(}-d_{k|t}^{1}\big{)}\bigg{)},$

where
$d_{k|t}^{1}:=\Big{(}\hat{\mu}_{k|t,k}^{\tilde{P}}+\mathcal{D}\big{[}\Sigma_{k|t}^{\tilde{P}}\big{]}-\ln(G_{k}^{*}\oslash
F_{k}^{*})\Big{)}\oslash\sqrt{\mathcal{D}\big{[}\Sigma_{k|t}^{\tilde{P}}\big{]}}$
and
$d_{k|t}^{2}:=d_{k|t}^{1}-\sqrt{\mathcal{D}\big{[}\Sigma_{k|t}^{\tilde{P}}\big{]}}$.

Consequently, in analogous to the call and put options, from equations (49)
and (50) net single premiums of the $T$–year life insurance products without
withdrawal and operational expenses, providing a benefit at the end of the
year of death (term life insurance) or the end of the year $T$ (pure endowment
insurance), are given by

  * 1.

for the $T$–year guaranteed term life insurance, corresponding to segregated
fund contract, it holds

$\displaystyle{S}{}_{x+t:{\mbox{$\scriptstyle\overline{T-t\,}$\kern-0.1pt\rule[-0.35pt]{0.3pt}{5.67776pt}}}}^{\makebox[16.41727pt]{\hfill$\scriptstyle
1$\hfill}}$ $\displaystyle=$
$\displaystyle\sum_{s}\int_{C,\Gamma,\mathsf{P}}\bigg{\\{}\sum_{k=t}^{T-1}F_{k+1}^{*}\odot
P_{k+1|t}(\mathcal{H}_{t})$ (51) $\displaystyle\times$
$\displaystyle{}_{k-t}p_{x+t}q_{x+k}\bigg{\\}}f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P};$

  * 2.

for the $T$–year guaranteed pure endowment insurance, corresponding to
segregated fund contract, it holds

$\displaystyle{S}{}_{x+t:{\mbox{$\scriptstyle\overline{T-t\,}$\kern-0.1pt\rule[-0.35pt]{0.3pt}{5.67776pt}}}}^{\hskip
19.19505pt\makebox[14.3683pt]{\hfill$\scriptstyle 1$\hfill}}$ $\displaystyle=$
$\displaystyle\sum_{s}\int_{C,\Gamma,\mathsf{P}}\Big{\\{}F_{T}^{*}\odot
P_{T|t}(\mathcal{H}_{t}){}_{T-t}p_{x+t}\Big{\\}}$ (52) $\displaystyle\times$
$\displaystyle
f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P};$

  * 3.

for the $T$–year guaranteed unit–linked term life insurance, it holds

$\displaystyle{U}{}_{x+t:{\mbox{$\scriptstyle\overline{T-t\,}$\kern-0.1pt\rule[-0.35pt]{0.3pt}{5.67776pt}}}}^{\makebox[16.41727pt]{\hfill$\scriptstyle
1$\hfill}}$ $\displaystyle=$
$\displaystyle\sum_{s}\int_{C,\Gamma,\mathsf{P}}\bigg{\\{}\sum_{k=t}^{T-1}\Big{[}F_{k+1}^{*}\odot
C_{k+1|t}(\mathcal{H}_{t})+B_{t,k+1}(\mathcal{H}_{t})G_{k+1}^{*}\Big{]}$ (53)
$\displaystyle\times$
$\displaystyle{}_{k-t}p_{x+t}q_{x+k}\bigg{\\}}f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P};$

  * 4.

for the $T$–year guaranteed unit–linked pure endowment insurance, it holds

$\displaystyle{U}{}_{x+t:{\mbox{$\scriptstyle\overline{T-t\,}$\kern-0.1pt\rule[-0.35pt]{0.3pt}{5.67776pt}}}}^{\hskip
19.19505pt\makebox[14.3683pt]{\hfill$\scriptstyle 1$\hfill}}$ $\displaystyle=$
$\displaystyle\sum_{s}\int_{C,\Gamma,\mathsf{P}}\Big{\\{}\Big{[}F_{T}^{*}\odot
C_{T|t}(\mathcal{H}_{t})+B_{t,T}(\mathcal{H}_{t})G_{T}^{*}\Big{]}{}_{T-t}p_{x+t}\Big{\\}}$
(54) $\displaystyle\times$ $\displaystyle
f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P}.$

##  5 Locally Risk–Minimizing Strategy

By introducing the concept of mean–self–financing, Follmer86 extended the
concept of the complete market into the incomplete market. If a discounted
cumulative cost process is a martingale, then a portfolio plan is called
mean–self–financing. In a discrete–time case, Follmer89 developed a locally
risk–minimizing strategy and obtained a recurrence formula for optimal
strategy. According to Schal94 (see also Follmer04), under a martingale
probability measure the locally risk–minimizing strategy and remaining
conditional risk–minimizing strategy are the same. Therefore, in this section,
we will consider locally risk–minimizing strategies, which correspond to the
Black–Scholes call and put options and Margrabe exchange options given in
Section 3 and the equity–linked life insurance products given in Section 4. In
the insurance industry, for continuous–time unit–linked term life and pure
endowment insurances with guarantee, locally risk–minimizing strategies are
obtained by Moller98.

To simplify notations we define: for $t=1,\dots,T$,
$\overline{P}_{t}:=(\overline{P}_{1,t},\dots,\overline{P}_{n,t})^{\prime}$ is
a discounted stock price process at time $t$,
$\overline{d}_{t}:=(\overline{d}_{1,t},\dots,\overline{d}_{n,t})^{\prime}$ is
a discounted dividend payment process at time $t$, and
$\Delta\overline{P}_{t}:=\overline{P}_{t}-\overline{P}_{t-1}$ is a difference
process at time $t$ of the consecutive stock price processes, where
$\overline{P}_{i,t}:=D_{t}P_{i,t}$ and $\overline{d}_{t}:=D_{t}d_{i,t}$ are
discounted stock price process and discounted dividend payment process,
respectively, at time $t$ of $i$–th stock. Note that
$\Delta\overline{P}_{t}+\overline{d}_{t}$ is a martingale difference with
respect to the filtration $\\{\mathcal{H}_{t}\\}_{t=0}^{T}$ and the
risk–neutral measure $\mathbb{\tilde{P}}$. Also, let $h_{t}$ be a proper
number of shares at time $t$ and $h_{t}^{0}$ be a proper amount of cash
(risk–free bond) at time $t$, which are required to successfully hedge a
generic contingent claim $H_{T}$, and $\overline{H}_{T}$ be a discounted
contingent claim, where we assume that the discounted contingent claim
$\overline{H}_{T}$ is square–integrable under the risk–neutral probability
measure. Then, following the idea in Follmer04 and Follmer89, one can obtain
that for a filtration $\\{\mathcal{F}_{t}^{x}\\}_{t=0}^{T}$ and the generic
discounted contingent claim $\overline{H}_{T}$, under the risk–neutral
probability measure $\mathbb{\tilde{P}}$, the locally risk–minimizing strategy
($h^{0},h$) is given by the following equations:

$h_{t+1}=\overline{\Omega}_{t+1}^{-1}\overline{\Lambda}_{t+1}~{}~{}~{}\text{and}~{}~{}~{}h_{t+1}^{0}=V_{t+1}-h_{t+1}^{\prime}(P_{t+1}+d_{t+1})$
(55)

for $t=0,\dots,T-1$ and $h_{0}^{0}=V_{0}-h_{1}^{\prime}P_{0}$, where
$\overline{\Omega}_{t+1}:=\tilde{\mathbb{E}}\big{[}(\Delta\overline{P}_{t+1}+\overline{d}_{t+1})(\Delta\overline{P}_{t+1}+\overline{d}_{t+1})^{\prime}\big{|}\mathcal{F}_{t}^{x}\big{]}$
is an $(n\times n)$ random matrix,
$\overline{\Lambda}_{t+1}:=\widetilde{\text{Cov}}\big{[}\Delta\overline{P}_{t+1}+\overline{d}_{t+1},\overline{H}_{T}\big{|}\mathcal{F}_{t}^{x}\big{]}$
is an $(n\times 1)$ random vector, and
$V_{t+1}:=\frac{1}{D_{t+1}}\mathbb{\tilde{E}}[\overline{H}_{T}|\mathcal{F}_{t+1}^{x}]$
is a value process of the contingent claim. Note that $h_{t}$ is a predictable
process, which means its value is known at time $t-1$, while for the process
$h_{t}^{0}$, its value is only known at time $t$, and if the contingent claim
$H_{T}$ is generated by stock prices $P_{t}$ and dividends $d_{t}$ for
$t=0,\dots,T$, then the process $h_{t}^{0}$ becomes predictable, see
Follmer89. Note that for $t=1,\dots,T$, since $\sigma$–fields
$\mathcal{H}_{T}$ and $\mathcal{T}_{T}^{x}$ are independent, if $X$ is any
random variable, which is independent of $\sigma$–field $\mathcal{T}_{T}^{x}$
and integrable with respect to the risk–neutral probability measure, then it
holds

$\mathbb{\tilde{E}}[X|\mathcal{H}_{t}^{x}]=\mathbb{\tilde{E}}[X|\mathcal{H}_{t}].$
(56)

To obtain the locally risk–minimizing strategy, we need a distribution of a
random variable, which is a sum of the price at time $t+1$ and the dividend at
time $t+1$ conditional on $\mathcal{H}_{t}$. It follows from the first line of
system (16) that the log stock price at time $t$ is given by

$\tilde{P}_{t}=G_{t}\bigg{(}\tilde{P}_{t-1}-\tilde{d}_{t}+\tilde{r}_{t}i_{n}-\frac{1}{2}\mathcal{D}\big{[}\Sigma_{uu,t}\big{]}\bigg{)}+\tilde{d}_{t}-h_{t}+G_{t}\tilde{u}_{t}$
(57)

under the risk–neutral probability measure $\mathbb{\tilde{P}}$. If we
substitute the above equation into the approximation equation (2), then we
have

$\ln\Big{(}\big{(}P_{t}+d_{t}\big{)}\oslash
P_{t-1}\Big{)}=\tilde{r}_{t}i_{n}-\frac{1}{2}\mathcal{D}[\Sigma_{uu,t}]+\tilde{u}_{t}$
(58)

under the risk–neutral probability measure $\mathbb{\tilde{P}}$. Consequently,
conditional on $\mathcal{H}_{t}$, approximated distribution of a log sum
random variable of the discounted stock price $\overline{P}_{t+1}$ and the
discounted dividend payment $\overline{d}_{t+1}$ is given by

$\ln\big{(}\overline{P}_{t+1}+\overline{d}_{t+1}\big{)}~{}|~{}\mathcal{H}_{t}\sim\mathcal{N}\bigg{(}\ln\big{(}\overline{P}_{t}\big{)}-\frac{1}{2}\mathcal{D}[\Sigma_{uu,t}],\Sigma_{uu,t}\bigg{)}$
(59)

under the risk–neutral probability measure $\mathbb{\tilde{P}}$. Therefore, it
follows from the fact that
$\tilde{\mathbb{E}}\big{[}\overline{P}_{t+1}+\overline{d}_{t+1}\big{|}\mathcal{H}_{t}\big{]}=\overline{P}_{t}$
and the well–known covariance formula of the multivariate log–normal random
vector that

$\displaystyle\overline{\Omega}_{t+1}(\mathcal{H}_{t})$ $\displaystyle:=$
$\displaystyle\tilde{\mathbb{E}}\big{[}(\Delta\overline{P}_{t+1}+\overline{d}_{t+1})(\Delta\overline{P}_{t+1}+\overline{d}_{t+1})^{\prime}\big{|}\mathcal{H}_{t}\big{]}$
(60) $\displaystyle=$
$\displaystyle\big{(}\exp\\{\Sigma_{uu,t}\\}-\mathsf{E}_{n}\big{)}\odot\overline{P}_{t}\overline{P}_{t}^{\prime},$

where $\mathsf{E}_{n}$ is an $(n\times n)$ matrix, whose elements are one,
see, e.g., Fang18. Consequently, by Lemmas 2 and 3 and the tower property of a
conditional expectation, we obtain that

$\displaystyle\overline{\Omega}_{t+1}=\tilde{\mathbb{E}}\big{[}\overline{\Omega}_{t+1}(\mathcal{H}_{t})\big{|}\mathcal{F}_{t}\big{]}=\sum_{s}\int_{C,\Gamma,\mathsf{P}}\overline{\Omega}_{t+1}(\mathcal{H}_{t})f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P}.$ (61)

On the other hand, according to equation (2), the sum of discounted price
process and dividend process, respectively, at time $t+1$ is represented by
the following equation

$\overline{P}_{t+1}+\overline{d}_{t+1}=D_{t}\exp\big{\\{}W_{t+1}x_{t+1}+G_{t+1}^{-1}h_{t+1}\big{\\}},$
(62)

where $W_{t+1}:=[G_{t+1}^{-1}:I_{n}-G_{t+1}^{-1}:-i_{n}]$ is an
$(n\times\tilde{n})$ matrix. By using the fact that
$\tilde{\mathbb{E}}\big{[}\overline{P}_{t+1}+\overline{d}_{t+1}\big{|}\mathcal{H}_{t}\big{]}=\overline{P}_{t}$,
a random vector
$\overline{\Lambda}_{t+1}(\mathcal{H}_{t}^{x}):=\widetilde{\text{Cov}}\big{[}\Delta\overline{P}_{t+1}+\overline{d}_{t+1},\overline{H}_{T}\big{|}\mathcal{H}_{t}^{x}\big{]}$
can be written by

$\displaystyle\overline{\Lambda}_{t+1}(\mathcal{H}_{t}^{x})=\tilde{\mathbb{E}}\big{[}\big{(}\overline{P}_{t+1}+\overline{d}_{t+1}\big{)}\overline{H}_{T}^{\prime}\big{|}\mathcal{H}_{t}^{x}\big{]}-\overline{P}_{t}\overline{V}_{t}^{\prime}(\mathcal{H}_{t}^{x})$
(63)

where
$\overline{V}_{t}(\mathcal{H}_{t}^{x}):=\tilde{\mathbb{E}}\big{[}\overline{H}_{T}\big{|}\mathcal{H}_{t}^{x}\big{]}$
is a discounted value process, corresponding to the contingent claim vector
$H_{T}$ for given information $\mathcal{H}_{t}^{x}$. Since the $\sigma$–fields
$\mathcal{H}_{T}$ and $\mathcal{T}_{T}^{x}$ are independent, due to
$t$–forward probability measure, the conditional covariance is

  * ($i$)

for the Black–Scholes call and put options and the Margrabe exchange options,

$\displaystyle\overline{\Lambda}_{t+1}(\mathcal{H}_{t}^{x})=D_{t}B_{t,T}(\mathcal{H}_{t})\hat{\mathbb{E}}_{t,T}\big{[}\big{(}\overline{P}_{t+1}+\overline{d}_{t+1}\big{)}H_{T}^{\prime}\big{|}\mathcal{H}_{t}\big{]}-\overline{P}_{t}\overline{V}_{t}^{\prime}(\mathcal{H}_{t}),$
(64)

  * ($ii$)

for the equity–linked pure endowment insurances,

$\displaystyle\overline{\Lambda}_{t+1}(\mathcal{H}_{t}^{x})=D_{t}B_{t,T}(\mathcal{H}_{t})\hat{\mathbb{E}}_{t,T}\big{[}\big{(}\overline{P}_{t+1}+\overline{d}_{t+1}\big{)}f(P_{T})^{\prime}\big{|}\mathcal{H}_{t}\big{]}{}_{T-t}p_{x+t}-\overline{P}_{t}\overline{V}_{t}^{\prime}(\mathcal{H}_{t}^{x}),$
(65)

  * ($iii$)

and for the equity–linked term life insurances,

$\displaystyle\overline{\Lambda}_{t+1}(\mathcal{H}_{t}^{x})=D_{t}\sum_{k=t}^{T-1}B_{t,k+1}(\mathcal{H}_{t})\hat{\mathbb{E}}_{t,k+1}\big{[}\big{(}\overline{P}_{t+1}+\overline{d}_{t+1}\big{)}f(P_{k+1})^{\prime}\big{|}\mathcal{H}_{t}\big{]}{}_{k-t}p_{x+t}q_{x+k}-\overline{P}_{t}\overline{V}_{t}^{\prime}(\mathcal{H}_{t}^{x}).$
(66)

In order to obtain the locally risk–minimizing strategies for the
Black–Scholes call and put options, the Margrabe exchange options, and the
equity–linked life insurance products, we need to calculate the conditional
covariances given in equations (64)–(66) for contingent claims
$H_{T}=[P_{T}-K]^{+}$, $H_{T}=[K-P_{T}]^{+}$, and
$H_{T}=[w_{i}P_{i,T}-w_{j}P_{j,T}]^{+}$ for $i,j=1,\dots,n$ and sum insureds
$f(P_{k})=F_{k}^{*}\odot[P_{k}^{*}-G_{k}^{*}\oslash F_{k}^{*}]^{+}+G_{k}^{*}$
and $f(P_{k})=F_{k}^{*}\odot[G_{k}^{*}\oslash F_{k}^{*}-P_{k}]^{+}$ for
$k=t+1,\dots,T$. Thus, we need Lemma 5, see Technical Annex.

Let us define vectors: $\pi:=W_{t+1}x_{t+1}+G_{t+1}^{-1}h_{t+1}$ is the
exponent of equation (62) and
$\phi_{i}:=\pi+(e_{i}^{\prime}\tilde{P}_{T})i_{n}$ is a sum of the vector
$\pi$ and a vector, consisting of the exponent, which is given in equation
(40). To apply the Lemma, we need conditional expectations and covariance
matrices of the random vectors $\pi$ and $\phi_{i}$ and conditional covariance
between the random vector $\pi$ and log price at time $k$ under the
$(t,u)$–forward probability measure. Since $x_{t+1}=J_{t+1|t}\bar{x}_{t}^{c}$
and $\tilde{P}_{k}=J_{P}J_{k|t}\bar{x}_{t}^{c}$ for $k=t+1,\dots,T$, according
to equations (24), (30), and (32), we have that

$\hat{\mu}_{t,u}^{\pi}:=\mathbb{\hat{E}}_{t,u}\big{[}\pi\big{|}\mathcal{H}_{t}\big{]}=W_{t+1}\hat{\mu}_{t+1|t,u}+G_{t+1}^{-1}h_{t+1},$
(67)
$\hat{\mu}_{t,u}^{\phi_{i}}:=\mathbb{\hat{E}}_{t,u}\big{[}\phi_{i}\big{|}\mathcal{H}_{t}\big{]}=W_{t+1}\hat{\mu}_{t+1|t,u}+G_{t+1}^{-1}h_{t+1}+i_{n}e_{i}^{\prime}J_{P}\hat{\mu}_{T|t,u},$
(68)
$\Sigma_{\pi}:=\widehat{\text{Var}}\big{[}\pi\big{|}\mathcal{H}_{t}\big{]}=W_{t+1}\Sigma_{t+1,t+1|t}W_{t+1}^{\prime},$
(69)
$\Sigma_{\phi_{i}}:=\widehat{\text{Var}}\big{[}\phi_{i}\big{|}\mathcal{H}_{t}\big{]}=\big{(}W_{t+1}J_{t+1|t}+i_{n}e_{i}^{\prime}J_{P}J_{T|t}\big{)}\Sigma_{t}^{c}\big{(}W_{t+1}J_{t+1|t}+i_{n}e_{i}^{\prime}J_{P}J_{T|t}\big{)}^{\prime},$
(70)

and

$\Sigma_{\pi,\tilde{P}_{k}}:=\widehat{\text{Cov}}\big{[}\pi,\tilde{P}_{k}\big{|}\mathcal{H}_{t}\big{]}=W_{t+1}J_{t+1|t}\Sigma_{t}^{c}J_{k|t}^{\prime}J_{P}^{\prime}.$
(71)

For the Margrabe exchange options, which gives owner a right to exchange
$w_{j}$ units of stock of $j$–th firm into $w_{i}$ units of stock of $i$–th
firm at time $T$, since its contingent claim is
$H_{T}=\big{(}w_{i}P_{i,T}-w_{j}P_{j,T}\big{)}^{+}$, due to equation (62), for
$k=1,\dots,n$, $k$–th element of the expectation in equation (64) is
represented by

$\displaystyle\hat{\mathbb{E}}_{t,T}\big{[}e_{k}^{\prime}\big{(}\overline{P}_{t+1}+\overline{d}_{t+1}\big{)}H_{T}\big{|}\mathcal{H}_{t}\big{]}$
$\displaystyle=D_{t}\Big{(}w_{i}\mathbb{\hat{E}}_{t,T}\big{[}\exp\\{e_{k}^{\prime}\pi\\}P_{i,T}1_{A}\big{|}\mathcal{H}_{t}\big{]}-w_{j}\mathbb{\hat{E}}_{t,T}\big{[}\exp\\{e_{k}^{\prime}\pi\\}P_{j,T}1_{A}\big{|}\mathcal{H}_{t}\big{]}\Big{)}$
(72)

for $i,j=1,\dots,n$ $(i\neq j)$. The first expectation of the above equation
can be written by

$\displaystyle\mathbb{\hat{E}}_{t,T}\big{[}\exp\\{e_{k}^{\prime}\pi\\}P_{i,T}1_{A}\big{|}\mathcal{H}_{t}\big{]}=\mathbb{\hat{E}}_{t,T}\Big{[}\exp\\{e_{k}^{\prime}\phi_{i}\\}1_{\\{(e_{i}-e_{j})^{\prime}\tilde{P}_{T}\geq\ln(w_{j}/w_{i})\\}}\Big{|}\mathcal{H}_{t}\Big{]}$
(73)
$\displaystyle=\mathbb{\hat{E}}_{t,T}\Big{[}\exp\Big{\\{}\big{(}e_{k}^{\prime}W_{t+1}J_{t+1|t}+e_{i}^{\prime}J_{P}J_{T|t}\big{)}\bar{x}_{t}^{c}+e_{k}^{\prime}G_{t+1}^{-1}h_{t+1}\Big{\\}}1_{\\{(e_{i}-e_{j})^{\prime}J_{P}J_{T|t}\bar{x}_{t}^{c}\geq\ln(w_{j}/w_{i})\\}}\Big{|}\mathcal{H}_{t}\Big{]}.$

Consequently, similarly to equations (41)–(43), it can be shown that the
conditional covariance, which is given by equation (64) and corresponds to the
Margrabe exchange option is given by the following equation

$\displaystyle\overline{\Lambda}_{t+1}^{ij}(\mathcal{H}_{t})$ $\displaystyle=$
$\displaystyle
D_{t}^{2}B_{t,T}(\mathcal{H}_{t})\bigg{[}w_{i}\exp\bigg{\\{}\hat{\mu}_{t,T}^{\phi_{i}}+\frac{1}{2}\mathcal{D}\big{[}\Sigma_{\phi_{i}}\big{]}\bigg{\\}}\Phi\Big{(}\tilde{d}_{T|t}^{ij,1}\Big{)}$
(74) $\displaystyle-$ $\displaystyle
w_{j}\exp\bigg{\\{}\hat{\mu}_{t,T}^{\phi_{j}}+\frac{1}{2}\mathcal{D}\big{[}\Sigma_{\phi_{j}}\big{]}\bigg{\\}}\Phi\Big{(}\tilde{d}_{T|t}^{ij,2}\Big{)}\bigg{]}-\overline{P}_{t}\overline{V}_{t}^{\prime}(\mathcal{H}_{t}),$

where the discounted value process is
$\overline{V}_{t}(\mathcal{H}_{t})=O_{T|t}^{ij}(\mathcal{H}_{t})$, see
equation (43),

$\displaystyle\tilde{d}_{T|t}^{ij,1}:=d_{T|t}^{ij,1}+\Big{(}\Sigma_{\pi,\tilde{P}_{T}}(e_{i}-e_{j})\Big{)}\oslash\sqrt{(e_{i}-e_{j})^{\prime}\Sigma_{T|t}^{\tilde{P}}(e_{i}-e_{j})i_{n}}$
(75)

and

$\tilde{d}_{T|t}^{ij,2}:=\tilde{d}_{T|t}^{ij,1}-\sqrt{(e_{i}-e_{j})^{\prime}\Sigma_{T|t}^{\tilde{P}}(e_{i}-e_{j})i_{n}}.$
(76)

As a result, it follows from the tower property of conditional expectation,
Lemmas 2, 3, and 5, and equations (LABEL:06050), (64)–(67), (69), (71), and
(74) that for $t=0,\dots,T-1$, $\overline{\Lambda}_{t+1}$s, which correspond
to the call and put options, the Margrabe exchange options, and the
equity–linked life insurance products are obtained by the following equations

  * 1.

for the dividend–paying Black–Scholes call option on the weighted asset price,
we have

$\displaystyle\overline{\Lambda}_{t+1}$ $\displaystyle=$
$\displaystyle\sum_{s}\int_{C,\Gamma,\mathsf{P}}\bigg{\\{}D_{t}^{2}B_{t,T}(\mathcal{H}_{t})\Psi^{+}\Big{(}K;\mu_{t,T}^{\pi};\hat{\mu}_{T|t,T}^{\tilde{P}};\Sigma_{\pi};\Sigma_{\pi,\tilde{P}_{T}};\Sigma_{T|t}^{\tilde{P}}\Big{)}\Big{)}\bigg{\\}}$
(77) $\displaystyle\times
f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P}-\overline{P}_{t}\overline{V}_{t}^{\prime},$

where the discounted value process is given by
$\overline{V}_{t}=D_{t}C_{T|t}$, see equation (37),

  * 2.

for the dividend–paying Black–Scholes put option on the weighted asset price,
we have

$\displaystyle\overline{\Lambda}_{t+1}$ $\displaystyle=$
$\displaystyle\sum_{s}\int_{C,\Gamma,\mathsf{P}}\bigg{\\{}D_{t}^{2}B_{t,T}(\mathcal{H}_{t})\Psi^{-}\Big{(}K;\mu_{t,T}^{\pi};\hat{\mu}_{T|t,T}^{\tilde{P}};\Sigma_{\pi};\Sigma_{\pi,\tilde{P}_{T}};\Sigma_{T|t}^{\tilde{P}}\Big{)}\Big{)}\bigg{\\}}$
(78) $\displaystyle\times
f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P}-\overline{P}_{t}\overline{V}_{t}^{\prime},$

where the discounted value process is given by
$\overline{V}_{t}=D_{t}P_{T|t}$, see equation (38),

  * 3.

for the Margrabe exchange options, we have

$\displaystyle\overline{\Lambda}_{t+1}$ $\displaystyle\approx$
$\displaystyle\sum_{s}\int_{C,\Gamma,\mathsf{P}}\overline{\Lambda}_{t+1}^{ij}f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P},$ (79)

where the conditional covariance
$\overline{\Lambda}_{t+1}^{ij}(\mathcal{H}_{t})$ is given by equation (74),

  * 4.

for the $T$-year guaranteed term life insurance, corresponding to a segregated
fund contract, we have

$\displaystyle\overline{\Lambda}_{t+1}$ $\displaystyle\approx$
$\displaystyle\sum_{s}\int_{C,\Gamma,\mathsf{P}}\bigg{\\{}D_{t}^{2}\sum_{k=t}^{T-1}B_{t,k+1}(\mathcal{H}_{t})\Psi^{-}\Big{(}G_{k+1}^{*}\oslash
F_{k+1}^{*};\mu_{t,k+1}^{\pi};\hat{\mu}_{k+1|t,k+1}^{\tilde{P}}(\mathcal{F});\Sigma_{\pi};$
(80)
$\displaystyle\Sigma_{\pi,\tilde{P}_{k+1}};\Sigma_{k+1|t}^{\tilde{P}}(\mathcal{F})\Big{)}{}_{k-t}p_{x+t}q_{x+k}\Big{)}\bigg{\\}}f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P}-\overline{P}_{t}\overline{V}_{t}^{\prime},$

where the discounted value process is given by
$\overline{V}_{t}=D_{t}{S}{}_{x+t:{\mbox{$\scriptstyle\overline{T-t\,}$\kern-0.1pt\rule[-0.35pt]{0.3pt}{5.67776pt}}}}^{\makebox[16.41727pt]{\hfill$\scriptstyle
1$\hfill}}$, see equation (51),

  * 5.

for the $T$-year guaranteed pure endowment insurance, corresponding to a
segregated fund contract, we have

$\displaystyle\overline{\Lambda}_{t+1}$ $\displaystyle\approx$
$\displaystyle\sum_{s}\int_{C,\Gamma,\mathsf{P}}\bigg{\\{}D_{t}^{2}B_{t,T}(\mathcal{H}_{t})\Psi^{-}\Big{(}G_{T}^{*}\oslash
F_{T}^{*};\mu_{t,T}^{\pi};\hat{\mu}_{T|t,T}^{\tilde{P}}(\mathcal{F});\Sigma_{\pi};$
(81)
$\displaystyle\Sigma_{\pi,\tilde{P}_{T}};\Sigma_{T|t}^{\tilde{P}}(\mathcal{F})\Big{)}{}_{T-t}p_{x+t}\Big{)}\bigg{\\}}f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P}-\overline{P}_{t}\overline{V}_{t}^{\prime},$

where the discounted value process is given by
$\overline{V}_{t}=D_{t}{S}{}_{x+t:{\mbox{$\scriptstyle\overline{T-t\,}$\kern-0.1pt\rule[-0.35pt]{0.3pt}{5.67776pt}}}}^{\hskip
19.19505pt\makebox[14.3683pt]{\hfill$\scriptstyle 1$\hfill}}$, see equation
(52),

  * 6.

for the $T$-year guaranteed unit–linked term life insurance, we have

$\displaystyle\overline{\Lambda}_{t+1}$ $\displaystyle\approx$
$\displaystyle\sum_{s}\int_{C,\Gamma,\mathsf{P}}\bigg{\\{}D_{t}^{2}\sum_{k=t}^{T-1}B_{t,k+1}(\mathcal{H}_{t})\bigg{[}\Psi^{+}\Big{(}G_{k+1}^{*}\oslash
F_{k+1}^{*};\mu_{t,k+1}^{\pi};\hat{\mu}_{k+1|t,k+1}^{\tilde{P}};$
$\displaystyle\Sigma_{\pi};\Sigma_{\pi,\tilde{P}_{k+1}};\Sigma_{k+1|t}^{\tilde{P}}\Big{)}+\exp\bigg{\\{}\mu_{t,k+1}^{\pi}+\frac{1}{2}\mathcal{D}\big{[}\Sigma_{\pi}\big{]}\bigg{\\}}(G_{k}^{*})^{\prime}\bigg{]}{}_{k-t}p_{x+t}q_{x+k}\Big{)}\bigg{\\}}$
$\displaystyle\times
f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P}-\overline{P}_{t}\overline{V}_{t}^{\prime},$

where the discounted value process is given by
$\overline{V}_{t}=D_{t}{U}{}_{x+t:{\mbox{$\scriptstyle\overline{T-t\,}$\kern-0.1pt\rule[-0.35pt]{0.3pt}{5.67776pt}}}}^{\makebox[16.41727pt]{\hfill$\scriptstyle
1$\hfill}}$, see equation (53),

  * 7.

and for the $T$-year guaranteed unit–linked pure endowment insurance, we have

$\displaystyle\overline{\Lambda}_{t+1}$ $\displaystyle\approx$
$\displaystyle\sum_{s}\int_{C,\Gamma,\mathsf{P}}\bigg{\\{}D_{t}^{2}B_{t,T}(\mathcal{H}_{t})\bigg{[}\Psi^{+}\Big{(}G_{T}^{*}\oslash
F_{T}^{*};\mu_{t,T}^{\pi};\hat{\mu}_{T|t,T}^{\tilde{P}};\Sigma_{\pi};$
$\displaystyle\Sigma_{\pi,\tilde{P}_{T}};\Sigma_{T|t}^{\tilde{P}}\Big{)}+\exp\bigg{\\{}\mu_{t,T}^{\pi}+\frac{1}{2}\mathcal{D}\big{[}\Sigma_{\pi}\big{]}\bigg{\\}}(G_{T}^{*})^{\prime}\bigg{]}{}_{T-t}p_{x+t}\Big{)}\bigg{\\}}$
$\displaystyle\times
f(\bar{s}_{t}^{c}|\mathcal{G}_{t})\tilde{f}(C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t})dCd\Gamma
d\mathsf{P}-\overline{P}_{t}\overline{V}_{t}^{\prime},$

where the discounted value process is given by
$\overline{V}_{t}=D_{t}{U}{}_{x+t:{\mbox{$\scriptstyle\overline{T-t\,}$\kern-0.1pt\rule[-0.35pt]{0.3pt}{5.67776pt}}}}^{\hskip
19.19505pt\makebox[14.3683pt]{\hfill$\scriptstyle 1$\hfill}}$, see equation
(54),

where the functions $\Psi^{+}$ and $\Psi^{-}$ are defined in Lemma 5. As a
result, by substituting equations (61) and (77)–(7.) into (55) we can obtain
the locally risk–minimizing strategies for the Black–Scholes call and put
options, the Margrabe exchange options, and the equity–linked life insurance
products, corresponding to the private companies.

##  6 Parameter Estimation

Recently, Battulga22a first time introduced parameter estimation approaches,
which are based on the maximum likelihood method for practically popular DDMs.
Also, parameter estimation methods can be found in Battulga22c and
Battulga23b. In this section, to simplify the estimation of parameters of the
model, we assume that there is $N=1$ regime and the coefficient matrices
$C_{k,1}$ and $C_{y,1}$ and covariance matrix $\Sigma_{1}$ are deterministic.
For the parameters, we introduce the following notations: $C_{k}:=C_{k,1}$,
$C_{y}:=C_{y,1}$, and $\Sigma:=\Sigma_{1}$. According to equation (5), we have
that

$\tilde{d}_{t}-\tilde{P}_{t}-h_{t}=G_{t}\big{(}\tilde{d}_{t-1}-\tilde{P}_{t-1}+\tilde{d}_{t}-\tilde{d}_{t-1}-C_{k}\psi_{k,t}\big{)}-G_{t}u_{t}.$
(84)

By taking expectation with respect to the real probability measure
$\mathbb{P}$, one finds that

$\mu_{t}-h_{t}=G_{t}\big{(}\mu_{t-1}+\mathbb{E}[\tilde{d}_{t}|\mathcal{F}_{0}]-\mathbb{E}[\tilde{d}_{t-1}|\mathcal{F}_{0}]-C_{k}\psi_{k,t}\big{)}.$
(85)

On the other hand, the definition of the linearization parameter $h_{t}$
implies that

$\mu_{t}-h_{t}=G_{t}\big{(}\mu_{t}-\ln(g_{t})\big{)}.$ (86)

Therefore, for successive values of the parameter $\mu_{t}$, it holds

$\mu_{t}=\mu_{t-1}+\ln(g_{t})+\mathbb{E}[\tilde{d}_{t}|\mathcal{F}_{0}]-\mathbb{E}[\tilde{d}_{t-1}|\mathcal{F}_{0}]-C_{k}\psi_{k,t}.$
(87)

Consequently, since $\mu_{0}=\tilde{d}_{0}-\tilde{P}_{0}$, we have that

$\mu_{t}=\sum_{i=1}^{t}\ln(g_{i})+\mathbb{E}[\tilde{d}_{t}|\mathcal{F}_{0}]-\tilde{P}_{0}-C_{k}\sum_{i=1}^{t}\psi_{k,i}.$
(88)

By repeating the second line equation of system (8) and taking expectation
with respect to the real probability measure, one gets that

$\mathbb{E}[\tilde{d}_{t}|\mathcal{F}_{0}]=J_{d}\mathbb{E}[y_{t}|\mathcal{F}_{0}]=J_{d}y_{0}+J_{d}C_{y}\sum_{i=1}^{t}\psi_{y,i},$
(89)

where $J_{d}:=[I_{n}:0]$ is a $(n\times[n+1])$ matrix, which is used to
extract the log dividend process $\tilde{d}_{t}$ from the process $y_{t}$,
that is, $\tilde{d}_{t}=J_{d}y_{t}$. As a result, equation (88) is represented
by

$\mu_{t}=\sum_{i=1}^{t}\ln(g_{i})+J_{d}y_{0}+J_{d}C_{y}\sum_{i=1}^{t}\psi_{y,i}-\tilde{P}_{0}-C_{k}\sum_{i=1}^{t}\psi_{k,i}.$
(90)

It can be shown that for $a=1,\dots,n$ and $t=1,\dots,T$, partial derivatives
from functions $\ln(g_{a,t})$, $g_{a,t}^{-1}$, and $g_{a,t}^{-1}h_{a,t}$ with
respect to parameter $\beta$ is given by the following equations

$\frac{\partial\ln(g_{a,t})}{\partial\beta}=\Big{(}1-g_{a,t}^{-1}\Big{)}\frac{\partial\mu_{a,t}}{\partial\beta},~{}~{}~{}\frac{\partial
g_{a,t}^{-1}}{\partial\beta}=-\frac{g_{a,t}-1}{g_{a,t}^{2}}\frac{\partial\mu_{a,t}}{\partial\beta},$
(91)

and

$\frac{\partial
g_{a,t}^{-1}h_{a,t}}{\partial\beta}=-\mu_{a,t}\frac{g_{a,t}-1}{g_{a,t}^{2}}\frac{\partial\mu_{a,t}}{\partial\beta},$
(92)

where $\beta$ is elements of the parameters $C_{k}$ and $c_{r}$. Consequently,
we have that

$\frac{\partial\ln(g_{t})}{\partial\beta}=(I_{n}-G_{t}^{-1})\frac{\partial\mu_{t}}{\partial\beta},~{}~{}~{}\frac{\partial\ln(|G_{t}|)}{\partial\beta}=(i_{n}-g_{t}^{-1})^{\prime}\frac{\partial\mu_{t}}{\partial\beta},$
(93) $\frac{\partial
g_{t}^{-1}}{\partial\beta}=-G_{t}^{-2}(G_{t}-I_{n})\frac{\partial\mu_{t}}{\partial\beta},~{}~{}~{}\text{and}~{}~{}~{}\frac{\partial
G_{t}^{-1}h_{t}}{\partial\beta}=-\text{diag}\\{\mu_{t}\\}G_{t}^{-2}(G_{t}-I_{n})\frac{\partial\mu_{t}}{\partial\beta},$
(94)

where $g_{t}^{-1}:=(1/g_{1,t},\dots,1/g_{n,t})$ is a reciprocal of the
linearization parameter vector $g_{t}$. Therefore, a partial derivative from
the mean log payment–to–value $\mu_{t}$ with respect to the parameter $\beta$
is given by

$\frac{\partial\mu_{t}}{\partial\beta}=\sum_{i=1}^{t}(I_{n}-G_{t}^{-1})\frac{\partial\mu_{i}}{\partial\beta}-\sum_{i=1}^{t}\frac{\partial
f_{i}}{\partial\beta},~{}~{}~{}t=1,\dots,T,$ (95)

where $f_{i}:=(\psi_{k,i}^{\prime}\otimes
I_{n})c_{k}-(\psi_{y,i}^{\prime}\otimes J_{d})c_{y}$ is an $(n\times 1)$
vector, which includes parameters of the model and $c_{k}:=\text{vec}(C_{k})$
and $c_{y}:=\text{vec}(C_{y})$ are the vectorizations of the parameter
matrices $C_{k}$ and $C_{y}$, respectively. It can be shown that a solution of
the above system of equations (95) is obtained by

$\frac{\partial\mu_{t}}{\partial\beta}=-\sum_{i=1}^{t}\Bigg{(}\prod_{j=i}^{t}G_{j}\Bigg{)}\frac{\partial
f_{i}}{\partial\beta},~{}~{}~{}t=1,\dots,T.$ (96)

Consequently, partial derivatives from the mean log payment–to–value $\mu_{t}$
with respect to the parameters $c_{k}$ and $c_{y}$ are given by the following
equations

$\frac{\partial\mu_{t}}{\partial
c_{k}^{\prime}}=-\sum_{i=1}^{t}\Bigg{(}\prod_{j=i}^{t}G_{j}\Bigg{)}(\psi_{k,i}^{\prime}\otimes
I_{n})$ (97)

and

$\frac{\partial\mu_{t}}{\partial
c_{y}^{\prime}}=\sum_{i=1}^{t}\Bigg{(}\prod_{j=i}^{t}G_{j}\Bigg{)}(\psi_{y,i}^{\prime}\otimes
J_{d})$ (98)

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

Let us denote a parameter vector of our model by
$\theta:=(c_{k}^{\prime},c_{y}^{\prime})^{\prime}$. Because an absolute value
of a determinant formed $\big{(}\tilde{n}T\times\tilde{n}T\big{)}$ matrix of
partial derivatives of the elements of a vector $(u^{\prime},v)^{\prime}$ with
respect to elements of a vector $(\tilde{V}^{\prime},y^{\prime})^{\prime}$
equals
$\Big{|}\frac{\partial(u^{\prime},\eta^{\prime})}{\partial(\tilde{V}^{\prime},y^{\prime})}\Big{|}=\prod_{t=1}^{T}|G_{t}^{-1}|$
and $\xi\sim\mathcal{N}(0,I_{T}\otimes\Sigma)$, the log–likelihood function of
the system (8) is given by

$\displaystyle\mathcal{L}(\theta)$ $\displaystyle=$
$\displaystyle-\frac{\tilde{n}T}{2}\ln(2\pi)-\frac{T}{2}\ln(|\Sigma|)-\frac{1}{2}\sum_{t=1}^{T}\text{tr}\big{(}\xi_{t}^{\prime}\Sigma^{-1}\xi_{t}\big{)}-\sum_{t=1}^{T}\ln(|G_{t}|)$
$\displaystyle=$
$\displaystyle-\frac{\tilde{n}T}{2}\ln(2\pi)-\frac{T}{2}\ln(|\Sigma|)-\frac{1}{2}\sum_{t=1}^{T}u_{t}^{\prime}\Omega_{uu}u_{t}-\sum_{t=1}^{T}u_{t}^{\prime}\Omega_{uv}\eta_{t}-\frac{1}{2}\sum_{t=1}^{T}\eta_{t}^{\prime}\Omega_{vv}\eta_{t}-\sum_{t=1}^{T}\ln(|G_{t}|),$

where for generic square matrix $O$, $\text{tr}(O)$ is the trace of the matrix
$O$, the matrices $\Omega_{uu}$, $\Omega_{uv}$, $\Omega_{vu}$, $\Omega_{vv}$
are partition, corresponding to the vector
$(u_{t}^{\prime},\eta_{t}^{\prime})^{\prime}$ of an inverse matrix
$\Sigma^{-1}$. Due to equations in system (8), white noise processes $u_{t}$
and $\eta_{t}$ are represented by

$u_{t}=\text{diag}\\{\tilde{P}_{t}-\tilde{d}_{t}\\}g_{t}^{-1}-\tilde{P}_{t-1}+\tilde{d}_{t}-(\psi_{k,t}^{\prime}\otimes
I_{n})c_{k}+G_{t}^{-1}h_{t}$ (100)

and

$\eta_{t}=y_{t}-(\psi_{y,t}^{\prime}\otimes J_{d})c_{y}-y_{t-1}.$ (101)

Consequently, it follows from equations (93), (94), (97), (98), (100), and
(101) that partial derivatives from the white noise processes $u_{t}$ and
$\eta_{t}$ with respect to parameters $c_{k}$ and $c_{y}$ are obtained by

$\displaystyle\frac{\partial u_{t}}{\partial
c_{k}^{\prime}}=-\text{diag}\big{\\{}\tilde{d}_{t}-\tilde{P}_{t}-\mu_{t}\big{\\}}\big{(}I_{n}-G_{t}^{-1}\big{)}\sum_{i=1}^{t}\Bigg{(}\prod_{j=i}^{t-1}G_{j}\Bigg{)}(\psi_{k,i}^{\prime}\otimes
I_{n})-(\psi_{k,t}^{\prime}\otimes I_{n}),$ (102) $\displaystyle\frac{\partial
u_{t}}{\partial
c_{y}^{\prime}}=-\text{diag}\big{\\{}\tilde{d}_{t}-\tilde{P}_{t}-\mu_{t}\big{\\}}\big{(}I_{n}-G_{t}^{-1}\big{)}\sum_{i=1}^{t}\Bigg{(}\prod_{j=i}^{t-1}G_{j}\Bigg{)}(\psi_{y,i}^{\prime}\otimes
J_{d}),$ (103)

and

$\frac{\partial\eta_{t}}{\partial c_{y}^{\prime}}=-(\psi_{y,i}^{\prime}\otimes
J_{d}).$ (104)

To simplify notations, let us define vectors
$u_{k,t}=\text{diag}\\{\tilde{P}_{t}-\tilde{d}_{t}\\}g_{t}^{-1}-\tilde{P}_{t-1}+\tilde{d}_{t}+G_{t}^{-1}h_{t}$
and $\eta_{y,t}=y_{t}-y_{t-1}$. Since
$u_{t}=u_{k,t}-(\psi_{k,t}^{\prime}\otimes I_{n})c_{k}$ and
$\eta_{t}=\eta_{y,t}-(\psi_{y,t}^{\prime}\otimes J_{d})c_{y}$, partial
derivatives from the log–likelihood function $\mathcal{L}(\theta)$ with
respect to parameters $c_{k}$ and $c_{y}$ are given by

$\frac{\partial\mathcal{L}(\theta)}{\partial
c_{k}^{\prime}}=-\sum_{t=1}^{T}\Big{(}u_{k,t}^{\prime}\Omega_{uu}+\eta_{t}^{\prime}\Omega_{\eta
u}\Big{)}\frac{\partial u_{t}}{\partial
c_{k}^{\prime}}+c_{k}^{\prime}\sum_{t=1}^{T}(\psi_{k,t}\otimes
I_{n})\Omega_{uu}\frac{\partial u_{t}}{\partial
c_{k}^{\prime}}-(i_{n}-g_{t}^{-1})\frac{\partial\mu_{t}}{\partial
c_{k}^{\prime}}$ (105)

and

$\displaystyle\frac{\partial\mathcal{L}(\theta)}{\partial c_{y}^{\prime}}$
$\displaystyle=$
$\displaystyle-\sum_{t=1}^{T}\Big{(}u_{t}^{\prime}\Omega_{uu}+\eta_{t}^{\prime}\Omega_{\eta
u}\Big{)}\frac{\partial u_{t}}{\partial
c_{y}^{\prime}}-\sum_{t=1}^{T}\Big{(}\eta_{y,t}^{\prime}\Omega_{\eta\eta}+u_{t}^{\prime}\Omega_{u\eta}\Big{)}\frac{\partial\eta_{t}}{\partial
c_{y}^{\prime}}$ (106) $\displaystyle+$ $\displaystyle
c_{y}^{\prime}\sum_{t=1}^{T}(\psi_{y,t}\otimes
J_{d}^{\prime})\Omega_{\eta\eta}\frac{\partial\eta_{t}}{\partial
c_{k}^{\prime}}-(i_{n}-g_{t}^{-1})\frac{\partial\mu_{t}}{\partial
c_{y}^{\prime}}.$

On the other hand, partial derivative from the log–likelihood function
$\mathcal{L}(\theta)$ with respect to the parameter $\Sigma$ is given by

$\frac{\partial\mathcal{L}(\theta)}{\partial\Sigma}=-\frac{T}{2}\Sigma^{-1}+\frac{1}{2}\Sigma^{-1}\bigg{(}\sum_{t=1}^{T}\xi_{t}\xi_{t}^{\prime}\bigg{)}\Sigma^{-1},$
(107)

see Lutkepohl05. By setting the above partial derivatives to zero, one obtains
that

$\hat{c}_{k}^{\prime}=\Bigg{(}\sum_{t=1}^{T}\Big{(}u_{k,t}^{\prime}\Omega_{uu}+\eta_{t}^{\prime}\Omega_{\eta
u}\Big{)}\frac{\partial u_{t}}{\partial
c_{k}^{\prime}}+(i_{n}-g_{t}^{-1})\frac{\partial\mu_{t}}{\partial
c_{k}^{\prime}}\Bigg{)}\Bigg{(}\sum_{t=1}^{T}(\psi_{k,t}\otimes
I_{n})\Omega_{uu}\frac{\partial u_{t}}{\partial c_{k}^{\prime}}\Bigg{)}^{-1},$
(108) $\displaystyle\hat{c}_{y}^{\prime}$ $\displaystyle=$
$\displaystyle\Bigg{(}\sum_{t=1}^{T}\Big{(}u_{t}^{\prime}\Omega_{uu}+\eta_{t}^{\prime}\Omega_{\eta
u}\Big{)}\frac{\partial u_{t}}{\partial
c_{y}^{\prime}}+\sum_{t=1}^{T}\Big{(}\eta_{y,t}^{\prime}\Omega_{\eta\eta}+u_{t}^{\prime}\Omega_{u\eta}\Big{)}\frac{\partial\eta_{t}}{\partial
c_{y}^{\prime}}$ (109) $\displaystyle+$
$\displaystyle(i_{n}-g_{t}^{-1})\frac{\partial\mu_{t}}{\partial
c_{y}^{\prime}}\Bigg{)}\Bigg{(}\sum_{t=1}^{T}(\psi_{y,t}\otimes
J_{d}^{\prime})\Omega_{\eta\eta}\frac{\partial\eta_{t}}{\partial
c_{y}^{\prime}}\Bigg{)}^{-1},$

and

$\hat{\Sigma}=\frac{1}{T}\sum_{t=1}^{T}\xi_{t}\xi_{t}^{\prime}.$ (110)

Under the suitable conditions, by substituting equations (97), (98), and
(100)–(104) and expressions for the processes $u_{k,t}$ and $v_{r,t}$ into
equation (108)–(110), the zig–zag iteration, corresponding to equations
(108)–(110) converges to maximum likelihood estimators of the model parameters
$c_{k}$, $c_{y}$, and $\Sigma$.

##  7 Technical Annex

Here we give the Lemmas, which are used in the paper.

######  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.

ee, e.g., Battulga23a and Battulga23e. ∎

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 $\mathbb{\tilde{P}}$, 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 two Lemmas holds.

######  Lemma 2.

For $t=0,\dots,T-1$, the following relation holds

$f(\bar{s}_{t}^{c}|\mathcal{G}_{t}):=\mathbb{\tilde{P}}\big{(}\bar{s}_{t}^{c}=\bar{s}_{t}^{c}|\mathcal{G}_{t}\big{)}=\frac{\displaystyle
f(C,\Gamma|\bar{s}_{t}^{c},\bar{s}_{t},\mathcal{F}_{0})\prod_{m=t+1}^{T}p_{s_{m-1}s_{m}}}{\displaystyle\sum_{\bar{s}_{t}^{c}}f(C,\Gamma|\bar{s}_{t}^{c},\bar{s}_{t},\mathcal{F}_{0})\prod_{m=t+1}^{T}p_{s_{m-1}s_{m}}},$

where $p_{s_{0}s_{1}}:=\mathbb{P}(s_{1}=s_{1}|\mathsf{P},\mathcal{F}_{0})$ and
$p_{s_{m-1}s_{m}}:=\mathbb{P}(s_{m}=s_{m}|\mathsf{P},s_{m-1}=s_{m-1},\mathcal{F}_{0})$
for $m=t+1,\dots,T$.

###### Proof.

See, Battulga23a. ∎

######  Lemma 3.

Conditional on $\mathcal{F}_{t}$, joint density of $\bar{\Pi}_{t}$,
$\bar{\Gamma}_{t}$, $\mathsf{P}$, and $\bar{s}_{t}$ is given by

$\tilde{f}\big{(}C,\Gamma,\mathsf{P},\bar{s}_{t}|\mathcal{F}_{t}\big{)}=\frac{\displaystyle\tilde{f}(\bar{y}_{t}|\mathcal{J}_{t})\bigg{(}\sum_{\bar{s}_{t}^{c}}f(C,\Gamma|\bar{s}_{t},\bar{s}_{t}^{c},\mathcal{F}_{0})\prod_{m=t+1}^{T}p_{s_{m-1}s_{m}}\bigg{)}\prod_{m=1}^{t}p_{s_{m-1}s_{m}}f(\mathsf{P}|\mathcal{F}_{0})}{\displaystyle\sum_{\bar{s}_{T}}\int_{C,\Gamma,\mathsf{P}}\tilde{f}(\bar{y}_{t}|\mathcal{J}_{t})f(C,\Gamma|\bar{s}_{T},\mathcal{F}_{0})\prod_{m=1}^{T}p_{s_{m-1}s_{m}}f(\mathsf{P}|\mathcal{F}_{0})dCd\Gamma
d\mathsf{P}},$

for $t=1,\dots,T$ with convention $p_{s_{1}}=p_{s_{0}s_{1}}$, where for
$t=1,\dots,T$,

$\tilde{f}(\bar{y}_{t}|\mathcal{J}_{t})=\frac{1}{(2\pi)^{nt/2}\prod_{m=1}^{t}|\Sigma_{m}|^{1/2}}\exp\Big{\\{}-\frac{1}{2}\big{(}\bar{x}_{t}-\tilde{\mu}_{t}(\mathcal{H}_{0})\big{)}^{\prime}\Sigma_{t}^{-1}(\mathcal{H}_{0})\big{(}\bar{x}_{t}-\tilde{\mu}_{t}(\mathcal{H}_{0})\big{)}\Big{\\}}$

with
$\tilde{\mu}_{t}(\mathcal{H}_{0}):=\big{(}\tilde{\mu}_{1|0}(\mathcal{H}_{0}),\dots,\tilde{\mu}_{t|0}(\mathcal{H}_{0})\big{)}^{\prime}$
and
$\Sigma_{t}(\mathcal{H}_{0}):=\big{(}\Sigma_{i_{1},i_{2}|0}(\mathcal{H}_{0})\big{)}_{i_{1},i_{2}=1}^{t}$.

###### Proof.

See, Battulga23a. ∎

######  Lemma 4.

Let $X\in\mathbb{R}^{n}$ be a random vector with a distribution
$X\sim\mathcal{N}(\mu,\Sigma)$, then for each deterministic matrix
$A\in\mathbb{R}^{m\times n}$ ($m\leq n$), vectors
$\alpha^{\prime}\in\mathbb{R}^{n}$ and $b,c\in\mathbb{R}^{m}$, and number
$\beta\in\mathbb{R}$, it holds

$\displaystyle\mathbb{E}\Big{[}\exp\big{\\{}\alpha
X+\beta\big{\\}}1_{\\{AX+b>c\\}}\Big{]}=\exp\bigg{\\{}\alpha\mu+\beta+\frac{1}{2}\alpha\Sigma\alpha^{\prime}\bigg{\\}}\mathbb{P}\Big{[}AX+b>c-A\Sigma\alpha^{\prime}\Big{]}$

where for each set $A\in\mathcal{F}_{T}$, $1_{A}$ is an indicator function,
corresponding to the set $A$.

###### Proof.

See, Battulga23e. ∎

######  Lemma 5.

Let $X_{1}\in\mathbb{R}^{n_{1}}$ and $X_{2}\in\mathbb{R}^{n_{2}}$ be random
vectors and their joint distribution is given by

$\begin{bmatrix}X_{1}\\\
X_{2}\end{bmatrix}\sim\mathcal{N}\bigg{(}\begin{bmatrix}\mu_{1}\\\
\mu_{2}\end{bmatrix},\begin{bmatrix}\Sigma_{11}&\Sigma_{12}\\\
\Sigma_{21}&\Sigma_{22}\end{bmatrix}\bigg{)}.$

Then, for all $L\in\mathbb{R}_{+}^{n_{2}}$, it holds

$\displaystyle\Psi^{+}\big{(}L;\mu_{1};\mu_{2};\Sigma_{11};\Sigma_{12};\Sigma_{22}\big{)}:=\mathbb{E}\Big{[}e^{X_{1}}\Big{(}\big{(}e^{X_{2}}-L\big{)}^{+}\Big{)}^{\prime}\Big{]}$
$\displaystyle=\Big{(}\mathbb{E}\big{[}e^{X_{1}}\big{]}\mathbb{E}\big{[}e^{X_{2}}\big{]}^{\prime}\Big{)}\odot
e^{\Sigma_{12}}\odot\Phi\Big{(}i_{n_{1}}\otimes
d_{1}^{\prime}+\Sigma_{12}\mathrm{diag}\big{\\{}\mathcal{D}\big{[}\Sigma_{22}\big{]}\big{\\}}^{-1/2}\Big{)}$
$\displaystyle-\Big{(}\mathbb{E}\big{[}e^{X_{1}}\big{]}L^{\prime}\Big{)}\odot\Phi\Big{(}\big{(}i_{n_{1}}\otimes
d_{2}^{\prime}\big{)}+\Sigma_{12}\mathrm{diag}\big{\\{}\mathcal{D}\big{[}\Sigma_{22}\big{]}\big{\\}}^{-1/2}\Big{)}$

and

$\displaystyle\Psi^{-}\big{(}L;\mu_{1};\mu_{2};\Sigma_{11};\Sigma_{12};\Sigma_{22}\big{)}:=\mathbb{E}\Big{[}e^{X_{1}}\Big{(}\big{(}L-e^{X_{2}}\big{)}^{+}\Big{)}^{\prime}\Big{]}$
$\displaystyle=\Big{(}\mathbb{E}\big{[}e^{X_{1}}\big{]}L^{\prime}\Big{)}\odot\Phi\Big{(}-i_{n_{1}}\otimes
d_{2}^{\prime}-\Sigma_{12}\mathrm{diag}\big{\\{}\mathcal{D}\big{[}\Sigma_{22}\big{]}\big{\\}}^{-1/2}\Big{)}$
$\displaystyle-\Big{(}\mathbb{E}\big{[}e^{X_{1}}\big{]}\mathbb{E}\big{[}e^{X_{2}}\big{]}^{\prime}\Big{)}\odot
e^{\Sigma_{12}}\odot\Phi\Big{(}-\big{(}i_{n_{1}}\otimes
d_{1}^{\prime}\big{)}-\Sigma_{12}\mathrm{diag}\big{\\{}\mathcal{D}\big{[}\Sigma_{22}\big{]}\big{\\}}^{-1/2}\Big{)},$

where for each $i=1,2$,
$\mathbb{E}\big{[}e^{X_{i}}\big{]}=e^{\mu_{i}+1/2\mathcal{D}[\Sigma_{ii}]}$ is
the expectation of the multivariate log–normal random vector,
$d_{1}:=\big{(}\mu_{2}+\mathcal{D}[\Sigma_{22}]-\ln(L)\big{)}\oslash\sqrt{\mathcal{D}[\Sigma_{22}]}$,
$d_{2}:=d_{1}-\sqrt{\mathcal{D}[\Sigma_{22}]}$, 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, Battulga23c. ∎

##  8 Conclusion

In this paper, we introduced a dynamic Gordon growth model, augmented by a
spot interest rate, which is modeled by the unit–root process with drift and
dividends, which are modeled by the Gordon growth model. It is assumed that
the regime–switching process is generated by a homogeneous Markov process.
Using the risk–neutral valuation method and locally risk–minimizing strategy,
we obtained pricing and hedging formulas for the dividend–paying European call
and put options, Margrabe exchange options, segregated funds, and unit–linked
life insurance products. Finally, we provided ML estimator of the model under
the assumption that there is one regime and coefficient matrices are
deterministic.

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