Prob lem Definition
Modeling Dynamics of Oil Supply – DemandShock Effects on Stock
Market Returns
Hassan Nojumi (author for all correspondence)
Department of Mathematical Sciences
Sharif University of Technology, Tehran, Iran
P.O.Box 11365-9415, Azadi Street, Tehran, Iran
E-Mail: [email protected]
[email protected]
Mobile: 0912 5937563
Tel: (+9821) 6616-5640
Fax: (+9821) 6600-5117
, Amirreza Shafaat, Hamidreza Golmakani, Hossein Salimi
Department of Industrial Engineering
Amir Kabir University of Technology, Tehran, Iran
Modeling Dynamics of Oil Supply – Demand
Shock Effects on Stock Market Returns
Hassan Nojumi*, Amirreza Shafaat**, Hamidreza Golmakani**,
Hossein Salimi**
* Department of Mathematical Sciences, Sharif University of
Technology, Tehran, Iran
** Department of Industrial Engineering, Amir Kabir University
of Technology, Tehran, Iran
Abstract
Throughout the industrial world, prices of financial products
are continually affected by various factors and events. One of
these significant factors is price of oil and oil products. This
papers aims at investigation of this connection through
mathematical modeling using stochastic analysis.
1. Introduction
Financial press has long asserted that stock prices and stock
market returns are affected by the oil price volatility. The Asian
financial crisis of 1997, the recession of 2001, and the
depreciation of the Dollar against the Euro in recent times can all
be read directly in the historical price of oil.
On the other hand, the resulting transfer of wealth from
industrialized nations to less developed oil-producing countries
has had a measurable effect on the financial system: From 2002 to
2003, the revenues accrued to the OPEC 10 members from sales jumped
from $180 billion to $240 billion, as the price of WTI averaged
$31.10/bushel (an increase of 30% from the previous year).
These revenues (the so-called petrodollars) have the potential
to move currencies, stocks and interest rates. For instance, the
market capitalizations of most Persian Gulf stock exchanges doubled
between January 2003 and September 2004, following the oil price
increase. As an illustration, Figure 1 plots the returns of major
equity indexes in the Persian Gulf, with the base set on January 1,
2003 and their correlation to the oil price. As it can be seen,
stock indexes of oil producing countries in are positively affected
by oil price growth.
Stock markets of 18 major countries (in local currencies) and 33
emerging markets have been analyzed in [6], in which, using
regression analysis, the stock price data with various oil price
indexes (Brent crude, WTI and Dubai ) have been compared to see if
there is any correlation. It turns out there is; in 12 of the 18
countries, changes in oil price have significantly predicted future
market returns on a lagging monthly basis. Not surprisingly, rise
in oil price suggests a lower stock market and a drop in oil price
infers rise in stock prices. The magnitude of the oil price shift
is also carried over into the magnitude of the expected
increase/decrease in stock prices.
As empirical studies have shown [10], we assume that there are
logical interconnections between stock market returns and oil price
volatility. With this assumption, what is sought is to model the
interconnections to capture and better estimate the behavior of
stock markets.
The paper continues by describing our model. A review of related
literature is presented in section 3. We model the two country
world in section 4; first with the world modeled in simple form
when market is complete and there are no jumps, then with the jumps
being incorporated. In both parts we consider a portfolio which is
risk neutral to non-oil effects, with price movement rela6ed to oil
volatility and shocks.
2. Model Description
We will model the oil supply-demand shock effects on stock
market returns in the form of system dynamics and stochastic
differential equations.
First we divide the world in to two parts: oil producers and oil
consumers. In these two categories the effect of oil price
volatility and jumps are opposite each other (see [10], [6]). A
simple two-country model is then developed. Using this model, stock
and oil prices are jointly determined.
The model identifies interconnections among crud oil price and
stock markets, and characterizes their joint dynamic as a
multi-factor model. These factors describe the international
financial equilibrium model and the interconnections between its
components (oil price and stock indexes). The model also contains
factors of oil supply and demand shocks for determining responses
of stock markets to these shocks.
We have assumed that crude oil price and stock indexes follow
Levy processes such as CGMY, Variance Gamma (VG), Normal Inverse
Gaussian (NIG), Log-Stable (LS) or some combined models such as
EJP, Bates and etc [4], [5].
Thus the conceptual system can be defined as follows:
Empirical validity of our model’s implications can be tested in
several dimensions. For this purpose one should run the model with
real data of some creditable financial markets such as NYMEX, WTI,
Brent Crude, Dubai Stock Exchange, London Stock Exchange, etc. Then
the results should be compared with the effects of oil price
volatility and jumps on stock market returns of both oil producer
and consumer in the real world.
3. Related Literatures
Various types of mathematical and statistical tests have been
used by analysts in determining the effective elements on an
international financial network.
Oil has important role in international finance and economy.
Also, stock market returns are affected by oil price volatility,
especially when the oil price trends include upward or downward
jumps [10], [6].
It has been shown [12] that oil price increases have been at
least partly responsible for every post-World War II US recession
except the one in the 1960’s. These basic findings have been tested
using alternative data and estimation procedures [11], [2], [19].
In the early 1990s the focus of research was on the role that
asymmetric oil price shocks have on the economy [20], [21], [17],
[9].
In sharp contrast to the volume of work investigating the link
between oil price shocks and macroeconomic variables, there has
been relatively little work done on the relationship between oil
price shocks and financial markets. Two notable exceptions are [14]
and [3].
In [9] quarterly data was used to test whether the reaction of
international stock markets to oil shocks could be justified by
current and future changes in real cash flows and/or changes in
expected returns. Using a standard cash-flow dividend valuation
model (see [3]), it was discovered that the reaction of Canadian
and US stock prices to oil price shocks can be completely accounted
for by the impact of these shocks on real cash flows. The results
for Japan and the UK are, however, not as strong.
A Vector Auto Regression (VAR) approach was used in [14] to
investigate the relationship between daily oil futures returns and
daily US stock returns. It was learned that oil futures returns do
lead to some individual oil company stock returns but oil future
returns do not have much impact on broad-based market indices like
the S&P 500.
Using quarterly data from 1947 to 1991, it was found that oil
prices do have an effect on aggregate stock returns [16]. In
contrast, daily data from 1979 to 1990 were used and no evidence
was found of a relationship between oil futures prices and
aggregate stock returns [14].
The dynamic interaction between oil prices and other economic
variables were examined in [23] using VAR approach. Results were
presented for oil price shocks, asymmetric oil price shocks, oil
price volatility shocks and asymmetric oil price volatility shocks.
It was found that oil price shocks have asymmetric effects on the
economy. In addition, it was found that the dynamics of oil price
shocks have changed across time. It was also found that oil price
shocks had a significant impact on real stock returns although this
impact was stronger after 1986.
The issue of asset pricing, in general, is controversial in
relation to equities. The development of alternative multi-factor
models, such as the Arbitrage Pricing Theory (APT), was a response
to the failure of traditional models, such as the Capital Asset
Pricing Model (CAPM), to adequately explain cross-sectional
variation in equity returns. Within this context, researchers have
sought to examine the sensitivity of equity returns to economic
factors.
For example, in [1] a two-factor version of the APT, using a
market and an oil price change factor was investigated. Based on a
sample of 29 NYSE-listed oil companies covering the period
1970–1978, it was found that the oil price risk premium was highly
unstable. Specifically, in the period immediately following the
OPEC oil price shock of 1973, equity returns of domestic oil
producers and large multinationals were associated with a
significant ex ante oil risk premia. While this investigation of
the asset pricing issue in the context of a small sample of oil
companies served the narrow purpose of this study, the question was
left open of how pervasive the oil price factor is across all
sectors. Similarly, in [16] an APT-type model was used in which
fundamental variables were augmented by an oil price change
variable.
Sensitivity of Australian industry equity returns to an oil
price factor over the period 1983–1996 was investigated in [8]
using the two-factor APT model. The followings were the key
findings. First, a degree of pervasiveness of an oil price factor,
beyond the influence of the market, was detected across some
Australian industries. Second, it was proposed and found that
significant positive oil price sensitivity existed between the Oil
and Gas, and Diversified Resources industries. Similarly,
significant negative oil price sensitivity was found in the paper
and packaging, and transport industries. Generally, it was found
that long-term effects persist, although we hypothesize that some
firms have been able to pass on oil price changes to customers or
hedge the risk.
More recently, using an error-correcting representation of a VAR
macroeconomic model and using a monthly data for Greece for the
period January 1989–June 1999, it was concluded that oil prices are
important in explaining stock price movements [22].
In [13] the spillover effects, day effects and dynamic
relationships among five S&P 500 oil sector stock indices and
five oil prices was examined for the U.S. oil markets using daily
data for the available period July 17, 1995 to October 10, 2001. It
was found that: (1) the oil price systems have common trends,
suggesting little potential for long-run portfolio diversification.
(2) In the S&P 500 oil sectors stock index system, the five
indices are not co-integrated, suggesting no index integration and
strong opportunities for gains from diversification. (3) On a daily
basis, none of the oil industry stock indices explains the future
movements of the NYMEX oil futures prices, while these prices can
explain the movements of independent oil companies engaged in
exploration, refining, and marketing; confirming the results that
the oil exploration companies and refiners take their cues from the
oil market. (4) The autoregressive conditional heteroskedasticity
(ARCH)/GARCH analysis suggests that the oil futures market has a
matching resonant or volatility-echoing effect on the stocks of the
oil exploration, production, and domestic integrated companies, and
a volatility-dampening effect on the stocks of oil international
integrated and oil and gas refining and marketing companies. (5)
The day effect with oil volatility transmission suggests that
Friday has a calming effect on the volatility of the oil stocks.
The policy implication is that, at times of oil volatility, traders
should choose the trading day and the S&P 500 sectors that
match their tolerance for volatility and use the right financial
derivative to profit from this volatility.
Although oil price is the considerable factor in international
finance, there is no idea what its real effect is and why the
financial markets react so slowly to the oil price volatility [10].
There has been little attempt to compare the effect of oil price
volatility on both oil producing and oil consuming countries
simultaneously, in which the oil price behavior and jump models are
considered. This has motivated the researchers to model the
co-movement among stock market prices and oil prices within a
multi-factor two-country dynamic equilibrium in which oil price
jumps are considered. The oil price effects on financial markets
can then be explicitly determined. This is the purpose of this
paper.
4. Modeling the Two Country World
4.1. Market Is Complete and There are No Jumps
Suppose that the process followed by country i oil output is
geometric Brownian motion [18]:
d Yi(t)/ Yi(t) = µi(t).dt +σi(t).dωi(t) (4.1)
where µi(t) and σi(t) >0 are arbitrary adapted processes, Yi
is country i oil output and
)
(
t
i
w
is Brownian motion.
Now if there are just two countries in world, one oil producer
(i=1) and the other oil consumer (i=2), then the total oil output
in the world will be:
å
å
=
=
þ
ý
ü
î
í
ì
+
÷
ø
ö
ç
è
æ
-
=
=
2
1
2
2
1
)
(
2
exp
)
0
(
)
(
)
(
i
i
i
i
i
i
i
i
Total
t
t
Y
t
Y
t
Y
v
s
s
m
(4.2)
This is equal to oil total consumption, and we have
i
i
i
i
i
i
i
i
i
i
i
Total
d
Y
dt
Y
t
dY
Y
d
t
dY
w
s
m
å
å
å
å
=
=
=
=
+
=
=
=
2
1
2
1
2
1
2
1
)
(
)
(
)
(
)
(
(4.3)
It is assumed that oil price is dependent only on
)
(
t
Y
Total
and t, in other words it is dependent only on
1,2
i
,
)
(
=
t
Y
i
and t. Therefore equation (4.4) shows process of the oil
price:
i
i
i
d
s
mdt
O
dO
w
å
=
+
=
2
1
(4.4)
Where O is the oil price, m is the expected return from oil,
i
d
w
i=1, 2 are the same as equation (4.1) and
2
,
1
i
,
=
i
i
d
s
w
are the components of the risk of return attributable to Yi,
i =1, 2. Thus it can be shown that
i
i
i
s
r
m
å
=
=
-
2
1
l
(4.5)
where r is risk free interest rate and λi is the market price of
risk for Yi.
Assume that the market index uncertainty in each of the consumer
and producer countries is separated to oil effects and non-oil
effects parameters. Thus we have:
P
P
i
i
i
P
P
P
p
d
S
d
S
dt
M
I
dI
w
w
¢
¢
+
+
=
å
=
2
1
,
C
C
i
i
i
C
C
C
C
d
S
d
S
dt
M
I
dI
w
w
¢
¢
+
+
=
å
=
2
1
,
(
)
0
)
,
(
and
0
,
>
<
O
I
COV
O
I
COV
P
C
(4.6)
where Ip is stock market index of the oil producer country, and
similarly IC is stock market index of the oil consumer country, MP
and MC are expected returns of indexes,
i
d
w
i=1, 2 are the same as equations (4.1) and (4.4),
2
,
1
i
,
=
i
i
d
S
w
are the components of the risk of return attributable to Yi, i
=1, 2 and
C
,
P
k
,
=
¢
¢
K
K
d
S
w
are the components of the risk of return attributable to non-oil
effects.
Now consider the portfolio
P
p
¢
which consists of all financial assets Ai i=1, 2,…, m in market
index of oil producer country with the same weight as in market
index which has the following property:
(
)
0
,
=
O
A
COV
i
(4.7)
Thus the uncertainty of this portfolio just refers to non-oil
effects and the process followed by
P
p
¢
is:
P
P
P
d
S
dt
M
d
P
P
w
p
p
p
p
¢
+
=
¢
¢
¢
¢
(4.8)
One can make the portfolio
P
p
risk neutral to non-oil effects by combining
P
S
p
¢
units of IP and -
P
S
¢
units of
P
p
¢
. This portfolio stochastic movement just refers to oil price
volatility so that one could analyze the oil effects on stock
market of oil producer country. Similarly, by making portfolios
C
p
¢
and
C
p
risk neutral, one can do the same analysis for stock market of
consumer country. Note that stochastic movements of portfolios
C
p
and
P
p
must be the same and in opposite directions; in other words,
(
)
1
,
-
=
C
P
Corr
p
p
.
Thus we have criteria and measures (
C
and
p
p
P
) for analyzing oil supply-demand shock effects on stock market
returns of each of oil producer and consumer country.
4.2. Market Is Complete and There are Jumps
Sometimes in the real world, because of unpredictable events,
oil supply or demand changes are high. Hence it is more accurate to
model processes with jumps so that equation (4.1) changes as
follows, where dJi’s are jump process which can follows CGMY, VG,
NIG and other Levy processes:
d Yi(t)/ Yi(t) = µi(t).dt +σi(t).dωi(t)+
i
a
dJi (4.9)
And world total oil output dynamics is then:
EMBED Equation.3
å
å
å
å
å
=
=
=
=
=
+
+
=
=
=
2
1
2
1
2
1
2
1
2
1
)
(
)
(
)
(
)
(
i
i
i
i
i
i
i
i
i
i
i
i
i
i
Total
dJ
d
Y
dt
Y
t
dY
Y
d
t
dY
a
w
s
m
(4.10)
Again if it is assumed that oil price is dependent only on
)
(
t
Y
Total
and t, then oil price dynamics change to:
å
å
=
=
+
+
=
2
1
,
2
1
i
i
O
i
i
i
i
dJ
d
s
mdt
O
dO
a
w
(4.11)
leading to market index processes as follows:
P
P
i
i
P
i
P
P
i
i
i
P
P
P
p
J
d
dJ
d
S
d
S
dt
M
I
dI
¢
¢
+
+
¢
¢
+
+
=
å
å
=
=
a
a
w
w
2
1
,
2
1
,
C
C
i
i
C
i
C
C
i
i
i
C
C
C
C
J
d
dJ
d
S
d
S
dt
M
I
dI
¢
¢
+
+
¢
¢
+
+
=
å
å
=
=
a
a
w
w
2
1
,
2
1
,
(
)
0
)
,
(
and
0
,
>
<
O
I
COV
O
I
COV
P
C
(4.12)
Where
i
dJ
and
2
,
1
i
,
=
i
i
d
S
w
are the components of the risk of return attributable to Yi, i
=1, 2 and,
K
J
d
¢
and
C
,
P
k
,
=
¢
¢
K
K
d
S
w
are the components of the risk of return attributable to non-oil
effects.
Now again consider the portfolio
P
p
¢
which consists of all financial assets Ai i=1, 2,…, m in market
index of oil producer country with the same weight as in market
index which has the property (4.7). This portfolio’s uncertainty
just refers to non-oil effects and the process followed by
P
p
¢
is:
P
P
P
P
J
d
d
S
dt
M
d
P
P
P
¢
+
¢
+
=
¢
¢
¢
¢
¢
p
p
p
a
w
p
p
(4.13)
Now if the ratio of Brownian changes in
P
I
and
P
p
¢
is equal to ratio of jumps in
P
I
and
P
p
¢
,
÷
÷
ø
ö
ç
ç
è
æ
¢
=
¢
¢
¢
P
P
P
P
S
S
p
p
a
a
, then we can make the portfolio risk neutral to non-oil
effects
P
p
by combining
P
S
p
¢
units of IP and -
P
S
¢
units of
P
p
¢
. Similarly we can make same portfolio
C
p
for consumer country. All the conditions mentioned in section
4.1 for
C
and
p
p
P
are also confirmed here.
5. Conclusion
In recent times, oil price shocks and energy price crises have
had a big role in international finance; thus financial analysts
should make more sophisticated instruments for modeling these
phenomenon. Here we introduced a criterion for measuring and
analyzing the impact of oil supply-demand shocks on stock market
returns of both oil producer and consumer countries. Possible
extensions of this model include extension into a multi-country
model, and extension by describing the jumps more clearly and with
more details.
Acknowledgements
The authors thank The Department of Mathematical Sciences of
Sharif University of technology, and The Department of Industrial
Engineering of Amir Kabir University, for their academic and
financial support.
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�
Figure1.Persian Gulf stock indexes and oil price
Figure2.Conceptual system for two-country model
� Organization of Petroleum Exporting Countries
� West Texas Intermediate
� In this paper recent fall in Persian Gulf stocks has not been
considered.
� Most of these18 countries are oil consumers.
� These models are made by combination of stochastic volatility
model, and jump models
� One can ask how increasing oil output could increase oil
price. This occurs because in the model it is assumed that world
total oil output is equal to world total oil consumption
� For more information about market price of risk, see [15]
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