The Log Private Company Valuation Model
Battulga Gankhuu∗
arXiv:2206.09666v1 [q-fin.MF] 20 Jun 2022
Abstract
For a public company, the option pricing models, hedging models, and pricing models of equity–
linked life insurance products have been developed. However, for a private company, because of
unobserved price, the option and the life insurance pricing, and the hedging are challenging tasks.
For this reason, this paper introduces a log private company valuation model, which is based on
the dynamic Gordon growth model. In this paper, we obtain closed–form option pricing formulas,
hedging formulas, and net premium formulas of equity–linked life insurance products for a private
company. Also, the paper provides ML estimators of our model, EM algorithm, and valuation
formula for the private company. The suggested model can be used not only by private companies
but also by public companies
Keywords: Private company, dynamic Gordon growth model, option pricing, life insurance,
locally risk–minimizing strategy, ML estimators.
1
Introduction
Dividend discount models (DDMs), first introduced by Williams (1938), are popular tools for stock
valuation. The basic idea is that the market price of a stock is equal to the present value of sum of
dividend paid by the firm and price of the firm, which correspond to the next period. As the outcome
of DDMs depends crucially on dividend forecasts, most research in the last few decades has been
around the proper model a dividend dynamics. An interesting review of some existing DDMs that
include deterministic and stochastic models can be found in d’Amico and De Blasis (2020).
Existing stochastic DDMs have one common disadvantage: If dividends have chances to take
negative values, then the stock price of the firm can takes negative value with a positive probability,
which is the undesirable property for a stock price. A log version of the stochastic DDM, which is
called by dynamic Gordon growth model was introduced by Campbell and Shiller (1988), 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.
Black–Scholes 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. For a public company, the European option pricing models have been developed. However,
since stock prices of a private company are unobserved, pricing of the European call and put options
is a challenging task for the private company. For the private company, to motivate employees to
adopt a personal investment in the company’s success, frequently offered the call option, giving the
holder the right to purchase shares of stock at a specified price of the company. Also, to motivate the
employee the private company may offer equity–linked life insurance products to the workers.
∗
Department of Applied Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia; E-mail: battulga.g@seas.num.edu.mn
1
In this paper, we aim to contribute the following fields: (i) to price the European options, (ii) to
connect the equity–linked life insurance products with the private company, (iii) to hedge the options
and life insurance products, and (iv) to estimate the model’s parameter and to value the private
company. To estimate a required rate of return of a private company, Battulga (2022b) considered
the (iv)th field. But the (i)–(iii) fields still have not been explored before. Therefore, the novelty of
the paper is threefold.
The rest of the paper is organized as follows: In Section 2, we introduce a log private company
valuation model. In Section 3, we obtain closed–form Black–Scholes call and put options pricing
formula. Section 4 provides connections between the log private company valuation model and equity–
linked life insurance products. In Section 6, we consider locally risk–minimizing strategies for the
options and equity–linked life insurance products. Section 7 is dedicated to ML estimators of the
model’s parameters and an estimator of the value of a private company. In Section 8, we conclude the
study. Finally, in Technical Annex, we provide Lemmas, which are useful for the pricing and hedging,
and their proofs.
2
Log Private Company Valuation Model
Let (Ω, GTx , P) be a complete probability space, where P is a given physical or real–world probability
measure, and GTx will be defined below. Dividend discount models (DDMs), first introduced by
Williams (1938), are a popular tool for stock valuation. In this paper, we assume that a firm will
not default in the future. For a DDM with default risk, we refer to Battulga, Jacob, Altangerel, and
Horsch (2022). The basic idea of all DDMs is that the market price at time t − 1 of stock of the
firm equals the sum of the stock price at time t and dividend at time t discounted at risk–adjusted
rates (required rate of return on stock). Therefore, for successive prices of the company, the following
relation holds
Pt = (1 + k)Pt−1 − dt , t = 1, 2, . . . ,
(1)
where k is the required rate of return, Pt is the stock price, and dt is the dividend, respectively, at
time t of the company. Note that following the idea in Battulga (2022b) one can model the required
rate of return by a linear equation, which depends economic variables.
As mentioned above, if dividends have chances to take negative values, then the stock price of
the firm can take negative value with a positive probability, which is the undesirable property for a
stock price. For this reason, we follow the idea in Campbell and Shiller (1988). As a result, the stock
price of the company takes positive value. Following the idea in Campbell and Shiller (1988), one can
obtain the following approximation
gt − 1 ˜
(2)
dt − P̃t − µt
ln 1 + exp{d˜t − P̃t } ≈ ln(gt ) +
gt
where P̃t := ln(Pt ) is a log price at time t, d˜t := ln(dt ) is a log dividend at time t, gt := 1 + exp{µt }
is a linearization parameter, and µt := E d˜t − P̃t F0 is a mean log dividend–to–price ratio at time t
of the company, where F0 is an initial information, see below. Therefore, we have
gt − 1 ˜
Pt + dt
≈ exp P̃t − P̃t−1 + ln(gt ) +
dt − P̃t − µt ,
(3)
exp{k̃} =
Pt−1
gt
where k̃ := ln(1 + k) is a log required rate of return. As a result, for the log price at time t, the
following approximation holds
P̃t ≈ (P̃t−1 − d˜t + k̃)gt + d˜t − ht .
(4)
where ht := gt ln(gt ) − µt + µt is a linearization parameter and the model is called by dynamic
Gordon growth model, see Campbell and Shiller (1988). To estimate parameters of the dynamic
2
Gordon growth model, to price Black–Scholes call and put options and equity–linked life insurance
products, and to hedge the options and life insurance products, which will appear in following sections
we must add a random component, namely, ut , to equation (4). In this case, equation (4) becomes
P̃t = (P̃t−1 − d˜t + k̃)gt + d˜t − ht + ut .
(5)
Let Bt be a book value of equity, Rt be a return of equity (ROE), bt be a book value growth rate,
and αt be a payout ratio, respectively, at time t of the company. Since the book value of equity at
time t − 1 grows at rate bt , its log value at time t becomes
ln(Bt ) = b̃t + ln(Bt−1 ).
(6)
b̃t := ln(1 + bt ) is a log book value growth rate. On the other side, as dividend payment at time t is
a product of the dividend payout ratio at time t, αt , and earning at time t, Rt Bt−1 , we have
˜ t + ln(Bt−1 ),
d˜t = ∆
(7)
˜ t := ln(αt Rt ) = d˜t −ln(Bt−1 ) is a log dividend–to–book ratio at time t. To obtain the dynamic
where ∆
Gordon growth model corresponding to a private company, let m̃t := ln(Pt /Bt ) be a log price–to–book
ratio at time t of the private company, and k̃ := ln(1 + k) be the log required rate of return on the
stock of the private company. It should be noted that by using the augmented Dickey–Fuller test
one can easily confirm that for the quarterly S&P 500 index, its log price–to–book ratio is governed
by the unit–root process with drift. For this reason, we assume that the log price–to–book ratio of
the company follows the unit–root process with drift, that is, m̃t = φ + m̃t−1 + vt . If we substitute
˜ t + ln(Bt−1 ) into equation (5),
equations P̃t = m̃t + b̃t + ln(Bt−1 ), P̃t−1 = m̃t−1 + ln(Bt−1 ), and d˜t = ∆
then the dynamic Gordon growth model for the private company is given by the following system
(
b̃t = −m̃t + gt m̃t−1 + ct + ut
for t = 1, . . . , T
(8)
m̃t = φ + m̃t−1 + vt
˜ t − ht is a deterministic process.
under the real probability measure P, where ct := k̃gt − (gt − 1)∆
Note that if the firm does not pay dividends, then the system is given by
(
b̃t = −m̃t + m̃t−1 + k̃ + ut
for t = 1, . . . , T
(9)
m̃t = φ + m̃t−1 + vt
under the real probability measure P, see Battulga (2022b). If we combine the systems, one obtains a
private company valuation model (system), which models a company whose dividends are sometimes
paid and sometimes not paid. Let us denote the indicator random variable by 1A for a generic event
A ∈ GTx . Then, the combined system is
(
b̃t = −m̃t + g̃t m̃t−1 + c̃t + ut
for t = 1, . . . , T
(10)
m̃t = φ + m̃t−1 + vt
under the real probability measure P, where g̃t = 1{dt =0} + gt 1{dt >0} and c̃t := k̃1{dt =0} + ct 1{dt >0} are
deterministic processes. Henceforth, we refer the system as a log private company valuation model
(system). The model can be extended by adding a equation, which models dividend–to–book ratio,
say, VAR(p) process that depend on economic variables in system (10), see Battulga (2022a).
The stochastic properties of the systems (8)–(10) are governed by the random variables {u1 , . . . , uT ,
v1 , . . . , vT , m0 }. Throughout the paper, we assume that the error random variables ut and vt for
t = 1, . . . , T and the initial log book–to–price ratio m̃0 are mutually independent, and follow normal
distribution, namely,
m̃0 ∼ N (µ0 , σ02 ),
ut ∼ N (0, σu2 ),
vt ∼ N (0, σv2 ),
under the real probability measure P.
3
for t = 1, . . . , T
(11)
3
Option Pricing
Black and Scholes (1973) 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
dividend into account. For public companies, most stock options traded on the option exchange pay
dividends at least once before they expire and for most private companies, they also pay dividends.
Therefore, it is important to develop formulas for the European call and put options on dividend–
paying stocks from a practical point of view. Merton (1973) 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 for private and public
companies, where the dividend process is discrete and correlated with the stock level. To price the
Black–Scholes call and put options, we use the risk–neutral valuation method.
Let T be a time to maturity of the Black–Scholes call and put options, and for t = 1, . . . , T ,
ξt := (ut , vt )′ be a (2 × 1) random error process of system (10). According to equation (11), ξ1 , . . . , ξT
is a random sequence of independent identically multivariate normally distributed random vectors
with means of (2 × 1) zero vector and covariance matrices of (2 × 2) matrix Σ := diag{σu2 , σv2 }.
Therefore, a distribution of a residual random vector ξ := (ξ1′ , . . . , ξT′ )′ is given by
ξ ∼ N 0, IT ⊗ Σ ,
(12)
where ⊗ is the Kronecker product of two matrices.
Let x := (x′1 , . . . , x′T )′ be a (2T × 1) vector, which consists of all book value growth rates and
dividend–to–book ratios of a company and whose t–th sub–vector is xt := (b̃t , m̃t )′ . We define σ–
˜ 1, . . . , ∆
˜ T ) and for t = 1, . . . , T , Ft :=
fields, which play major roles in the paper: F0 := σ(B0 , ∆
F0 ∨ σ(b̃1 , . . . , b̃t ) and Gt := Ft ∨ σ(m̃0 , . . . , m̃t ), where for generic σ–fields O1 and O2 , O1 ∨ O2 is
the minimal σ–field containing them. Note that σ(B0 , b̃1 , . . . , b̃t ) = σ(B0 , B1 , . . . , Bt ), the σ–field Ft
represents available information at time t for a private company, the σ–field Gt represents available
information at time t for a public company, and the σ-fields satisfy Ft ⊂ Gt for t = 0, . . . , T . For the
public company, to price and hedge we have to use the information Gt . Therefore, one obtain pricing
and hedging formulas for the public company by replacing the information Ft , corresponding to the
private company by the information Gt . It follows from equation (12) that a joint density function of
the random vector x is given by
fx (x) = c exp
T
−1
1X
Qxt − Qt xt−1 − qt Σ Qxt − Qt xt−1 − qt
−
2
(13)
t=1
under the real probability measure P, where the constant is c := (2π)T 1|Σ|T /2 , the coefficient matrices
of the vectors xt and xt−1 are
1 1
0 g̃t
c̃
Q :=
and Qt :=
, and qt := t .
0 1
0 1
φ
To price the call and put options and equity–linked life insurance products, we need to change
from the real probability measure to some risk–neutral measure. Let Dt := 1/(1 + r)t be a discount
process, where r is a risk–free rate. According to Pliska (1997) (see also Bjork (2020)), a conditional
expectation of a return process (Pt + dt )/Pt−1 − 1 must equal the risk–free rate r under some risk–
neutral probability measure P̃ and a filtration {Gt }Tt=0 . Thus, it must hold
Pt + dt
Gt−1 = 1 + r
(14)
Ẽ
Pt−1
4
for t = 1, . . . , T , where Ẽ denotes an expectation under the risk–neutral probability measure P̃. If
we substitute equation (5) into approximation equation (3), then condition (14) is equivalent to the
following condition
(15)
Ẽ exp ut /g̃t − (r̃ − k̃) Gt−1 = 1,
where r̃ := ln(1 + r) is a log risk–free rate. It should be noted that condition (15) corresponds only
to the error random variable ut . Thus, we need to impose a condition on the error random variable
vt under the risk–neutral probability measure. This condition is fulfilled by Ẽ[exp{vt }|Gt−1 ] = θ̂t for
Gt−1 measurable any random variable θ̂t . Because for any admissible choices of θ̂t , condition (15)
holds, the market is incomplete. But prices of the options and equity–linked life insurance products
are still consistent with the absence of arbitrage. In this paper, we assume that a joint distribution of
the state variables m̃t , t = 0, . . . , T is the same for the real probability measure P and the risk–neutral
measure P̃. Thus, we require that
(16)
Ẽ exp vt − σv2 /2 Gt−1 = 1.
If we combine conditions (15) and (16), then we have
Ẽ exp Rt (ξt − θt ) Gt−1 = 1,
(17)
′
where Rt := diag{1/g̃t , 1} is a (2 × 2) diagonal matrix and θt := g̃t (r̃ − k̃), σv2 /2 is a (2 × 1)
deterministic Girsanov kernel process. To obtain the risk–neutral probability measure, we define the
following state price density:
t
Y
1
′ −1
′ −1
exp (θm − αm ) Σ ξm − (θm − αm ) Σ (θm − αm )
Lt :=
(18)
2
m=1
′
for t = 1, . . . , T, where αm := 21 σu2 /g̃m , σv2 is a (2 × 1) deterministic vector. Then, Lt is a martingale
with respect to the filtration {Gt }Tt=0 and the real probability measure P. Since LT > 0 and E[LT |F0 ] =
1, we can define the following new probability measure:
Z
LT (z)fx (x)dx1 , . . . dxT
P̃ x ∈ B =
=
Z
c exp
B
B
T
−1
1X
−
Qxt − Qt xt−1 − q̃t + αt Σ Qxt − Qt xt−1 − q̃t + αt
2
(19)
t=1
where B ∈ B(R2T ) is an any Borel set, fx (x) is the joint density function of the random vector x
given by equation (13), and LT is the state price density process at time T given by equation (18).
Therefore, the log private company valuation system (10) becomes
(
b̃t = −m̃t + g̃t m̃t−1 + c̃∗t + ũt
for t = 1, . . . , T,
(20)
m̃t = φ + m̃t−1 + ṽt
˜ t + r̃g̃t −ht − 1 σu2 is a deterministic
under the risk–neutral probability measure P̃, where c̃∗t := (1−g̃t )∆
2g̃t
′
′
T
′
˜
˜
˜
process, and a residual random vector ξ := (ξ̃ , . . . , ξ ) with ξt := (ũt , ṽt ) , t = 1, . . . , T has the same
1
T
distribution as the residual random vector ξ, that is,
ξ˜ ∼ N 0, IT ⊗ Σ
(21)
under the risk–neutral probability measure P̃. Comparing the two systems (10) and (20), one can
deduce that the log required rate of return changed from k̃ to the log risk–free rate r̃, and an additional
term 2g̃1t σu2 arises. Observe that the first line of system (20) is equivalent to
P̃t = (P̃t−1 − d˜t + r̃)g̃t + d˜t − ht 1{dt >0} −
5
1 2
σ + ut .
2g̃t u
(22)
Let us rewrite system (20) in the following form
(
b̃t = ψt′ zt + c̃∗t + ũt
zt = Azt−1 + a + ηt
for t = 1, . . . , T,
(23)
where zt := (m̃t , m̃t−1 )′ is a (2 × 1) state vector of the log price–to–book ratios at times t and t − 1,
ψt := (−1, g̃t )′ is a (2 × 1) deterministic vector, a := (φ, 0)′ is a (2 × 1) constant vector, ηt := (ṽt , 0)′
is a (2 × 1) random vector, and
1 0
A :=
1 0
is a (2 × 2) constant matrix. Observe that m̃t = e′1 zt , where e1 = (1, 0)′ is a (2 × 1) unit vector. If we
repeatedly use the process zt , which is given in system (23), then one get that
zt+i = Ai zt + (I2 − A)−1 (I2 − Ai )a + Ai−1 ηt+1 + · · · + Aηt+i−1 + ηt+i
for i ≥ 1, where for the matrix A and vector a, the following equations hold
1 0
iφ
i
−1
i
A =
and (I2 − A) (I2 − A )a =
1 0
(i − 1)φ
(24)
(25)
for i ≥ 1. As a result, for each i, j = 1, 2, . . . and the information Gt , a conditional expectation of the
state vector at time t + i and a covariance matrix between the state vectors at times t + i and t + j
are obtained as
m̃t + iφ
Ẽ[zt+i |Gt ] =
(26)
m̃t−1 + (i − 1)φ
and
"

γij γij



 γ
g t+i , zt+j |Gt ] = σ 2 × " ij γij
Cov[z
v

γij



 γ −1
ij
#
−1
−1
γij − 1
γij − 1
if
#
if
i 6= j
,
(27)
i=j
where γij := min(i, j) is a minimum of i and j. Consequently, as m̃t = e′1 zt , it follows from system
(23) and equation (26) that for each i = 1, . . . , T − t, conditional on Ft expectations of the log state
variable and the log book value growth rate are given by
Ẽ[m̃t+i |Gt ] = m̃t + iφ
(28)
and
Ẽ[b̃t+i |Gt ] = m̃t (g̃t+i − 1) + φ (i − 1)g̃t+i − i + c̃∗t+i
(29)
The log price at time T can be written in terms of the log price–to–book ratio at time T , the log
book value growth rates at times t + 1, . . . , T , and the log book value at time t, that is,
P̃T = m̃T + b̃t+1 + · · · + b̃T + ln(Bt )
(30)
′ , . . . , ξ )′ follows multivariate normal distribution
for t = 0, . . . , T − 1. Since a random vector (ξt+1
T
with mean zero and covariance matrix IT −t ⊗ Σ, conditional on the information Gt the log price at
time T follows normal distribution:
P̃T | Gt ∼ N µT |t (m̃t ), σT2 |t ,
(31)
6
where due to equations (28) and (29),
µT |t (m̃t ) := α̃T |t m̃t + β̃T |t
(32)
is a conditional expectation of the log price at time T , where
α̃T |t :=
and
β̃T |t :=
and
σT2 |t :=
X
T
T
X
i=t+1
g̃i − T + t + 1
T
X
(T − t − 1)(T − t)
(i − t − 1)g̃i −
c̃∗i + ln(Bt ),
φ+
2
i=t+1
T
X
i=t+1
(33)
(34)
i=t+1
g T , zi |Gt ]ψi +
e′1 Cov[z
T
T
X
X
i=t+1 j=t+1
g i , zj |Gt ]ψj + (T − t)σu2
ψi′ Cov[z
(35)
is a conditional variance of the log price at time T .
Therefore, according to equations (32) and (35), and Lemma 3 that is used to price the Black–
Scholes call and put options, see Technical Annex, conditional on the information Gt prices at time t
of the Black–Sholes call and put options are given by
+ CT |t (m̃t ) = e−(T −t)r̃ Ẽ PT − K
Gt
σT2 |t Φ(d1T |t ) − e−(T −t)r̃ KΦ(d2T |t ),
(36)
= exp µT |t (m̃t ) − (T − t)r̃ +
2
and
+ Gt
PT |t (m̃t ) = e−(T −t)r̃ Ẽ K − PT
−(T −t)r̃
= e
KΦ(−d2T |t )
σT2 |t Φ(−d1T |t ),
− exp µT |t (m̃t ) − (T − t)r̃ +
2
(37)
where d1T |t := µT |t (m̃t ) + σT2 |t − ln(K) /σT |t and d2T |t := d1T |t − σT |t . Note that equation (36) and
(37) are used to price call and put options for public companies because their price–to–book ratios
are known.
Since µT |t (m̃t ) ∼ N µ̃T |t , α̃2T |t (σ02 + tσv2 ) , where µ̃T |t := α̃T |t (µ0 + tφ) + β̃T |t is a mean of the
random variable µT |t (m̃t ) or equivalently, a mean of the random price P̃T given the information Ft ,
by using Lemma 1 for the call and put option formulas, which are given by equations (36) and (37),
we obtain that for the private company, prices at time t of the Black–Scholes call and put options are
given by the following equations
+ CT |t (K) = e−(T −t)r̃ Ẽ PT − K
Ft
2
σ̃T |t
Φ(d˜1T |t ) − e−(T −t)r̃ KΦ(d˜2T |t ),
(38)
= exp µ̃T |t − (T − t)r̃ +
2
and
+ PT |t (K) = e−(T −t)r̃ Ẽ K − PT
Ft
σ̃T2 |t −(T −t)r̃
2
Φ(−d˜1T |t ),
(39)
=e
KΦ(−d̃T |t ) − exp µ̃T |t − (T − t)r̃ +
2
respectively, where σ̃T2 |t := σT2 |t + α̃2T |t (σ02 + tσv2 ), d˜1T |t := µ̃T |t + σ̃T2 |t − ln(K) /σ̃T |t , and d˜2T |t :=
d˜1 − σ̃T |t .
T |t
7
4
Life Insurance Products
Now we consider a pricing of some equity–linked life insurance products using the risk–neutral measure.
Here we will price segregated fund contract with guarantee, see Hardy (2001) and unit–linked life
insurance with guarantee, see Aase and Persson (1994) and Møller (1998). We suppose that stock
represents some fund and an insured receives dividends from the fund. Let Tx be x aged insured’s
future lifetime random variable, Ttx = σ(1{Tx >s} : s ∈ [0, t]) be σ–field, which is generated by a
death indicator process 1{Tx ≤t} , Ft be an unit of the fund, and Gt be an amount of the guarantee,
respectively, at time t. We assume that the σ–fields GT and TTx are independent, and operational
expenses, which are deducted from the fund and withdrawals are omitted from the life insurance
products. A common life insurance product in practice is endowment insurance, and combinations of
a term life insurance and pure endowment insurance lead to interesting endowment insurances, see
Aase and Persson (1994). 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 form f (PT ) for some Borel function f . 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,
f (x) = 1, 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 Aase and Persson (1994), Bowers, Gerber, Hickman, Jonas, and Nesbitt (1997), and
Hardy (2001). As a result, a discounted contingent claim of the T –year pure endowment insurance
can be represented by the following equation
H T := DT f (PT )1{Tx >T } .
(40)
To price the contingent claim we define σ–fields Ftx := Ft ∨ Ttx for t = 1, . . . , T . Since the σ–fields
GT and TTx are independent, one can obtain that value at time t of a contingent claim f (PT )1{Tx >T }
is given by
1
1
Ẽ[H T |Ftx ] =
Ẽ[DT f (PT )|Ft ]T −t px+t ,
(41)
Vt =
Dt
Dt
where t px := P[Tx > 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 the pure endowment insurance, the term life insurance’s sum insured
depends on time t, that is, its sum insured form is g(Pt ) because random death occurs at any time in
T years. Therefore, a discounted contingent claim of the T –term life insurance is given by
H T := DKx +1 f (PKx +1 )1{Kx +1≤T } =
T
−1
X
k=0
Dk+1 f (Pt+k )1{Kx =k} ,
(42)
where Kx := [Tx ] 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 Gt and Ttx are independent that a value process at time t of the term insurance is
Vt =
T
−1
X
1
1
Ẽ[H T |Ftx ] =
Ẽ[Dk+1 f (Pk+1 )|Ft ]k−t px+t qx+k .
Dt
Dt
(43)
k=t
where t qx := P[Tx ≤ 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 (Pk ) = Fk Gk /Fk − Pk
8
for k = 1, .. . , T . On the
+ other hand, the sum insured forms of the unit–linked life insurance are
f (Pk ) = Fk Pk − Gk /Fk + Gk for k = 1, . . . , 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 Gk /Fk
and maturity k for k = t + 1, . . . , T .
To price the options for given Gt , we need conditional distributions of the prices at times k =
t + 1, . . . , T . If we replace T in equation (31) with k = t + 1, . . . , T , then one get that
2
P̃k | Gt ∼ N µk|t(m̃t ), σk|t
, k = t + 1, . . . , T,
(44)
2 are conditional mean and variance of the random variable P given information
where µk|t(m̃t ) and σk|t
k
Gt and are obtained by replacing T in equations (32) and (35) with k, respectively. Therefore,
analogous to equations (38) and (39), one can obtain that for k = t + 1, . . . , T ,
+ Ck|t Gk /Fk = e−(k−t)r̃ Ẽ Pk − Gk /Fk
Ft
2
σ̃k|t
Φ(d˜1k|t ) − e−(T −t)r̃ Gk /Fk Φ(d˜2k|t ),
= exp µ̃k|t − (T − t)r̃ +
2
(45)
and
+ Pk|t Gk /Fk = e−(k−t)r̃ Ẽ Gk /Fk − PT
Ft
2 σ̃k|t
−(k−t)r̃
2
˜
Φ(−d˜1k|t ),
=e
Gk /Fk Φ(−dT |t ) − exp µ̃k|t − (k − t)r̃ +
2
(46)
2 :=
respectively, where µ̃k|t := α̃k|t(µ0 + tφ) + β̃k|t is a mean of the random variable µk|t(m̃t ), σ̃k|t
2
2 + α̃2 (σ 2 + tσ 2 ), d˜1 := µ̃
˜1
˜2
σk|t
k|t + σ̃k|t − ln(g̃k /Fk ) /σ̃k|t , and dk|t := dk|t − σ̃k|t .
v
k|t
k|t 0
Consequently, from equations (45) and (46) 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
T
−1
X
1
S x+t:T
(47)
Fs+1 Pk+1|t Gk+1 /Fk+1 k−t px+t qx+k ;
=
−t
k=t
2. for the T –year guaranteed pure endowment insurance, corresponding to segregated fund contract, it holds
1
= FT PT |t GT /FT T −t px+t ;
(48)
S x+t:T −t
3. for the T –year guaranteed unit–linked term life insurance, it holds
1
=
U x+t:T
−t
T
−1
X
k=t
Fk+1 Ck+1|t Gk+1 /Fk+1 + Gk+1 k−t px+t qx+k ;
4. for the T –year guaranteed unit–linked pure endowment insurance, it holds
1
U x+t:T −t
= FT CT |t GT /FT + g̃T T −t px+t .
9
(49)
(50)
5
Locally Risk–Minimizing Strategy
By introducing the concept of mean–self–financing, Föllmer and Sondermann (1986) 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, Föllmer and
Schweizer (1989) developed a locally risk–minimizing strategy and obtained a recurrence formula for
optimal strategy. According to Schäl (1994) (see also Föllmer and Schied (2004)), 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 given in Section 4 and the equity–linked life
insurance products given in Section 5. In the insurance industry, for continuous–time unit–linked term
life and pure endowment insurances with guarantee, locally risk–minimizing strategies are obtained
by Møller (1998).
For t = 0, 1, . . . , T , let us denote a discounted stock price at time t by P t := Dt Pt , discounted
dividend at time t by dt := Dt dt , and a proper number of shares at time t by ht and a proper
amount of cash (risk–free bond) at time t by h0t , which are required to successfully hedge a generic
contingent claim HT , where we assume that the contingent claim HT is square–integrable under the
risk–neutral probability measure. Also, we denote a discounted contingent claim by H T := DT HT .
Then, according to Föllmer and Schied (2004) and Föllmer and Schweizer (1989), for the filtration
{Ftx }Tt=0 and the contingent claim HT , under the risk–neutral measure P̃ the locally risk–minimizing
strategy (h0 , h) is given by the following equations:
ht+1 =
Λt+1
σ 2t+1
and h0t+1 = Vt+1 − ht+1 (Pt+1 + dt+1 )
(51)
1
x
is a value process of the
for t = 0, . . . , T − 1 and h00 = V0 − h1 P0 , where Vt+1 := Dt+1
Ẽ H T Ft+1
2
g H T , P t+1 + dt+1 − P t Ftx .
contingent claim HT , σ 2t+1 := Ẽ P t+1 + dt+1 − P t Ftx and Λt+1 := Cov
Note that ht is a predictable process, which means its value known at time t − 1, while for the process
h0t , its value only known at time t, and if the contingent claim HT is generated by stock prices Pt
and dividends dt for t = 0, . . . , T , then the process h0t becomes predictable, see Föllmer and Schweizer
(1989). For t = 1, . . . , T , since σ–fields Gt and Ttx are independent, if X is any random variable, which
is independent of σ–field Ttx and integrable with respect to the risk–neutral measure, then it holds
E[X|Gtx ] = E[X|Gt ],
(52)
where Gtx := Gt ∨ Ttx is the minimal σ–field that contains the σ–fields Gt and Ttx . 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, namely, πt+1 := Pt+1 + dt+1 conditional on Gt . If we
substitute equation (22) into the approximation equation (3), then we have
σu2 σu2
σu2
ũt
Pt + dt
∼ N r̃ − 2 , 2
≈ r̃ − 2 +
(53)
ln
Pt−1
g̃t
2g̃t
2g̃t g̃t
under the risk–neutral probability measure P̃. Therefore, one get that
2 2
2 Pt+1 + dt+1
Pt
2
Gt
Ẽ
− (1 + r)
σ t+1 (m̃t ) := Ẽ P t+1 + dt+1 − P t Gt =
(1 + r)2
Pt
2 n
o
σu
= D2t exp
.
−
1
exp
2
m̃
+
ln(B
)
t
t
2
g̃t+1
Consequently, as 2m̃t ∼ N 2(µ0 + tφ), 4(σ02 + tσv2 ) , one obtain that
2 o
n σu
2
2
2
.
σ t+1 = D2t exp
+
tσ
−
1
exp
2
µ
+
tφ
+
ln(B
)
+
σ
0
t
0
v
2
g̃t+1
10
(54)
(55)
g H T , P t+1 + dt+1 − P t G x for
We define a conditional covariance given Gtx : Λt+1 (Gt ) := Cov
t
the generic contingent claim HT . Then, because under the risk–neutral probability measure P̃, a
conditional expectation of the sum random variable πt+1 is (1 + r)Pt , and the σ–fields Gt and Ttx are
independent, the conditional covariance is
a) for the Black–Scholes call and put options,
1
Λt+1 (Gt ) = D2t+1 Ẽ
H T Pt+1 + dt+1 − (1 + r)Pt Gt ,
Dt
b) for the equity–linked pure endowment insurances,
DT
f (PT ) Pt+1 + dt+1 − (1 + r)Pt Gt T −t px+t ,
Λt+1 (Gt ) = D2t+1 Ẽ
Dt
(56)
(57)
c) and for the equity–linked term life insurances,
Λt+1 (Gt ) = D2t+1
T
−1
X
k=t
Dk+1
f (Pk+1 ) Pt+1 + dt+1 − (1 + r)Pt Gt k−t px+t qx+k .
Ẽ
Dt
(58)
In order to obtain the locally risk–minimizing strategies for the Black–Scholes call and put options,
and the equity–linked life insurance products, we need to calculate the conditional covariances given
in equations (56)–(58) for contingent claims HT = [PT − K]+ and HT = [K − PT ]+ , and sum insureds
f (Pk ) = Fk [Pk − Gk /Fk ]+ + Gk and f (Pk ) = Fk [Gk /Fk − Pk ]+ for k = t + 1, . . . , T . Thus, we need
Lemma 4, see Technical Annex.
To use Lemma 4 for the options and equity–linked life insurance products, for given Gt , we need
conditional covariances between the log price at time k, k = t + 1, . . . , T and a log of the sum
random variable at time t + 1. Since the random variable ũt+1 is independent of the random variables
ũt+2 , . . . , ũT and ṽt+1 , . . . , ṽT and log price at time k is represented by P̃k = m̃k + b̃t+1 +· · ·+ b̃k +ln(Bt ),
according to system (20) and equation (24), we get that
g P̃k , ln Pt+1 + dt+1 |Gt = σu2 /g̃t+1
σP̃k ,π̃t+1|t := Cov
(59)
where for k = t + 1, . . . , T .
Consequently, it follows from Lemma 4 and equations (56)–(58) that for t = 0, . . . , T − 1, Λt+1 s,
which are correspond to the call and put options and 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
σ̃T2 |t σu2
σu2
1
˜
+
Φ dT |t +
Λt+1 = D2t exp µ̃T |t − (T − t)r̃ +
g̃t+1
2
g̃t+1
2
σ
−CT |t (K) − e−(T −t)r̃ KΦ d˜2T |t + u
;
g̃t+1
2. for the dividend–paying Black–Scholes put option on the weighted asset price, we have
σu2
−(T −t)r̃
2
˜
− PT |t (K)
Λt+1 = D2t e
KΦ − dT |t −
g̃t+1
2
σ̃T2 |t σu2
σ
u
1
+
;
− exp µ̃T |t − (T − t)r̃ +
Φ − d˜T |t −
g̃t+1
2
g̃t+1
11
(60)
(61)
3. for the T -year guaranteed term life insurance, corresponding to a segregated fund contract, we
have
σu2
2
˜
− Pk+1|t Gk+1 /Fk+1
Λt+1 = −D2t
Fk+1 e
KΦ − dk+1|t −
g̃t+1
k=t
2
σ̃k+1|t
σu2
σu2
1
˜
− exp µ̃k+1|t − (k + 1 − t)r̃ +
+
Φ − dk+1|t −
k−t px+t qx+k ; (62)
g̃t+1
2
g̃t+1
T
−1
X
−(k+1−t)r̃
4. for the T -year guaranteed pure endowment insurance, corresponding to a segregated fund contract, we have
σu2
−(T −t)r̃
2
˜
− PT |t GT /FT
Λt+1 = −D2t FT e
KΦ − dT |t −
g̃t+1
2
σ̃T |t
σu2
σu2
1
˜
(63)
+
Φ − dT |t −
− exp µ̃T |t − (T − t)r̃ +
T −t px+t ;
g̃t+1
2
g̃t+1
5. for the T -year guaranteed unit–linked term life insurance, we have
T
−1 X
2
σ̃k+1|t
σu2
σu2
1
˜
Λt+1 = D2t
Fk+1 exp µ̃k+1|t − (k + 1 − t)r̃ +
+
Φ dk+1|t +
g̃t+1
2
g̃t+1
k=t
σ2
−Ck+1|t Gk+1 /Fk+1 − e−(k+1−t)r̃ KΦ d˜2k+1|t + u
+ Gk+1 k−t px+t qx+k ;
(64)
g̃t+1
6. and for the T -year guaranteed unit–linked pure endowment insurance, we have
σ̃T2 |t σ2
σ2
Λt+1 = D2t FT exp µ̃T |t − (T − t)r̃ + u +
Φ d˜1T |t + u
g̃t+1
2
g̃t+1
2
σ
+ GT T −t px+t ,
−CT |t GT /FT − e−(T −t)r̃ KΦ d˜2T |t + u
g̃t+1
(65)
where for equations (60) and (61), d˜1T |t := µ̃T |t + σ̃T2 |t − ln(K) /σ̃T |t and d˜2T |t := d˜1T |t − σ̃T |t , and for
2 − ln G /F
˜2
˜1
equations (62)–(65), d˜1k|t := µ̃k|t + σ̃k|t
k
k /σ̃k|t and dk|t := dk|t − σ̃k|t for k = t + 1, . . . , T .
As a result, by substituting equations (55) and (60)–(65) into (51) we can obtain the locally risk–
minimizing strategies for the Black–Scholes call and put options and the equity–linked life insurance
products corresponding to the private company.
6
The Kalman Filtering
Let us reconsider the log private company valuation model (10). The system can be written by
(
b̃t = ψt′ zt + c̃t + ut
for t = 1, . . . , T,
(66)
zt = Azt−1 + a + ηt
′
where zt := (m̃
of the price–to–book ratios at times t and t − 1,
′ t , m̃t−1 ) is a (2 × 1) state vector
ψt := − 1, g̃t is a (2 × 1) vector, a := (φ, 0)′ is a (2 × 1) vector, ηt := (vt , 0)′ is a (2 × 1) random
vector, and
1 0
A :=
1 0
12
is a (2 × 2) matrix. For system (66), its first line determines the measurement equation and the
second line determines the transition equation. For each t = 0, . . . , T , conditional on the information
Ft , conditional expectations and covariance matrices of the log book value growth rates and the state
vectors are recursively obtained by the Kalman filtering (see Hamilton (1994) and Lütkepohl (2005)):
• Initialization:
– Expectation
z0|0 := E[z0 |F0 ] = (µ0 , µ0 )′
(67)
Σ(z0 |0) := Cov[z0 |F0 ] = diag{σ02 , σ02 }
(68)
– Covariance
• Prediction step: for t = 1, . . . , T ,
– Expectations
zt|t−1 := E(zt |Ft−1 ) = Azt−1|t−1 + a
b̃t|t−1 := E(b̃t |Ft−1 ) =
ψt′ a
+ c̃t +
(69)
ψt′ Azt−1|t−1
(70)
– Covariances
Σ(zt |t − 1) := Cov[zt |Ft−1 ] = AΣ(zt−1 |t − 1)A′ + Ση
Σ(b̃t |t − 1) := Var[b̃t |Ft−1 ] =
ψt′ Σ(zt |t
− 1)ψt +
σu2
(71)
(72)
• Correction step: for t = 1, . . . , T ,
– Expectations
zt|t := E[zt |Ft ] = zt|t−1 + Kt b̃t − b̃t|t−1
– Covariances
Σ(zt |t) := Cov[zt |Ft ] = Σ(zt |t − 1) − Kt Σ(b̃t |t − 1)Kt′ ,
(73)
(74)
where Kt := Σ(zt |t − 1)ψ/Σ(b̃t |t − 1) is the Kalman filter gain.
For each t = T + 1, T + 2, . . . , conditional on the information FT , conditional expectations and
covariance matrices of the log book value growth rates and the state vectors are recursively obtained
by (see Hamilton (1994) and Lütkepohl (2005)):
• Forecasting step: for t = T + 1, T + 2, . . . ,
– Expectations
zt|T
b̃t|T
:= E[zt |FT ] = Azt−1|T + a
(75)
:= E[b̃t |FT ] = ψt′ zt|T + c̃t
(76)
– Covariances
Σ(zt |T ) := Cov[zt |FT ] = AΣ(zt−1 |T )A′ + Ση
Σ(b̃t |T ) := Var[b̃t |FT ] =
ψt′ Σ(zt |T )ψt
+
σu2
(77)
(78)
The Kalman filtering, which is considered above provides an algorithm for filtering of the state
vector of price–to–book ratio zt , which is unobserved variable. To estimate parameters of our model
(10), in addition to the Kalman filtering, we also need to make inference about the state vector of price–
to–book ratio zt for each t = 1, . . . , T based on the full information FT , see below. Such an inference
is called the smoothed estimate of the state vector of price–to–book ratio zt . The smoothed inference
of the state vector can be obtained by the following Kalman smoother recursions, see Hamilton (1994)
and Lütkepohl (2005).
13
• Smoothing step: for t = T − 1, T − 2, . . . , 0,
– Expectations
zt|T := E[zt |FT ] = zt|t + St zt+1|T − zt+1|t
– Covariances
Σ(zt |T ) := Cov[zt |FT ] = Σ(zt |t) − St Σ(zt+1 |t) − Σ(zt+1 |T ) St′ ,
(79)
(80)
where St := Σ(zt |t)A′ Σ−1 (zt+1 |t) is the Kalman smoother gain. Also, it can be shown that
Σ(zt , zt+1 |T ) := Cov[zt , zt+1 |FT ] = St Σ(zt+1 |T ),
(81)
see Battulga (2022b).
In the EM algorithm, one considers a joint density function of a random vector, which is composed
of observed variables and state variables. In our cases, the vectors of observed variables and the state
variables correspond to a vector of the dividend–to–book ratios, b̃ := (b̃1 , . . . , b̃T )′ , and a vector of the
price–to–book ratios, m̃ := (m̃0 , . . . , m̃T )′ , respectively. Interesting usages of the EM algorithm in
econometrics can be found in Hamilton (1990) and Schneider (1992). Let us denote the joint density
function by fb̃,m̃ (b̃, m̃). The EM algorithm consists of two steps. In the expectation (E) step of the
EM algorithm, one has to determine a form of an expectation of log of the joint density
given the
full
information FT . We denote the expectation by Λ(θ|FT ), that is, Λ(θ|FT ) := E ln fb̃,m̃ (b̃, m̃) |FT .
For our log private company valuation model (10), one can show that the expectation of log of the
joint density of the vectors of the dividend–to–book ratios b̃ and the price–to–book ratios m̃ is
Λ(θ|FT ) = E ln fb̃,m̃ (b̃, m̃) FT
h
i
X
2
T
1
2T + 1
ln(2π) − ln(σu2 ) − 2
E b̃t + m̃t − m̃t−1 − k̃ FT
=−
2
2
2σu
{t|dt =0}
i
h
X
1
˜ t − (m̃t−1 − ∆
˜ t + k̃)gt + ht 2 FT
− 2
(82)
E b̃t + m̃t − ∆
2σu
{t|dt >0}
−
T
i 1
i
2
2
1 X h
1 h
T
E mt − φ − mt−1 FT − ln(σ02 ) − 2 E m0 − µ0 FT ,
ln(σv2 ) − 2
2
2σv
2
2σ0
t=1
′
where θ := k, φ, µ0 , σu2 , σv2 , σ02 is a (6 × 1) vector, which consists of all parameters of the model (10).
If the dividend is paid at time t, observe that the log dividend–to–price ratio is given by d˜t −
˜ t − b̃t − m̃t . Thus, due to system (8), the log dividend–to–price ratio is represented by
P̃t = ∆
˜
˜ t + k̃t )gt − ut . Therefore, as µt = E[d˜t − P̃t |F0 ], we get that
dt − P̃t = ht − (m̃t−1 − ∆
˜ t − k̃t − E[m̃t−1 |F0 ] gt .
µt = ht + ∆
(83)
Consequently, because ht = gt ln(gt ) − µt + µt and E[m̃t−1 |F0 ] = µ0 + (t − 1)φ, one obtain that
gt =
1
,
1 − exp{ϕt }
(84)
˜ t − k̃ − µ0 + (t − 1)φ . Further, since µt = ln(gt − 1) = ϕt + ln(gt ), we get that
where ϕt := ∆
ϕt exp{ϕt }
ht = −
+ ln 1 − exp{ϕt } .
(85)
1 − exp{ϕt }
14
Thus, partial derivatives of the linearization parameters gt and ht are obtained as
∂gt
∂ϕt
= (gt − 1)gt
∂α
∂α
and
∂ht
∂ϕt
= −ϕt (gt − 1)gt
,
∂α
∂α
(86)
where α ∈ {k̃, µ0 , φ}. The partial derivatives of the linerization parameters will be used to obtain
parameter estimators.
In the maximization (M) step of the EM algorithm, we need to find a maximum likelihood estimator
θ̂ that maximizes the expectation, which is determined in the E step. Taking partial derivatives from
Λ(θ|FT ) with respect to the parameters and setting these partial derivatives to zero gives the maximum
likelihood estimators as
P
P
{t|dt >0} ft (gt − 1) + ut|T gt +
{t|dt =0} bt + m̃t|T − m̃t−1|T
ˆ
P
k̃ :=
,
(87)
2 {t|dt >0} µ0 + (t − 1)φ − m̃t−1|T (gt − 1)gt2 + T0
φ̃ˆ :=
σ2
1
m̃T |T − m̃0|T + v2
T
σu
µ̂0 := m̃0|T +
and
σ̂u2 :=
σ02
σu2
X
{t|dt >0}
X
{t|dt >0}
ft (t − 1)(gt − 1)gt ,
(88)
2
σ̂02 := σ0|T
,
(89)
ft (gt − 1)gt ,
T
1X
E u2t |FT ,
T t=1
σ̂v2 :=
T
1X
E vt2 |FT ,
T t=1
where T0 := # t|dt = 0, t = 1, . . . , T is a number of non dividend paying times of the firm,
m̃t−1,t−1|T
m̃t−1,t|T
˜
˜
− ∆t −
− ∆t + k̃t gt + ht m̃t−1|T ,
ft := ut|T µ0 + (t − 1)φ − b̃t +
m̃t−1|T
m̃t−1|T
(90)
(91)
ut|T := b̃t − ψt′ zt|T − c̃t is a smoothed residual of the random error ut , m̃t|T := e′1 zt|T is the first
component of the smoothed inference vector zt|T , m̃t−1,t−1|T := E[m̃t−1 m̃t−1 |FT ] = e′1 (Σ(zt−1 |T ) +
′
zt−1|T zt−1|T
)e1 is an expectation of a product of the state variables, m̃t−1 m̃t−1 , given the full infor′ )e is
mation FT , and due to equation (81), m̃t−1,t|T := E[m̃t−1 m̃t |FT ] = e′1 (St−1 Σ(zt |T ) + zt−1|T zt|T
1
an expectation of a product of the state variables, m̃t−1
m̃
,
given
the
full
information
F
.
t
T
To calculate the conditional expectations E u2t |FT and E vt2 |FT , let vt|T = e′1 (zt|T − Azt−1|T − a)
be a smoothed residual at time t of the error random variable vt . The random errors at time t of the
log book value growth rate and the log price–to–book ratio can be represented by
ut = ut|T − ψt′ (zt − zt|T )
vt = vt|T + e′1 (zt − zt|T ) − e′1 A(zt−1 − zt−1|T ).
(92)
Therefore, as ut|T and vt|T are known at time T (measurable with respect to the full information FT ),
it follows from equations (81) that
= u2t|T + ψt′ Σ(zt |T )ψt
E u2t |FT
2
= vt|T
+ e′1 Σ(zt |T )e1 + e′1 AΣ(zt−1 |T )A′ e1 − 2e′1 ASt−1 Σ(zt |T )e1 ,
(93)
E vt2 |FT
c.f. Schneider (1992). If we substitute equation (93) into (90), then under suitable conditions the
zig–zag iteration that corresponds to equations (67)–(74), (79), (80), and (87)–(90) converges to the
15
maximum likelihood estimators of our log private company valuation model. As a result, an estimator
of the value at time t of the private company is calculated by the following formula
P̂t = mt|T Bt ,
t = 0, 1, . . . , T,
(94)
where mt|T = exp{m̃t|T } is a smoothed price–to–book ratio at time t. Also, an analyst can forecast the
value of the private company by using equations (75) and (76). It should be noted that the log private
company valuation model we consider in this section can be used not only by private companies but
also by public companies.
7
Conclusion
For a private company, because of unobserved price, the option and the life insurance pricing, and the
hedging are challenging tasks. For this reason, this paper introduces a log private company valuation
model, which is based on the dynamic Gordon growth model. To the best of our knowledge, this is
the first attempt to introduce an option pricing model for a private company. For a public company,
the traditional option valuation model concentrate on observed price. In this paper, we introduce an
option pricing model that is based on book values and price–to–book ratios for the public company.
Since most private (public) companies pay dividends at least once before they expire, we consider the
dividend–paying Black–Scholes call and put options and obtain closed–form pricing formulas for the
options. Next, under the assumption that the unobserved (observed) stock price represents a fund
and an insured receives dividends from the fund, we obtain pricing formulas for the equity–linked
life insurance products, which consist of the unit–linked term life and pure endowment insurance
products, and the term life and pure endowment insurances, which correspond to segregated fund
contract. Because hedging is an important concept for options and life insurance products, we derived
locally risk–minimizing strategies for the Black–Scholes call and put options and the equity–linked
life insurance products. Finally, to use the model, we provide maximum likelihood estimators of the
model’s parameters and EM algorithm.
Further extensions of the model are as follows: (i) error term should be modeled by correlated
conditional heteroscedastic models, (ii) price–to–book ratio should be modeled by AR(p) process with
unit root, (iii) risk–free rate should be modeled by a model that varies over time, and (iv) as mentioned
before dividend–to–book ratio should be modeled by some model.
8
Technical Annex
Here we give the Lemmas and their proofs.
Lemma 1. Let X ∼ N (µ, σ 2 ), α1 , α2 , α3 > 0, and β1 , β2 , β3 ∈ R. Then, it holds
α1 µ + β1
E Φ(α1 X + β1 ) = Φ p
1 + α21 σ 2
(95)
and
h
E exp α2 X + β2
where Φ(x) =
Rx
−∞
i
α22 σ 2
α3 µ + β3 + α2 α3 σ 2
p
Φ(α3 X + β3 ) = exp α2 µ + β2 +
,
Φ
2
1 + α23 σ 2
2
√1 e−u /2 du
2π
(96)
is the cumulative standard normal distribution function.
Proof. Let Z be a standard normal random variable, which is independent of the random variable X.
Then, we have
E Φ(α1 X + β1 ) = E P[Z ≤ α1 X + β1 |X] = P[Z − α1 X ≤ β1 ].
16
Because the random variable Z − α1 X follows normal distribution with mean −α1 µ and variance
1 + α21 σ 2 , we prove equation (95). For equation (96) of the Lemma, observe that
α2 x + β2 −
α22 σ 2 (x − µ − α2 σ 2 )2
(x − µ)2
=
α
µ
+
β
+
−
.
2
2
2σ 2
2
2σ 2
As a result, we get that
h
E exp α2 X + β2
i
α22 σ 2
E Φ(α3 Y + β3 ) ,
Φ(α3 X + β3 ) = exp α2 µ + β2 +
2
where Y ∼ N (µ + α2 σ 2 , σ 2 ). By using equation (95), one prove equation (96). That completes the
proof.
Lemma 2. Let X1 and X2 be real random variables and their joint distribution is given by
2
X1
µ1
σ1 σ12
.
∼N
,
σ12 σ22
X2
µ2
Then, it holds
h
X1
E e
X2 +
−e
i
σ22
µ1 − µ2 + σ12 − σ12
µ1 − µ2 + σ12 − σ22
σ12
p
p
− exp µ2 +
.
Φ
Φ
= exp µ1 +
2
2
σ12 − 2σ12 + σ22
σ12 − 2σ12 + σ22
Proof. The tower property of a conditional expectation implies that
h
h +
+ i
i
E eX1 − eX2
= E E eX1 − eX2
X1 .
(97)
At the first step, let us consider the conditional expectation of equation (97). According to the well
known conditional distribution formula of the multivariate normal distribution, we have
2
X2 | X1 ∼ N µ2.1 (X1 ), σ22.1
,
2 /σ 2 is a conditional
2
:= σ22 −σ12
where µ2.1 (X1 ) := µ2 +σ12 /σ12 (X1 −µ1 ) is a conditional mean and σ22.1
1
variance of the random variable X2 given X1 . Let us denote a density function of the random variable
X2 given X1 by φ(x2 |X1 ). Then, one get that
+
E eX1 − eX2
X1 = eX1
Z
X1
−∞
φ(x2 |X1 )dx2 −
Z
X1
−∞
eX1 φ(x2 |X1 )dx2 .
For the first and second term of the right–hand side of the above equation, by using the partial
integration formula and the completing the square method one can show that
Z X1
X1 − µ2.1 (X1 )
X1
X1
φ(x2 |X1 )dx2 = e Φ
e
√
σ22.1
−∞
and
Z
X1
−∞
X1
e
σ22.1
X1 − µ2.1 (X1 ) − σ22.1
φ(x2 |X1 )dx2 = exp µ2.1 (X1 ) +
.
Φ
√
2
σ22.1
At final step, since µ2.1 (X1 ) is the linear function for its argument X1 , using Lemma 1 one completes
the proof of the Lemma.
17
Lemma 3. Let X ∼ N (µ, σ 2 ). Then for all K > 0,
X
+ σ2
E e −K
= exp µ +
Φ(d1 ) − KΦ(d2 )
2
and
(98)
σ2
Φ(−d1 ),
(99)
E K −e
= KΦ(−d2 ) − exp µ +
2
Rx
2
where d1 := µ+σ 2 −ln(K) /σ, d2 := d1 −σ, and Φ(x) = −∞ √12π e−u /2 du is the cumulative standard
normal distribution function.
X +
Proof. By taking X1 = X and X2 = ln(K) for equation (98) and X1 = ln(K) and X2 = X for
equation (99) in Lemma 2, one obtain the proof of the Lemma.
Lemma 4. Let X1 and X2 be real random variables and their joint distribution is given by
2
X1
µ1
σ12
σ
∼N
.
, 1
X2
µ2
σ12 σ22
Then, for all K > 0, it holds
h
+ X2
i
E eX1 − K
e − E eX2
X X σ
σ12
σ12
2
1
12
− Φ(d1 ) − K Φ d2 +
− Φ(d2 )
=E e
E e
e Φ d1 +
σ1
σ1
and
h
+ X
i
E K − eX1
e 2 − E eX2
σ12
σ12
= E eX2 K Φ − d2 −
− Φ(−d2 ) − E eX1 eσ12 Φ − d1 −
− Φ(−d1 ) ,
σ1
σ1
2
where for each i = 1, 2, E eXi = eµi +σi /2 is the expectation of the log–normal random variable,
Rx
2
d1 := µ1 + σ12 − ln(K) /σ1 , d2 := d1 − σ1 , and Φ(x) = −∞ √12π e−u /2 du is the cumulative standard
normal distribution function.
Proof. The proof of the Lemma follows from Lemmas 2 and 3.
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