J. Service Science & Management, 2010, 3, 383-389
doi: 10.4236/jssm.2010.34045 Published Online December 2010 (http://www.SciRP.org/journal/jssm)
383
Volatility Forecasting of Market Demand as Aids
for Planning Manufacturing Activities
Jean-Pierre Briffaut1,3, Patrick Lallement2,3
1Institut Telecom-Telecom & Management SudParis, Evry, France; 2Université de Technologie de Troyes (UTT), Troyes, France;
3Charles Delaunay Institute, Troyes, France.
Received August 20th, 2010; revised October 12th, 2010; accepted November 11th, 2010.
ABSTRACT
The concepts and techniques designed and used for pricing financial options have been applied to assist in scheduling
manufacturing activities. Releasing a manufacturing order is viewed as an investment opportunity whose properties are
similar to a call option. Its value can be considered as the derivative of the market demand mirrored in the selling price
of the manufactured products and changes over time following an Itô process. Dynamic programming has been used to
derive the optimal timing for releasing manufacturing orders. It appears advisable to release a manufacturing when the
unit selling price come to a threshold P* given by the relation P* = β/(β – 1) C with C = unit cost price. β is a parame-
ter whose value depends on the trend parameter α and the volatility σ of the selling price, the discount rate ρ applicable
to the capital appreciation relevant to the business context under consideration. The results have been successfully ap-
plied to the evolution of the quarterly construction cost index in France over ten years.
Keywords: Forecasting, Uncertainty, Stochastic process, Dynamic Programming, Manufacturing Activities
1. Introduction
Since the dawn of entrepreneurship men have been cal-
culating the chances of success in future actions by as-
sessing the available statistics, if any, on past actions.
But entrepreneurs1 are confronted with their interpreta-
tion in terms of what is actually measured and how they
are reliable.
A wide range of mathematical strands has been devel-
oped to deal with the ongoing uncertainty of the eco-
nomic environment in the field of financial economics-
options, futures, other derivatives - as well as capital
budgeting [1]. In this paper we aim to demonstrate how
the concepts and techniques used in these fields can con-
tribute to scheduling manufacturing activities. Schedul-
ing the manufacturing of a batch of products is consid-
ered as an option analogous to a financial call option on a
common stock market. When deciding on a schedule, the
option to manufacture is “killed” giving up the possibil-
ity of waiting for complementary information about the
market.
The objectives of this paper are threefold:
1) to produce the understanding and definitions of
several concepts associated with the way uncertainty is
dealt with in financial economics;
2) to demonstrate how this mindset can be used to en-
gineer the schedule of manufacturing activities in vola-
tile contexts;
3) to show how the technique explained can be applied
on the basis of an example.
In Section 2 is explained the way the timing of a deci-
sion impacts its relevance in a perspective of manufac-
turing activities. Section 3 describes the mathematical
background pertaining to the application of stochastic
processes to deal with risk. Section 4 focuses on the
paradigm of uncertainty engineered to cancel the risk in
the domain of financial options and its practical use for
manufacturing project. In Section 5 an example is pre-
sented on the basis of data relating to the construction
cost index in France. Section 6 touches on the main as-
sumptions underlying the Itô process and on the rele-
vance of the option pricing model for forecasting pur-
poses.
2. Forecasting and the Option Approach in a
Perspective of Manufacturing Activities
1Entrepreneur: one who undertakes a business or enterprise with chance
of profit or loss (Oxford Dictionary) How should a manufacturing company, facing uncer-

Volatility Forecasting of Market Demand as Aids for Planning Manufacturing Activities
384
tainty over future market conditions, decide on its pro-
duction schedule? To keep on top of demand, a company
has to avoid shortages on one hand and too much invent-
tory on the other. This means that forecasting is a key
issue, whether the company builds its own products in its
own factories or outsources its production. Short of being
able to control market, demand planning and scheduling
manufacturing operations depend on an ability to predict
what will be the market demand and how it will fluctuate
during the execution of plans generally frozen over a
certain horizon.
Forecasting by extrapolating historical data (linear re-
gression, simple and double exponential smoothing, de-
composition of time-series in trends, cyclical variations
and randomness …..) [2] is a technique over-described in
textbooks and research papers. It is straightforward to
apply and could be justified in the postwar “fordist” era.
Today volatility is the key feature of the global economy:
the fluttering of a butterfly in Bejing can trigger a tor-
nado in Cuba.
The irreversibility of a decision and the possibility of
delaying its implementation are two important charac-
teristics when converting forecasts into scheduled
courses of action. A manufacturing company with an
opportunity to schedule production is holding an “op-
tion” analogous to a financial call option on a common
stock - it has the right and not the obligation, at some
future time of its choosing, to make an expenditure
(manufacturing a batch of products) and receive a
revenue (the proceeds of their sale) [3]. When such a
company decides on a schedule, it exercises, or “kills”,
its option to manufacture. It gives up the possibility of
waiting for complementary information to arrive that
might affect the desirability or timing of the expendi-
ture incurred by the scheduled production versus the
market value of the manufactured products, value
which fluctuates stochastically. The option can be “in
the money” meaning that when it is exercised it would
yield a positive payoff. It is said to be “out of the
money” if exercising yields a negative payoff. It cannot
disinvest2 that is get back the money already spent,
should market conditions change adversely.
An investment project, such as manufacturing a batch
of products, can be made more valuable if it gives the
manufacturer the option to expand when economic con-
ditions become favorable. That option to expand in the
future clearly has a value. The higher the volatility of the
underlying asset - the market demand - the higher value
is given to the option - the market value of the batch of
products manufactured to sell. Economists assume that
the selling price of a product depends on the market de-
mand expressed in terms of quantity of products so that
the proceeds of a sale increase along with market de-
mand.
3. Market Volatility and the Associated
Risks
3.1. Volatility
Volatility is the relative rate at which market demand
moves up and down. It is found by calculating, over a
period of time, the standard deviation of daily, weekly or
monthly change in market demand. The time intervals
considered are impelled to be chosen by the behavior of
market demand. If it moves up and down rapidly over
short periods of time, it is highly volatile. On the con-
trary if market demand is almost never changed it has
low volatility.
The risk that the scheduler of manufacturing activities
is exposed to, is based on the potential for the volatility
of the underlying market demand or the market’s percep-
tion of that volatility to change. It is generally assumed
that uncertainty in the future depends on the time span.
The longer the time interval, the greater is uncertainty
about the situation by the end of this time interval and, as
a consequence, the volatility of a variable z under con-
sideration. To match with this commonly accepted belief,
the time-dependent probability density function of the
variable z at a fixed time, when normally distributed, is
[4]

2
1
,exp
2
2.
z
fzt t
t 





(1)
The mean value of at any time is
zt


,0Ezt

and the variance of at time is
zt

t t,zVar

.
In many contexts of business life changes in the values
of a variable are not measured in absolute terms but as
increments with respect to its values, that is in relative
terms. That is why volatility is calculated as the variance
of a log function representing the values of a variable at
different times.
Denoting the market demand as for the time
interval i, the volatility of this variable is defined as the
standard deviation of the function

Di


/ 1Di Diln



with 1in

.
3.2. Wiener Process
A Markov process is a particular type of stochastic proc-
ess where only the present value of a variable is relevant
for predicting the future. The past history of the variable
is supposed to be embedded in its present value. Predic-
tions for the future are uncertain and must be expressed
in terms of probability distribution. If the weak form of
market efficiency were not true, analysts could make
above-average returns on stocks by interpreting the cha-
rts of the past history.

Volatility Forecasting of Market Demand as Aids for Planning Manufacturing Activities385
The variable following a Wiener process has a
mean change of zero and a variance rate of 1 per time
unit. The Wiener process has been used in physics to
describe what is referred to as geometric Brownian mo-
tion [5]. The change during a period of time
z
zt

is
z   t (2)
where

has a standard normal distribution N(0,1).
The values of for any two different are sta-
tistically independent. If a finite time interval is bro-
ken down into time intervals so that
z
N
t
T
tNt

,
, the difference is described by the
sum of statistically independent increments
tT
 
0zT z

1
.
n
i
i
t 


 (3)
The volatility of over the time span T is the
variance of , that is T.
z

0z
 
zT 
When the market demand D is assumed to follow the
rules of the geometric brownian motion, it evolves ac-
cording the equation
dDDdt Ddz

  (4)
where is the increment of a Wiener process. dz
A variable evolving in this way is said to follow an Itô
process. This is precisely the model chosen by [6] for the
contingent claims analysis of options.
Equation (4) implies that the current value of is
known, but future values are log-normally distributed
with a variance that grows linearly with the time horizon.
Thus although information arrives over time ( is
changing) the future market value of the production re-
mains uncertain. Equation (4) describes the percentage
changes in,, which are the changes in the
natural logarithm of .
D
D
D/dD D
D
3.3. Itô’s Lemma
Suppose that a variable x follows an Itô process such that
 
,,dxxtxdtx txdz   (5)
Itô’s lemma [7] shows that the differential of a func-
tion


,Gxt is characterized by (4), it will be necessary
to compute variations . We can expand variations
by using the Itô’s lemma which leads to
dV
dV
2
22
2
1
2
GGG
dGxx dt
xt x










G
x
dz
x


 (6)
4. Uncertainty and the Analogy to Financial
Options
4.1. General Considerations
It is generally accepted that there is an immediate corre-
spondence between selling price and market demand. In
fact the selling price of manufactured products increases
along with the market demand when a specified constant
quantity of products is made available in the market
place. If the unit cost price is higher than the unit selling
price the payoff is negative. If the unit cost price is lower
than the unit selling price, then the payoff turns positive.
Demand volatility describes the randomness of market
behaviour and selling price. Therefore the selling price P
and, as a consequence, the value of a manufacturing
batch are determined by the market demand in terms
of the quantity of products asked by customers.
D
Two types of reasoning can be envisaged when sched-
uling manufacturing activities. The first one is developed
by insurers. The selling price is forecast for future time
periods with an expected value (mean) and a standard
deviation that is regarded as the measure of the risk in-
curred. The quantity of products to manufacture is ad-
justed to the expected selling prices. When the manufac-
tured products become available in the market place with
some manufacturing lead times, the market prices turn
higher or lower than the cost prices. In this situation the
attitude of the forecaster is to consider that over a large
number of time periods the cumulative payoffs turn posi-
tive.
A second type of reasoning is followed by traders.
Their strategy is to follow the market dynamics not to
minimize risks but to cancel them for each course of ac-
tion. This is engineered by a hedging mechanism. Within
a manufacturing context the tactics taken by a scheduler
can be to choose the timing to release manufacturing
orders and the quantity to manufacture intended to satisfy
a specified proportion of customers. In this paper we
assume that deciding on releasing a manufacturing order
depends only on the unit selling price and the unit
cost price .
P
C
Let us see how this technique can be applied to sched-
uling manufacturing activities. The opportunity value of
the option to release a manufacturing order today de-
pends on a stochastic process for . The problem is to
find the future value V of the option today. To solve
this problem, two approaches can be called on, i.e. the
technique of portfolio and dynamic programming [8,9].
Both of them yield the same result [10]. In our case the
solution by dynamic programming is more straightfor-
ward and is chosen.
P
4.2. Pricing a Manufacturing Option V(t)
The value


Vt is a growing function of the selling
price P. The selling price is supposed to follow a
Wiener process as (4) and then its derivative writes

Pt
2To disinvest: to realize invested assets of a person, institution…
(Oxford Dictionary)

Volatility Forecasting of Market Demand as Aids for Planning Manufacturing Activities
386
dPPdt Pdz

  (7)
In the first step let us consider the deterministic evolu-
tion of , i.e. P

= 0. The value of the manufacturing
opportunity at time T is

,(0). .
T
VPTP eCeT

 



(8)
with

 , otherwise


,VPT would increase in-
definitely without delivering an optimal value forT.

is the discount rate.
Differentiating


,VPT with respect to T yields
the value maximizing
*T


,VPT

*1/ ln/0TCP    

(9)
It leads to
*T=0 when =1

/0CP  

C
or
 
0/PC   (10)
That means it is advisable to launch the manufacturing
of products when the selling price goes beyond


/C

 . Then,


0P is considered as the optimal
price to release a manufacturing order.
*
P
In the second step we have to consider the situation for
σ ≠ 0. To calculate the value of the option two situations
are taken into account. After the manufacturing order has
been released the value of the option equals the unit sell-
ing price minus the unit cost price incurred , i.e.
. Before the manufacturing order has not
been released the manufacturing opportunity yields a
return from its capital appreciation at the discount rate
C

VPP C

.
When applying the Itô’s lemma (6) it appears that
, the derivative of the underlying selling price ,
is also governed by a Wiener process (4). The expected
value of induced by

Vt P
dV


Pt during the time interval
writes dt

EdVVdt   (11)
Expanding using Itô’s lemma (6) gives dV

2
22
2
1
2
VV
EdV PPdt
PP










(12)
Hence, using (11,12), it leads to
2
22
2
1
2
VV
PP
PPV





 
1
(13)
Equation (13) as to be solved with the boundary con-
ditions

00V (14)

**VPP C (15)
/VP when (16) *PP
(16) expresses the continuity condition at the junction
point *PP

with


VPP C

VP
). The solution of the
differential equation (13) is .AP

 such that
is a root of the quadratic equation
P


21/2 0
 

 (17)
Two roots are found. The negative one does not fulfil
the boundary condition (14) so that the other one whose
value is higher than 1 is left. The valid value solution for

is given by
2
22 2
112
1
22
 
 


 

 (18)
From the other boundary conditions (15,16) it is de-
duced that
*1 P


C (19)
and

*
*
PC
A
P


 (20)
It turns out that the solutions are

*/VPCPP

*

 when *PP
and
VPC

 when (21) *PP
It can be shown that /1

 increases as

in-
creases. As

 we have 1

 and
/1


. In other words, ceteris paribus, the higher
the uncertainty about the market price measured by

,
the higher is the selling required to release a manu-
facturing order. This conclusion is fully in line with what
the intuitive common sense expects.
*P
4.3. Concluding Remarks
In this section, we have shown that the optimal selling
price to release a manufacturing order in a determi-
nistic situation contrasts strongly with this that is valid in a
stochastic situation characterized by its volatility
*P

. We
think that this result can assist efficiently in scheduling
manufacturing activities because the number of parameters
required is limited and can be easily deduced from past
observation on the supposition that the future behaviour of
the market demand can be deduced from the past.
5. Example of Application
5.1. Variations in the Construction Cost Index
Derived from Observations in the
Construction Sector
The quarterly variations in the French construction cost
index over 10 years (1999-2008) are presented Table 1.

Volatility Forecasting of Market Demand as Aids for Planning Manufacturing Activities387
These figures reflect the market price for products and
services linked with this economic sector. In one way or
the other the values of materials and services provided in
this economic sector are strongly correlated to this con-
struction cost index.
Launching a development project in the construction
sector depends heavily on the economic context. When
the economy is sluggish, the investor-readiness to start a
project is restrained. On the contrary, when the economic
conditions are favorable, investors are inclined towards
undertaking capital-intensive projects such as developing
building sites. Their attitude is similar to that of option
traders. On the other hand the construction sector moni-
tors the activities of many upstream manufacturing in-
dustries (metallurgical, electrical, mechanical…). This is
why the construction index was chosen as a relevant
point of reference.
Figure 1 gives the scatter diagram of indexes X and
time intervals in an attempt to gain an impression of the
relation between the construction cost indexes and time
intervals. It shows a positive correlation. As the Itô proc-
ess refers to a deterministic exponential function com-
bined with “noisy” variations constituting the stochastic
part of the index variations we tried to fit the set of data
by an exponential regression.
The form of the exponential curve is
t
X

  (22)
where γ and δ are parameters to be estimated from the
data. Denoting these estimates by and respect-
tively, we can estimate
cd
X
by
X

from the data regres-
sion curve
.t
X
cd

(23)
Taking natural logarithms we obtain the regression
curve
Table 1. Quarterly variations in the construction cost index
in France from 1999 till 2008 (source: INSEE).
Year 1st Q 2nd Q 3rd Q 4th Q
1999 1071 1074 1080 1065
2000 1083 1089 1093 1127
2001 1125 1139 1145 1140
2002 1159 1163 1170 1172
2003 1183 1202 1203 1214
2004 1225 1267 1272 1269
2005 1270 1276 1278 1332
2006 1362 1366 1381 1406
2007 1385 1435 1443 1474
2008 1497 1562
Figure 1. Scatter diagram of construction cost indexes over
time (derived from Table 1).

ˆ
lnln ln
X
cd t

(24)
Each pair of observations

,
ii
X
t satisfies the rela-
tion


lnln ln
iiiii
X
c dteabte

 (25)
where is a stochastic variation at time .
i
The least-squares fitting procedure gives
ei t
X

= 1036* (1.0095)
t = 1036*


exp t

where

= (1,0095) = 0,00945516 ln
A measure of the quality of fit is given by the correla-
tion coefficient RE square which is here 0,967.
The standard deviation ofi egives the parameter

,
and we obtain

= 0,016.
The relative variations per quarter in /
X
X are plot-
ted Figure 2 showing the spreads of i

 for each quar-
ter with respect to

. /
X
X

for time interval
writes when
i
t

= 1(quarter):


/i
i
XX

 
5.2. Uses of the Previous Results
The forecasted values of
X
for quarters after the 2nd
quarter of 2008 valued 1562 are given by:
X

(t) = 1562*(1.0095) t
where is measured in quarters starting in the 3rd quar-
ter 2008.
t
Since the standard deviation of changes in
X
grows
with the square root of the time horizon, the upper and
lower bounds of a 68-percent forecast confidence interval
(Gaussian distribution) write respectively plus and minus
1562*(1.0095) t (1.0255)√t .
The optimal value of X is derived from the equation


/1C  . The only unknown in the expression of


Volatility Forecasting of Market Demand as Aids for Planning Manufacturing Activities
388
Figure 2. Scattergr aph of dX/X around

.
0
2
4
6
8
10
12
14
16
18
20
0,010,0150,020,0250,03 0,035 0,04 0,045 0,05 0,055 0,06
discount rate
Beta/(Beta -1)
Figure 3.

versus

1/  

.
is the discount rate per quarter

. Figure 3 portrays the
variations in as a function of

/  

1

.

and

are given the values derived from the exponential regres-
sion we carried out.
5.3. Application to Scheduling Manufacturing
Activities
The application of the model we presented requires the
availability of a very limited number of parameters. As a
consequence it should be easily used by scheduling de-
partments of manufacturing companies. On one hand two
of them reflect the market’s behaviour, i.e. its evolution
given by α characterizing its exponential change over
time and σ its volatility. They can be easily derived from
historical market data that is generally easily accessible.
On the other hand the discount rate

and the unit cost
price have to be derived from managerial data. They
are easier to describe than to determine.
C
We have seen that the discount rate ρ must be higher
than the trend parameter

of the selling price. Other-
wise the manufacturing project would never be profitable.
This constraint makes sense economically speaking. In
fact the discount rate

is definitely not the ruling
market interest rate but the current or targeted return on
investment (ROI) available on the operations run by a
business. The constraint means that manufacturing a
batch of products is worthwhile when the anticipated
yield is higher than the increase rate of the market de-
mand in terms of selling price. Evaluating

for a
company is quite an exercise and a lot of discussion has
taken place to devise practical approaches. Calculating a
relevant cost price is also another difficult issue: the ad-
vent of what is called ABC (Activity-Based Costing)
proves the shortcomings of traditional methods.
6. Conclusions
It is important to bear in mind that the Itô process we
presented and used in this paper rests on many assump-
tions of which the main ones are mentioned hereafter.
The first one refers to the type of randomness we dealt
with. The validity of the Itô process we worked with de-
pends entirely on whether market demand and selling
prices really do fit the Gaussian curve. That means that
uncertainty is proportional to t1/2. Thus in other words it
is assumed that the deviation of market demand and cor-
related selling prices is directly proportional to the square
root of time. This is strongly challenged by Mandelbrot
and his followers [11,12] in many circumstances. The
bell-shaped Gaussian curve reflects a form of chance
called “mild” by Mandelbrot because it does not ac-
count for extreme events.
The second assumption we have to consider is the
current relevance of what is called a Markov process. It
is supposed that the increments for any two differ-
ent time intervals
z
t

are statistically independent,
which proves wrong in many situations because long-
term correlations exist [11].
In addition to these restrictive conditions that com-
mand the validity of the option pricing model, another
question can be raised: is the option pricing model more
suitable than other models to produce relevant conclu-
sions? The model described here is based on the analysis
of historical series, like other standard forecasting tech-
niques (exponential smoothing, Bass and Gompertz dif-
ferential equations, Fourier series…). In business prac-
tice, several forecasting models are used concurrently for
two main reasons:
1) achieving the adjustment of model parameters does
not prove that the model is valid: least-square methods
do well;
2) arguments advanced to decide which forecasting
model is best are difficult to be put forward. Furthermore
what was valid in the past does not preclude that the fu-
ture may behave according another model.
The option pricing model is eligible for forecasting
economic activities such as manufacturing along with
other standard models when the decision-making context
is comparable with that found for financial option trad-

Volatility Forecasting of Market Demand as Aids for Planning Manufacturing Activities
389
ing.
REFERENCES
[1] R. Elliott and E. Kopp, “Mathematics of Financial Mar-
kets,” Springer Finance, 2004.
[2] S. Thomassey, “Méthodologies de la Prévision des
Ventes Appliquée à la Distribution Textile,” (In French)
Thèse D’automatique et Informatique Industrielle, Lille 1,
2002.
[3] J. C. Hull, “Options, Futures and Other Derivatives,”
(Chapter 12), Prentice Hall, 2005.
[4] T. L. Saaty and J. M. Alexander, “Thinking with Models
(Chapter 3) ,” Pergamon Press, Oxford, England, 1981.
[5] P. Lévy, “Processus Stochastiques et Mouvement Brow-
nien,” 2ème Edition, Gauthier Villard, Paris, 1965.
[6] F. Black and M. Scholes, “The Pricing of Options and
Corporate Liabilities,” Journal of Political Economy, Vol.
81, No. 3, May - June 1973.
[7] K. Itô, “On Stochastic Differential Equations,” Memoirs
American Mathematical Society, 1951.
[8] R. E. Bellman, “Dynamic Programming,” Princeton Un-
versity Press, New Jersy, 1957, Dover Paperback Edition
2003.
[9] A. K. Dixit, “Optimization in Economic Theory (Chapter
11),” Oxford University Press, 1990.
[10] A. K. Dixit and R. S. Pindyck, “Investment under Uncer-
tainty (Chapter 5),” Princeton University Press, New
Jersy, 1994.
[11] B. B. Mandelbrot and R. L. Hudson, “The (mis) Behav-
iours of Markets (Chapter 2),” Profile Books Ltd., Lon-
don, 2004.
[12] J. Lévy Véhel et Chr. Walter, “Les Marchés Fractals (In
French), ” PUF 2002.