African Journal of Business Management Vol. 5(12), pp. 4916-4933, 18 June, 2011

Available online at http://www.academicjournals.org/AJBM

DOI: 10.5897/AJBM10.1598

ISSN 1993-8233 ©2011 Academic Journals

Full Length Research Paper

Valuation of investment on a firm with trade credit

under uncertainty: A real options approach

Po-Yuan Chen

1,2

*, Horng-Jinh Chang

1

and I-Ming Jiang

2

1

Graduate Institute of Management Sciences, Tamkang University, 151 Ying-chuan Road, Tamsui,

Taipei County, Taiwan.

2

Department of Financial and Tax Planning, Jinwen University of Science and Technology, 99 Anchung Road, Sindian

District, New Taipei City, Taiwan, R. O. C.

2

Faculty of Finance, School of Management, Yuan Ze University, 135 Yuan-Tung Road, Chungli, Taoyuan, Taiwan.

Accepted 18 January, 2011

This paper intends to propose a corporate valuation framework by incorporating both the stochastic

product price and the stochastic interest rate in a delay payment context. By using Ito-Isometry

theorem, we derived the analytical solution for corporate value, based on which sensitivity analyses

and further simulations for the real option value are performed. Some critical factors were considered in

the sensitivity analysis of corporate value: the drift and the volatility in the price and in the interest rate

processes, the price elasticity of demand, the cost rate, the market share, as well as the length of time

period for delay payments. To valuate an opportunity of investment on a firm with trade credit under

uncertainty, a real options model was employed. The simulation results for the real option value

indicated that increasing demand elasticity, market share, number of time periods for delay payments,

interest rate drift, price volatility, and interest rate volatility all contribute to increasing real option value,

whereas the increasing cost rate and price drift lead to the decreasing real option value as expected.

Key words: Trade credit, stochastic price, stochastic interest rate, real option.

INTRODUCTION

The mechanism of trade credits is frequently adopted in

business-to-business (B2B) transactions to provide

buyers with more flexibility in their working capital

management and thereby enhance their financial liquidity.

The payment terms in B2B transactions are also

employed to facilitate the billing system and attract the

buyers to maintain their loyalty. Such mechanism permits

buyers to arrange a more flexible payment scheduling.

From the perspective of buyers, the payment terms of

accounts payable are considered as trade credits, which

allow them to defer the payable amounts until the

maturity of the credit period and hence increase their

financial ability in management of net working capitals.

Besides, the delayed cash outflows in the procurement

activities could also be invested in financial instruments

with guaranteed returns and safety over the credit period

between the invoice date of merchandise purchases and

*Corresponding author. E-mail: mikecpy@ms7.hinet.net Tel:

+886-2-8212-2000 ext. 6375. Fax: +886-2-8212-2399.

the payment date of account payable settlements. Robb

and Silver (2006) classify the supplier trade credits into

two categories: date term and day term. Date-term entails

buyer’s payments on a specific date (on 20

th

or 30

th

in

each month), while day-term permits the buyer to

postpone its payments without incurring any interest

charges for a specified period of time (20 or 30 days).

Based on a day-term credit policy, the optimal payment

period in a cost minimization model is derived by Jamal,

Sarker, and Wang (2000), who divide the payment period

into two sub periods: the credit period incurring no

interest charge and the overdue period incurring some

interest charges. They conclude that buyer’s payment

decisions should depend on the multiplier of unit price

relative to unit cost. The smaller the multiplier, the earlier

the buyer should pay. However, the optimal payment date

should not be earlier than the maturity of a credit period

in terms of inventory cost minimization. Bhaba et al.

(1994) investigate the optimal order quantity and

maximum allowable shortage in terms of inflation and

time value of money. They conclude that the decisions on

order quantity and allowable shortage depend evidently

Chen et al. 4917

Figure1. The monthly price and quantity processes for the LCD product in a 10-year period.

on the difference between the inflation and discount rate.

Dave and Patel (1981) optimize the total costs to obtain

the optimal number of replenishments with a time linearly

dependent demand over a finite time horizon. Kim et al.

(1995) propose a profit maximization model from the

perspective of the supplier to obtain the optimal length of

credit period by using decreasing demand functions of

the retail price. In their model, the supplier and the buyer

should jointly determine the retail price and order size.

Petersen and Rajan (1997) conduct an empirical study on

the importance of trade credit for 3404 small firms in the

USA from the dataset of the National Survey of Small

Business Finances (NSSBF). They find out that larger

firms borrow and extend more trade credit, which serving

as a strategic tool to increase business scale. NG et al.

(1999) empirically investigate determinants of trade credit

from a survey of credit managers from 2538 firms, drawn

from COMPUSTAT files. They classify trade credit into

two categories: net term (net 30) and two-part term (10/2

net 30). Factors impacting the choice of different term are

discussed. Fatoki and Odeyemi (2010) conduct a

questionnaire survey and conclude that 331 SMEs

(small- or medium-sized enterprise) out of total 417

respondents in South Africa apply for trade credits from

their suppliers (creditors), but only 71 respondents are

approved. The result reveals that most SMEs consider

trade credit as a better financing channel than ordinary

bank loan. However, the suppliers choose the trade-credit

receivers in a strict and secure way.

Once a profit-maximizing firm receives trade credits, it

should actively invest the delayed dollars in instruments

with higher and guaranteed returns to increase the

corporate value even only for a short period. In view of

this, we intend to formulate a stochastic model to valuate

such a firm facing price and interest rate risks. The

approaches of corporate valuation fall into two

categories: market-based and DCF-based (Discounted

Cash Flow) models. The market-based model dates back

to the work of Merton (1974), who proposes the

stochastic asset value in a call option, whose exercise

price is the contingent claim on debt. Based on his option

framework, many related models have been developed

by Black and Cox (1976), and Landes and Loistl (1992),

etc. On the other hand, the frameworks employing the

DCF to calculate the corporate value can be found in the

works by Rappaport (1986), and Copeland et al. (1996).

The main theme of these works lies in the discounted

present value of all future cash inflows. Casey (2001)

computes the discounted value from a series of uncertain

future payments until the date of insolvency and then

obtains the present value distribution.

In this paper, we propose a model for corporate

valuation from the perspective of a buyer with day-term

trade credits under the stochastic interest rates, based on

the short rate of the Merton model (1973). According to

the time series data acquired from the Taiwan Economic

Journal (TEJ) in the period of 2001 to 2010, we found

that the monthly price for the LCD component evolves

stochastically as seen in Figure 1. The interest rate of 3-

month T-bill also evolves stochastically as shown in

4918 Afr. J. Bus. Manage.

Figure 2. The monthly interest rate process for 3-month T-bill in a 10-year period.

Figure 2. The quantity demanded demonstrates a

negative correlation with the price. Therefore, the

inversed demand function is adopted in our model. The

corporate value can then be derived by discounting all

the future profits, which also evolve as a stochastic

process due to the randomness in the interest rate and in

the price. The sensitivity of corporate value was analyzed

to provide readers with more insights into the interactions

among the following factors: the drift and the volatility for

the stochastic interest rate and for the price, the price

elasticity of demand, the market share, the cost rate, the

length of each period, and the number of delay periods.

METHODOLOGY

Assumption and notation

In this model, we calculate the expected discounted value of future

cash flows for a buyer. In the context of trade credits offered by the

supplier, the buyer’s cash outflows can be delayed for

k

periods

with the length of each period denoted by

δ

. The buyer has an

opportunity to invest the deferred amounts payable in financial

instruments with safety and guaranteed returns until the maturity

date of credit period, e.g. bank savings in LIBOR market, Floating

Rate Note, etc. We consider the rate of return as a stochastic

process and adopt the short rate structure proposed by Merton

(1973). The short rate in his framework evolves stochastically on

the continuous time basis. The product price also evolves

stochastically. Because of the stochastic effects, the uncertainties

are propagated into the future cash flows and then into the

discounted present value, whose expected value represents the

corporate value in our model. According to the Merton model of

short rate, we denote the instantaneous interest rate at time t as

t

r

,

which stochastically evolves as

t r r t

dr dt dW

α σ= +

(1)

or equivalently,

0 0 0

( ), 0

t r r t

r r t W W W

α σ

= + + − =

(2)

where

r

α

1

is a constant drift,

r

σ

is a constant volatility, and

t

W

is a standard Brownian motion with a filtration

{

}

( );0

t t

≥

F,

t

dW

is the normally distributed increment with the expected value

0 and the variance 0

= − =

dt t t

.

The product price also evolves stochastically as:

t

p p t

t

d P

d t d Z

P

α σ= +

(3)

where

p

α

is a constant drift,

p

σ

is a constant volatility, and

t

Z

is

1

Tuckman (2002) divides

α

into two components: the expected growth and

the risk premium.

a standard Brownian motion with a filtration

{

}

( );0

t t

≥

F,

t

dZ

is

the normally distributed increment with the expected value 0 and

the variance 0

= − =

dt t t

. It is assumed that

t

dW

and

t

dZ

are

independent.

In order to calculate the time value of money, we denote a

continuous compounding money market account as

,

t k

β

, and

define it as follows:

,

t k

s

t

r d s

t k

e

δ

β

+

=

. (4)

where

δ

is the constant length of each period between two

consecutive transactions.

We assume that the market demands are price sensitive and can

be defined by an inversed demand function of Carruth et al. (2001)

as follows:

1

−

=

t t

P Q

ε

. (5)

The buyer’s marginal cost is assumed proportional to its revenue

according to the assumption made by Schwartz and Moon (2001),

who divide the cost structure into two components: a fixed term and

a variable term, which is proportional to the revenue. We also

assume that the buyer could control its cost rate at a fixed level

relative to the revenues through its managerial activities such as its

bargaining power to negotiate with the supplier. In the context of

trade credits offered by the supplier, the payments for marginal

costs incurred at time

t

can be delayed for

k

periods. We denote

the costs incurred at time

t

, but actually paid at time

t k

δ

+

as

,

t k

C

, which is defined as follows:

,

t k t

C P

γ

= ⋅

, (6)

where

γ

is the constant cost rate,

0,1,2,

k

=

.

Framework

It is assumed that the market share of the firm in question is

constant and denoted by

s

. The firm’s revenue at time t, denoted

by

t

R

, is therefore as follows:

1

−

= ⋅ ⋅ =

t t t t

R P s Q sP

ε

. (7)

Because the firm can defer the payments of accounts payable

for

k

periods and invest the delayed amounts in interest-bearing

financial instruments, the buyer’s cash profits at time t, denoted

by

t

π

, can be expressed as:

,,

1

,

( )

t t k t k k t k t k k

t k t k k

R s Q C

sP

δ δ δ δ

ε

δ δ

π β

β γ

− − − −

−

− −

= ⋅ − ⋅ ⋅

= −

(8)

Chen et al. 4919

We assume that the transactions occur at the time when it is the

multiple of the fixed time length

δ

. That is, the buyer periodically

purchases merchandises from the supplier at time

1

δ

,

2

δ

,

and

3

δ

, etc, while the supplier allows the buyer to postpone its

amounts payable for

k

periods. The buyer can simultaneously sell

and deliver the goods it received to the final customers on the cash

basis. Therefore, no inventory-related costs are considered in this

model. We denote

t

V

as the corporate value at time

t

, and define it

over an infinite time horizon as follows:

n

n

T

t t t T

n k

V e

ρ

π

∞

−

+

=

=

E

(9)

where

, 0,1,2,.

= =

n

T n nδ and

{

}

t

⋅

E is the conditional

expectation given the information set at time

t

. The discount rate is

assumed exogenous and denoted by

ρ

.

Substitute Equations (2), (4), and (8) into (9), we have (proved in

Appendix A):

1

1 2

( ) ( )

t t

V s P I I

ε

γ

−

= ⋅ − ⋅

where

2

(1 )( )

2

1

=

t T

n p

s n p n k

t T

n k

r ds T T

t

n k

I e

σ

ρ ε α

+

−

+

−

∞

− + − −

=

E

, (10)

and

2

(1 )( )

2

2

=

p

n p n k

T T

t

n k

I e

σ

ρ ε α

−

∞

− + − −

=

E

. (11)

Equations (10) and (11) can be simplified as follows (proved in

Appendix B): .

2 2 3

2

2 2 2 3

1

( ) ( ) ( )

2 6

1

1

(1 )( ) ( )

2 2

1

r

t r

p

r p r

r k k k

k k

e

I

e

α

ρ δ δ σ δ

σ

ρδ α δ ε α δ σ δ

− + +

− + + − − +

=

−

(12)

22

((1 )( ) )

2

1

p

p

k

e

I

e

ρδ

σ

ε α ρ δ

−

− − −

=

−

(13)

The following constraint should also be satisfied due to the

convergence condition:

2

2 2 2

1

( (1 )( ) )

2 2

p

r p r

k k

σ

ρ α δ ε α σ δ

> + − − +.

4920 Afr. J. Bus. Manage.

Figure 3. The sensitivity of corporate value to drift and volatility of interest rate under parameters.

p p

t t

0.01,0.001,s 0.2,1.2,1/52,

k 4,0.2,r 0.065,P 0.008,0.12

α = σ = = ε = δ =

= γ = = = ρ =

By combining Equations (12) and (13), we can finally derive the analytical solution for corporate value as follows:

2 2 3

2 2

2 2 2 3

1

( ) ( ) ( )

2 6

1

1

(1 )( ) ( ) ((1 )( ) )

2 2 2

( ) ( )

1 1

− + +

−

−

− + + − − + − − −

= ⋅ − ⋅

− −

r

t r

p p

r p r p

r k k k

k

t t

k k

e e

V s P

e e

α

ρ δ δ σ δ

ρδ

ε

σ σ

ρδ α δ ε α δ σ δ ε α ρ δ

γ

(14)

where parameters should satisfy

2

2 2 2

1

( (1 )( ) )

2 2

p

r p r

k k

σ

ρ α δ ε α σ δ

> + − − +

.

According to equation (14), corporate value at time t (

t

V

) is a

function of stochastic price (

t

P

) and interest rate (

t

r

) at time t. In

our formulation, an opportunity of investment on a firm with trade

credit under uncertainty is taken into account. A real option model

for such an investment opportunity is adopted. Subsequently, we

can simulate the process of corporate value and then obtain the

numerical solution for the real option value as follows:

(

)

(

)

0 0 0

(;,)

−

= −

t

t t

t

F V P r Max E V I e

ρ

, (15)

where

F

denote the real option value, defined by Pindyck (1991),

and

I

denotes the constant sunk cost for the investment

opportunity.

Analytical solution

The sensitivity of corporate value based on equation (14) is

analyzed and illustrated in Figures 3 through 8. It can be seen in

Figure 3 that the corporate value increases with increasing interest

rate drift and volatility. However, in Figure 4 the price drift has a

negative impact on corporate value, while the price volatility

demonstrates a positive impact on corporate value, even though

such an impact is not so significant.

Figure 5 illustrates the relationship among the cost rate, the

demand elasticity, and the corporate value, while Figure 6

demonstrates the relationship among the cost rate, the market

share and the corporate value. They confirm the business practice

that higher cost reduces the cash inflows generated from

operational profits and hence lowers corporate value. Nonetheless,

the cost rate affects corporate value insignificantly when demand

elasticity is as low as 1.1 and the market share approaches 0.1.

The impact of the cost rate becomes more significant with the

increase in price elasticity and market share. On the other hand, the

impacts of the demand elasticity and the market share on corporate

value become more significant with the decreases in the cost rate.

However, corporate value is a convex function of price elasticity in

Figure 5, whereas it is a concave function of the market share in

Chen et al. 4921

Figure 4. The sensitivity of corporate value to drift and volatility of price under parameters:

r r

t t

0.01,0.001,s 0.2,1.2,1/52,

k 4,0.2,r 0.065,P 0.008,0.12

α = σ = = ε = δ =

= γ = = = ρ =

Figure 5. The sensitivity of corporate value to cost rate and to demand elasticity under parameters:

r r p p

t t

0.01,0.001,0.01,0.001,s 0.2,

1/52,k 4,r 0.065,P 0.008,0.12

α = σ = α = σ = =

δ = = = = ρ =

Figure 6. It can be inferred that small change in price causes large

change in quantities when the price elasticity is larger. As a result,

more revenue and more corporate value are anticipated in Figure 5.

In Figure 7, the firm obtains more value when it can defer its

payments for the purchased inventory as late as possible due to the

trade credits it receives form the supplier. It can be shown that

corporate value increases with longer time to maturity of payment,

indicated by the increasing number of delay periods (

k

) in our

framework. However, the length of each time period (

δ

)

represents the time interval between two consecutive transactions.

The shorter length of each period implies that transactions occur

4922 Afr. J. Bus. Manage.

Figure 6. The sensitivity of corporate value to cost rate and to market share under parameters:

r r p p

t t

0.01,0.001,0.01,0.001,1.2,

1/52,k 4,r 0.065,P 0.008,0.12

α = σ = α = σ = ε =

δ = = = = ρ =

Figure 7. The sensitivity of corporate value to time length and to number of delay periods under

r r p p

t t

0.01,0.001,0.01,0.001,0.2,

1.2,S 0.2,r 0.065,P 0.008,0.12

α = σ = α = σ = γ =

ε = = = = ρ =

more frequently and hence more cash flows are generated. As a

result, a corporation with higher transaction frequency, indicated by

shorter length of each period in our framework, results in higher

corporate value. That accounts for the reason why corporate value

is negatively correlated with the length of each period. That is, the

shorter the length of each period, the higher the corporate value.

Chen et al. 4923

Figure 8. The sensitivity of corporate value to initial interest rate and price under parameters:

r r p p

0.01,0.001,0.01,0.001,0.2,

1.2,S 0.2,k 4,1/52,0.12

α = σ = α = σ = γ =

ε = = = δ = ρ =

Because corporate value is a convex function of the time period

length, it becomes smoother when the length of each period is

larger. In Figure 8, the initial price demonstrates more significant

impacts on corporate value than the initial interest rate. It can be

inferred that the lower price induces more market demand, and thus

generates more revenues and more corporate value. As a result,

the impact of the price on corporate value is negative. Contrarily,

the impact of the initial interest rate is slightly positive as expected.

RESULTS

We considered the 3-month T-bill rate as a proxy for the

short rate of the Merton model and approximated the

random fluctuations of the short rate ranging from 0.00 to

0.0625 in the period of 2000 to 2010 as shown in Figure

2. We therefore adopted 0.065 as initial interest rate for

the simulations of the interest rate process. Moreover,

Chan et al. (1992) estimate the drift and volatility for the

interest rate process to be 0.005 and 0.02 respectively.

We therefore chose 0.005 and 0.015 to be benchmark

parameters for interest rate drift and volatility respectively.

As seen in Figure 1, the parameters of drift and volatility

for LCD price were estimated to be -0.039 and 0.350,

respectively. Based on such parameter estimations, we

set the benchmark parameters for the stochastic price

process in our simulations as follows: ±0.01, ±0.05, ±0.09

for the price drift and price volatility. In order to investigate

the impacts of different parameters on real option value,

we simulated by using different values of those

parameters, including the number of delay periods, the

cost rate, the market share, and the price elasticity.

Simulation results

Similar to the stochastic dominance methodology in the

study of new product development by Kleczyk (2008)

who adopts Monte Carlo simulation, the stochastic

programming by simulation approach was adopted by

this work to obtain numerical solution for equation (15) in

a 10-year period with 10,000 paths for each replication.

For each sample path, we generated two independent

and random standard normal values for

t

dW

and

t

dZ

for

equations (1) and (3). Based on those two random

values, the price and the interest rate processes were

then simulated for the computations of the corporate

value and those of the real option value in equations (14)

and (15). The simulated sample paths for the product

price, the interest rate, and the corporate value, as well

as the discounted investment value are illustrated in

Figures 9 through 12. The results calculated from the

10,000 paths are presented in Tables 1 through 4. The

numbers in each cell are the mean and the standard

deviation (in parentheses) from the 10,000 sample paths

under different parameter combinations. It can be seen in

4924 Afr. J. Bus. Manage.

Figure 9. Simulated sample paths for the stochastic interest rate process.

Figure 10 that the price drift is negative, causing the price

to have a downtrend. However, such a downtrend in price

may induce more demand and thus generates more

value for the firm as shown in Figure 11. However, the

discounted value for such an investment opportunity is

shown in Figure 12 as decreasing with time passage.

The numerical results can be found in the following

tables. In Table 1, the real option value increases with

increasing number of delay periods. Even though the

price drift is positive in Table 2, the real option value still

increases with increasing number of delay periods. It

implies that when the interest rate process has an

uptrend, the firm should seek for the trade credit

opportunity to obtain the higher option value. Besides, it

can be shown that the real option value increases with

the increasing price elasticity of demand whatever the

drifts for the interest rate and for the price are. Similar to

the effects of volatility in financial options, the impact of

volatility for both the interest rate and the price on real

option value are also positive. The cost rate is not

surprisingly a negative factor to the option value as seen

in the third column of those tables, while the market share

has a positive impact on the option value as we can see

in the last column. It can be seen that Tables 3 and 4

almost demonstrate the same results as Figures 1 and 2

do. However, two major differences can be found: the

effect of interest rate drift becomes positive and the effect

of interest rate volatility becomes insignificant.

Conclusion

This paper integrated the stochastic interest rate model

proposed by Merton (1973) and a stochastic price

process into corporate valuation to investigate the

impacts of the drift and the volatility for the interest rate

and for the price, the cost rate, time to maturity of delay

payments, the price elasticity of demand, and the market

Chen et al. 4925

Figure10. Simulated sample paths for the stochastic price process.

Figure 11. Simulated sample paths for the derived stochastic corporate value.

4926 Afr. J. Bus. Manage.

Figure 12. Simulated sample paths for the discounted values of (Vt-I).

flows were then discounted and summed up as expected

corporate value. An analytical solution for corporate value

was obtained and further sensitivity analyses were

performed. Based on the analytical results, we concluded

that corporate value increases with the increasing interest

rate drift and volatility, but decreases with the increasing

price drift. Besides, the price elasticity of demand has a

positive impact on corporate value. Not surprisingly, the

cost rate, a percentage of the selling price, for purchased

inventory has a negative impact on corporate value.

However, corporate value is less sensitive to the cost rate

when price elasticity is smaller. Similarly, the market

share has less significant impacts on corporate value

when the cost rate approaches 1.0. It suggests that the

profit-maximizing firm in our setting should give first

priority to entering a market with higher price elasticity

and reinvesting its delayed payments in a financial

instrument with higher growth rate. Higher volatility of the

interest rate is taken into account as second priority while

the profit margins for the goods sold are lastly considered

if the price elasticity and the market share are smaller.

Based on the analytical solution for corporate value,

real option values were obtained through further

simulations. The numerical results, presented in Tables 1

through 4, demonstrate that the drift of the interest rate is

quite critical in the derivations of the real option value.

The real option value increases with decreasing price drift

and increasing interest rate drift. Regarding the volatility

for both the interest rate and the price, their impact on

real option value is consistently positive. The price

elasticity of demand and the market share have positive

impacts on real option value, while the cost rate has

expectedly negative impacts.

The framework discussed in this paper has its

implications: the reinvestment opportunities due to the

delay payments of trade credit offered by the supplier

allow the firm to obtain higher corporate value, once it

correctly chooses a niche market to enter, such as the

product market with higher demand elasticity and higher

market share, as well as the financial market with positive

growth and higher volatility for returns. The analytical

solution obtained in this paper could be also employed as

the basis for the strategic investment decisions, such as

market entry and exit policies in further studies. Other risk

sources such as foreign exchange rate in the financial

market could be also taken into account in the extension

price. The interest rate was applied to the cash

reinvestments due to the delay payments. An inversed

demand function proposed by Carruth et al. was adopted

in our formulation.

To construct the value dynamics for the firm in ques-

tion, we derived a stochastic profit process. The profit

share on corporate value and on the real option value in a

context where a firm receives trade credits from its

supplier. In our framework, the uncertainty of the profits

comes from the stochastic interest rate and the stochastic

of this model to explore the effects of other

uncertainties.

Chen et al. 4927

Table 1. Sensitivity of real option value obtained from simulated 10,000 sample paths under parameters: = 1 week, = 0.12, P0= 0.008, r0= 0.065,r = 0.005,r r= 0.005.

Option value (F)

Delay periods () Price elasticity () Cost rate () Market share (S)

4 8 12 1.2 1.6 2.0 0.2 0.5 0.8 0.2 0.5 0.8

Price drift (p) -0.09

121.9

(0.004)

129.3

(0.020)

136.7

(0.05)

121.9

(0.004)

2290.1

(0.470)

37541.2

(65.35)

759.4

(0.010)

440.6

(0.010)

121.9

(0.004)

-9.8

(0.170)

38.7

(0.030)

121.9

(0.004)

-0.05

105.5

(0.002)

112.1

(0.020)

118.6

(0.04)

105.5

(0.002)

1639.6

(0.170)

154368.4

(9.03)

696.7

(0.010)

401.1

(0.010)

105.5

(0.002)

-12.7

(0.150)

28.4

(0.030)

105.5

(0.002)

-0.01

91.4

(0.001)

97.3

(0.010)

103.2

(0.03)

91.4

(0.001)

1267.5

(0.070)

9688.1

(2.56)

642.5

(0.010)

366.9

(0.010)

91.4

(0.001)

-15.1

(0.130)

19.6

(0.020)

91.4

(0.001)

Price volatility

(p)

0.01

91.4

(0.001)

97.3

(0.010)

103.2

(0.03)

91.4

(0.001)

1267.5

(0.070)

9688.1

(2.56)

642.5

(0.010)

366.9

(0.010)

91.4

(0.001)

-15.1

(0.130)

19.6

(0.020)

91.4

(0.001)

0.05

91.9

(0.150)

97.8

(0.150)

103.8

(0.20)

91.9

(0.150)

1285.1

(5.76)

9992.7

(115.2)

644.4

(0.200)

368.1

(0.150)

91.9

(0.150)

-15.1

(0.490)

20.2

(0.340)

91.9

(0.150)

0.09

93.3

(0.560)

99.3

(0.580)

105.3

(0.62)

93.3

(0.560)

1331.6

(20.27)

10831.7

(447.8)

649.0

(0.970)

371.1

(0.700)

93.3

(0.560)

-14.9

(0.880)

21.5

(0.970)

93.3

(0.560)

Table 2. Sensitivity of real option value obtained from simulated 10,000 sample paths under parameters: = 1 week, = 0.12, P0= 0.008, r0= 0.065,r= 0.005,r= 0.005.

Option value (F)

Delay periods () Price elasticity () Cost rate () Market share (S)

4 8 12 1.2 1.6 2.0 0.2 0.5 0.8 0.2 0.5 0.8

Price drift

(p)

+0.01

85.0

(0.010)

90.6

(0.010)

96.2

(0.030)

85.0

(0.010)

1135.5

(0.040)

8160.8

(1.300)

618.1

(0.020)

351.5

(0.010)

85.0

(0.010)

-16.2

(0.120)

15.6

(0.020)

85.0

(0.010)

+0.05

73.5

(0.010)

78.6

(0.010)

83.7

(0.020)

73.5

(0.010)

935.5

(0.010)

6196.8

(0.480)

573.8

(0.010)

323.6

(0.010)

73.5

(0.010)

-18.1

(0.100)

8.4

(0.020)

73.5

(0.010)

+0.09

63.3

(0.010)

68.0

(0.010)

72.6

(0.020)

63.3

(0.010)

768.4

(0.080)

4769.5

(0.580)

534.6

(0.010)

299.5

(0.010)

63.3

(0.010)

-19.7

(0.090)

2.5

(0.010)

63.3

(0.010)

Price volatility

(p)

0.01

85.0

(0.010)

90.6

(0.010)

96.2

(0.030)

85.0

(0.010)

1135.5

(0.040)

8160.8

(1.300)

618.1

(0.020)

351.5

(0.010)

85.0

(0.010)

-16.2

(0.120)

15.6

(0.020)

85.0

(0.010)

0.05

85.5

(0.140)

91.1

(0.160)

96.8

(0.190)

85.5

(0.140)

1149.6

(4.530)

8373.3

(82.90)

619.8

(0.180)

352.6

(0.140)

85.5

(0.140)

-16.2

(0.450)

16.2

(0.320)

85.5

(0.140)

0.09

86.8

(0.500)

92.5

(0.560)

98.2

(0.580)

86.8

(0.500)

1186.5

(16.30)

8948.7

(301.6)

624.2

(0.910)

355.4

(0.650)

86.8

(0.500)

-15.9

(0.821)

17.4

(0.910)

86.8

(0.500)

4928 Afr. J. Bus. Manage.

Table 3. Sensitivity of real option value obtained from simulated 10,000 sample paths under parameters: = 1 week= 0.12, P0= 0.008, r0= 0.065, p =-0.01, p= 0.01.

Option value (F)

Delay periods () Price elasticity () Cost rate () Market share (S)

4 8 12 1.2 1.6 2.0 0.2 0.5 0.8 0.2 0.5 0.8

Interest rate drift (r)

+0.005

91.4

(0.01)

97.3

(0.01)

103.2

(0.04)

91.4

(0.01)

1267.6

(0.05)

9688.1

(2.39)

642.5

(0.01)

366.9

(0.01)

91.4

(0.01)

-15.2

(0.13)

19.6

(0.02)

91.4

(0.01)

+0.010

94.4

(0.01)

103.4

(0.01)

112.5

(0.03)

94.4

(0.01)

1290.1

(0.06)

9855.5

(2.52)

645.6

(0.02)

370.0

(0.01)

94.4

(0.01)

-14.7

(0.13)

21.5

(0.02)

94.4

(0.01)

+0.015

97.5

(0.01)

109.7

(0.01)

122.0

(0.04)

97.5

(0.01)

1312.8

(0.07)

10024.0

(2.82)

648.2

(0.01)

373.1

(0.01)

97.5

(0.01)

-14.2

(0.14)

23.4

(0.02)

97.5

(0.01)

Interest rate volatility

(r)

0.005

91.4

(0.01)

97.3

(0.01)

103.2

(0.04)

91.4

(0.01)

1267.6

(0.05)

9688.1

(2.39)

642.5

(0.01)

366.9

(0.01)

91.4

(0.01)

-15.2

(0.13)

19.6

(0.02)

91.4

(0.01)

0.010

91.4

(0.01)

97.3

(0.07)

103.2

(0.19)

91.4

(0.01)

1267.6

(0.11)

9688.3

(2.91)

642.5

(0.01)

366.9

(0.01)

91.4

(0.01)

-15.2

(0.20)

19.6

(0.06)

91.4

(0.01)

0.015

91.4

(0.04)

97.3

(0.19)

103.2

(0.43)

91.4

(0.04)

1267.6

(0.19)

9688.7

(3.42)

642.5

(0.01)

366.9

(0.01)

91.4

(0.04)

-15.2

(0.28)

19.7

(0.12)

91.4

(0.04)

Table 4. Sensitivity of real option value obtained from simulated 10,000 sample paths under parameters:= 1 week= 0.12, P0

= 0.008, r

0= 0.065,p = 0.01,p = 0.01.

Option Value (F)

Delay Periods () Price Elasticity () Cost Rate () Market Share (S)

4 8 12 1.2 1.6 2.0 0.2 0.5 0.8 0.2 0.5 0.8

Interest rate

drift (r)

+0.005

85.0

(0.01)

90.6

(0.01)

96.2

(0.03)

85.0

(0.01)

1135.5

(0.02)

8160.8

(1.30)

618.1

(0.01)

351.5

(0.01)

85.0

(0.01)

-16.2

(0.12)

15.6

(0.02)

85.0

(0.01)

+0.010

87.8

(0.01)

96.4

(0.01)

104.9

(0.03)

87.8

(0.01)

1153.9

(0.04)

8280.5

(1.50)

620.6

(0.02)

354.4

(0.01)

87.8

(0.01)

-15.8

(0.12)

17.4

(0.02)

87.8

(0.01)

+0.015

90.7

(0.01)

102.2

(0.01)

113.8

(0.03)

90.7

(0.01)

1172.5

(0.04)

8400.9

(1.52)

623.8

(0.01)

357.3

(0.01)

90.7

(0.01)

-15.3

(0.12)

19.2

(0.02)

90.7

(0.01)

Interest rate

volatility (r)

0.005

85.0

(0.01)

90.6

(0.01)

96.2

(0.03)

85.0

(0.01)

1135.5

(0.02)

8160.8

(1.30)

618.1

(0.01)

351.5

(0.01)

85.0

(0.01)

-16.2

(0.12)

15.6

(0.02)

85.0

(0.01)

0.010

85.0

(0.01)

90.6

(0.06)

96.3

(0.07)

85.0

(0.01)

1135.5

(0.06)

8160.9

(1.76)

618.1

(0.01)

351.5

(0.01)

85.0

(0.01)

-16.2

(0.18)

15.7

(0.06)

85.0

(0.01)

0.015

85.0

(0.04)

90.7

(0.18)

96.5

(0.40)

85.0

(0.04)

1135.5

(0.13)

8161.2

(2.16)

618.1

(0.01)

351.5

(0.01)

85.0

(0.04)

-16.2

(0.26)

15.7

(0.11)

85.0

(0.04)

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nd

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4930 Afr. J. Bus. Manage.

APPENDIX

Appendix A

Substitute Equations (2), (4), and (8) into (9), we have:

2

1

,

1 1

(1 )( )

1 1

2

( )

( )

+

+

−

−

+

−

+

−

∞

−−

+ − + −

=

∞ ∞

−

−− −

+ + −

= =

− + − −

− +

− −

= −

−

−

n

n n

t T

n

s n

t T

n k n

n k n

t T

n p

s n p n k

n

t T

n k

T

t t t T k t T k k

n k

r ds T

T

t t T t t T k

n k n k

r ds T T

T

t t t t

V sP e

sP e sP e

s P e sP e

ρε

δ δ

ρ

ρε ε

δ

σ

ρ ε α

ρ

ε ε

β γ

γ

γ

E

E E

E E

2

2 2

(1 )( )

2

(1 )( )

(1 )( )

1 1

2

2

1 1

1 2

1

1 2

( )

( ) ( )

( ) ( )

−

+

−

−

+

−

∞ ∞

− −

= =

∞ ∞

− + − −

− + − −

− −

= =

− −

−

−

= ⋅ − ⋅

= ⋅ − ⋅

s

p

p n k

t T

n

p p

s n p n k

n p n k

t T

n k

T

n k n k

r ds T T

T T

t t t t

n k n k

t t

t

P e sP e

s P I s P I

s P I I

σ

ε α

σ σ

ρ ε α

ρ ε α

ε ε

ε ε

ε

γ

γ

γ

E E

where

2

(1 )( )

2

1

=

t T

n p

s n p n k

t T

n k

r ds T T

t

n k

I e

σ

ρ ε α

+

−

+

−

∞

− + − −

=

E,

and

2

(1 )( )

2

2

=

p

n p n k

T T

t

n k

I e

σ

ρ ε α

−

∞

− + − −

=

E.

Appendix B

To simplify Equations (10) and (11), we firstly calculate the expected exponential function with an Ito integral in the

power. According to Stochastic Fubini Theorem, we can rewrite

t T s

n

r u

t T t T

n k n k

dW ds

t

e

σ

+

+ +

− −

E as

t T t T

n n

r u

t T u

n k

dsdW

t

e

σ

+ +

+

−

E. Then

integrating the inner part in the exponential function gives the result as follows:

( )

t T

n

r n u

t T

n k

t T u dW

t

e

σ

+

+

−

+ −

E

. (B1)

We define an Ito integral as

0

( ) ( )= Δ

t

u

I t u dW

where ( )

Δ = + −

n

u t T u

is a nonrandom function. Because

( )

Δ

u

is

( )

u

F -measurable for each time index

0

≥

u and it satisfies

2

0

( )

Δ < ∞

t

E u du, Equation (12) can be rewritten in the

form of an Ito integral as follows:

Chen et al. 4931

( )

( ) ( )

r n n k

I t T I t T

t

e

σ

−

+ − +

E

.

It can be shown that for each

0

≥

t,

( )

I t

is a normally distributed random variable with the expected value 0 and the

variance

2

0

( )

Δ

t

u du

.Subsequently, we calculate the variance for the Ito integral as follows.

( ) ( )

( )

2

2

2

( ( ) ( ))

[ ( ) ( ) ] [ ( ) ( ) ]

[ ( ) ( ) ]

−

− −

−

+ − +

= + − + − + − +

= + − +

n n k

t n n k t n n k

t n n k

Var I t T I t T

I t T I t T I t T I t T

I t T I t T

E E

E

By applying the Ito-Isometry theorem, we have

(

)

2

2

2

[ ( ) ( ) ]

[ ( ) ]

( )

n

n k

n

n k

t n n k

t T

t

t T

t T

n

t T

I t T I t T

u du

t T u du

−

−

−

+

+

+

+

+ − +

= Δ

= + −

E

E

2 2

3

2 2

3 3

[( ) 2( ) ]

[( ) ( ) ]

3

.

3

n

n k

nn k

t T

n n

t T

t T

n n t T

t T t T u u du

u

t T u t T u

k δ

−

−

+

+

++

= + − + +

= + − + +

=

Therefore,

( ( ) ( ))

−

+ − +

n n k

I t T I t T is a normal distribution with the expected value 0 and the variance

3 3

3

k

δ

. Similarly,

( ( ) ( ))

r n n k

I t T I t Tσ

−

+ − + is also a normal distribution with the expected value 0 and the variance

2 3 3

3

r

k

σ δ

when

r

σ

is a constant volatility.

The following lemma (Shreve, 2000) is employed in our derivation for the corporate value.

Lemma 1.

Let the random variable

ξ

has the Gaussian law

2

(,)

Normal

µσ

. Then the random variable

=

e

ξ

ς has the expected

value

2

1

2

+

e

µ σ

and the variance

2 2

2

( 1)

+

−

e e

µ σ σ

.

Based on Lemma 1, Equation (B1) can be simplified as follows:

( )

2 3 3

2 3 3

1 1

2 3 6

0( ) ( )

t T s

n

k

r

r u

t T t T r

r n n kn k n k

dW ds

kI t T I t T

t t

e e e e

σ δσ

σ δ

σ

+

+ +

−− −

+ ⋅+ − +

= = =

E E. (B2)

By substituting Equation (B2) into Equation (10), we can derive the analytical solution for

1

I

as follows:

4932 Afr. J. Bus. Manage.

( )

2

2

2

(1 )( )

2

1

( ( )) ( (1 )( )

2

(1 )( ) ( ( ))

2

t T

n p

s n p n k

t T

n k

t T

n

p

t r n k r s t T n p n k

T n k

n k

t T

n k

p

n p n k t T r n k

n k

r ds T T

t

n k

r a s t T W W ds T T

t

n k

T T r ds a s t T d

t

I e

ee

σ

ρ ε α

σ

σ ρ ε α

σ

ρ ε α

+

−

+

−

+

+ − + −

−

−

+

−

− + −

−

∞

− + − −

=

∞

+ − + + − − + − −

=

− + − − + + − +

=

=

=

E =

EE

( )

t T t T t T

n n n

r s t T

n k

t T t T t T

n k n k n k

s W W ds

n k

σ

+ + +

+

−

+ + +

− − −

∞

+ −

=

2

2 2

2

2 2

(1 )( ) ( ) ( )

2 2

( (1 )( )( )

2 2

t T

n

p

r

n p n k t T n n k r s t T

n k n k

t T

n k

t T s

n

p

r

r u

p t T

n k

t T t T

n k n k

T T r T T k W W ds

t

n k

dW ds

n n k r k k

t t

n k

e

e e

σ

α

ρ ε α δ σ

σ

α

σ

ρ δ ε α δ δ δ

+

− + − +

− −

+

−

+

+

−

+ +

− −

∞

− + − − + − + + −

=

∞

− + − − − + +

=

=

= ⋅

E

E E

2

2 2 2 3

1

( (1 )( )( ) ( )

2 2 6

p

r

p r

t T

n k

n n k k k

r k

t

n k

e e

σ

α

ρ δ ε α δ δ σ δ

δ

+

−

∞

− + − − − + +

=

= ⋅

E. (B3)

According to the definition of the stochastic short rate process in Equation (1), we have

( )

( ).

t T

n k

t r n k r t T

n k

t r n k r t T

n k

r k

r T dW k

r T k k dW

δ

α σ δ

α δ σ δ

+

−

− +

−

− +

−

= + +

= + ⋅ + ⋅

Because

+

−

t T

n k

dW is the increment for the standard Brownian motion and is normally distributed, we can obtain another

normal distribution with the mean and the variance as follows:

(

)

( )

2 2 2 2 2 2

( ) ( ( ) )

( )

t T t r n k t r

n k

t T r n k r

n k

r k r T k r n k k

t

r k k T k n k

t

E

Var

δ α δ α δ δ

δ δ σ δ σ δ

+ −

−

+ −

−

+ ⋅ + − ⋅

−

= =

= =

Again, according to Lemma 1, we have

2 2 2

1

( ( ) ) ( )

2

( ).

t r r

t T

n k

r n k k k n k

r k

t

E e e

α δ δ δ σ δ

δ

+

−

+ − ⋅ + −

=

Equation (B3) then becomes

2

2 2 3 2 2 2 3

1 1

( ( ) ) ( ) ( (1 )( )( ) ( )

2 2 2 6

p

r

t r r p r

r n k k n k k n n k k k

n k

e

σ

α

α δ δ σ δ ρ δ ε α δ δ σ δ

∞

+ − + − + − + − − − + +

=

. (B4)

Simplify Equation (B4), we can derive the analytical solution for

1

I

as follows.

Chen et al. 4933

2

2 2 3 2 2 2 3

2

2 2 3 2 2 2 3

1 1

( ( ) ) ( ) ( (1 )( )( ) ( )

2 2 2 6

1

1 1

( (0) ) (0)( ) ( ( ) (1 )( )(0) ( )

2 2 2 6

1

( (1) ) (1)(

2

p

r

t r r p r

p

r

t r r p r

t r

r n k k n k k n n k k k

n k

r k k k k k

r k

I e

e

e

σ

α

α δ δ σ δ ρ δ ε α δ δ σ δ

σ

α

α δ δ σ δ ρ δ ε α δ δ σ δ

α δ δ

∞

+ − + − + − + − − − + +

=

+ + + − + − − + +

+ +

=

=

+

2

2 2 3 2 2 2 3

2

2 2 3 2 2 2 3

2 2 3

2

2

1

) ( ( 1) (1 )( )(1) ( )

2 2 6

1 1

( (2) ) (2)( ) ( ( 2) (1 )( )(2) ( )

2 2 2 6

1

( ) ( ) ( )

2 6

1

(1 )( )

2 2

1

p

r

r p r

p

r

t r r p r

r

t r

p

r p

k k k k

r k k k k k

r k k k

k

e

e

e

σ

α

σ δ ρ δ ε α δ δ σ δ

σ

α

α δ δ σ δ ρ δ ε α δ δ σ δ

α

ρ δ δ σ δ

σ

ρδ α δ ε α δ

+ − + + − − + +

+ + + − + + − − + +

− + +

− + + − − +

++

=

−

2 2 3

( )

r

kσ δ

In order to obtain the converged solution for the above infinite series, parameters in our model should satisfy the

following constraint:

2

2 2 2

1

( (1 )( ) )

2 2

p

r p r

k k

σ

ρ α δ ε α σ δ

> + − − +.

Similarly, simplifying equation (11) can solve for

2

I

as follows.

2

2 2 2 2

0 1 1 2 2 3 3

2

(1 )( )

2

2

(1 )( ) (1 )( ) (1 )( ) (1 )( )

2 2 2 2

( ) (1 )( )(0 ) ( 1) (1

2

...

p

n p n k

p p p p

k p k p k p k p

p

p

T T

t

n k

T T T T T T T T

t

k k

t

I e

e e e e

e e

σ

ρ ε α

σ σ σ σ

ρ ε α ρ ε α ρ ε α ρ ε α

σ

ρ δ ε α δ ρ δ

−

+ + +

∞

− + − −

=

− + − − − + − − − + − − − + − −

− ⋅ + − − − ⋅ + +

=

= + + + +

= +

E

E

E

2 2 2

2

)( )(1 ) ( 2) (1 )( )(2 ) ( 3) (1 )( )(3 )

2 2 2

((1 )( ) )

2

...

1

p p p

p p p

p

p

k k

k

e e

e

e

σ σ σ

ε α δ ρ δ ε α δ ρ δ ε α δ

ρδ

σ

ε α ρ δ

− − − ⋅ + + − − − ⋅ + + − −

−

− − −

+ + +

=

−

The following constraint should also be satisfied due to the convergence condition:

2

(1 )( )

2

p

p

σ

ε α ρ

− − <

.

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