Page 1 of 17
Journal for Studies in Management and Planning
Available at http://internationaljournalofresearch.org/index.php/JSMaP
e-ISSN: 2395-0463
Volume 01 Issue 07
August 2015
Available online: http://internationaljournalofresearch.org/ P a g e | 109
Empirical Verification of Heston Model of Stock
Market Prices
1E.O.Ogbaji, 2E.S.Onah,3T.Aboiyar and 4A.R.Kimbir
1Department of Mathematics and Statistics, Federal University Wukari
(Ogbajieka@yahoo.com)
2,3,4Department of Mathematics/Statistics/Computer Science University Of Agriculture, Makurdi
Abstract
Volatility is a great concern to investors.
Investors like to know how much volatility or
risk that they are exposed to before they can
invest in a stock. Potential investors are
advised to invest in companies that exhibit
relative calm or high stability. In [1] where
they used geometric Brownian motion model
to study the behaviour of stock market prices.
In their study, volatility was considered over a
very long of time in such case, it is difficult for
investors to predict the behaviour of stock
price for a short period. Also [47] used Heston
model, where volatility was considered over a
short period. This work involves solving the
geometric Brownian motion model used by [1]
and Heston model by transforming it into
parabolic partial differential equation and to
show that the analytical solution satisfy the
parabolic differential equation. We used
selected companies from Nigerian Stock
Exchange to empirically verified Heston
model. A Pascal programming language was
used to code the Euler’s –maruyama method
of the solution of the Heston model. We
conclude that Stock price is randomly
distributed, positive return on investment and
low volatility of stock price imply viable and
growing company at a particular period,
negative return on investment and high
volatility of stock price imply non-viable and
collapsing company at that particular period,
volatility and return on investment are also
randomly distributed. In view of the above
results, we recommend that companies with
low stock price volatility and positive return
on investment are advised to investment
in.Heston model can be extended to three and
four compartments by incorporating return on
investment and return on investment volatility.
Key words: Stock price, voltality of stock
price, rate of return and return on investment
INTRODUCTION
In finance, the Heston model, named after
Steven Heston, is a mathematical model
describing the evolution of the volatility of an
underlying asset. It is a stochastic volatility
model: such a model assumes that the
volatility of the asset is not constant, nor even
deterministic, but follows a random process.
The Nigerian Stock Exchange (NSE)
commenced operation in 1961with only 19
companies worth N80million. As at May 2009,
the number of listed companies had increased
to 294, made up of 86 Government Stocks
with Industrial Loans Stocks and 208 Equity/
Ordinary Shares (including emerging market)
with a total market capitalization of N9.45
trillion.However, the Nigerian Stock Exchange
Page 2 of 17
Journal for Studies in Management and Planning
Available at http://internationaljournalofresearch.org/index.php/JSMaP
e-ISSN: 2395-0463
Volume 01 Issue 07
August 2015
Available online: http://internationaljournalofresearch.org/ P a g e | 110
still seems to have a long way to go when
compared with developed stock markets.
The volatility of each stock as estimated by the
conditional variance shows some reasonable
level of persistence and reverts to the mean
value very slowly. This could arise as a result
of misspecification of the conditional variance
function or the ignorance of structural break(s)
in the unconditional variance which has not
been established.
The ability of financial system and markets to play these roles hinges on the stability in the system particularly the stock prices. The major component of instability in stock prices is exhibited by the varying conditional variance (volatility) of the stock prices. Hence, to be able to establish and maintain a viable stock market that could enhance the growth of
the economy, there must be an in depth and
comprehensive understanding of the volatility
of stock price[28].
Volatility is one of the important aspects of
financial market developments providing an
important input for portfolio management,
option pricing and market regulations[39].
Stock returns volatility differs dramatically
across international markets
([46],[43],[43],[29,30],[10]and[3]) and have
received a great attention from both
academicians and practitioners over the last
two decades because it can be used as a
measure of risk in financial markets. Volatility
of stock returns has long been an issue of
interest in financial literature. A wide variety
of research has been conducted on stock
returns volatility in developed and emerging
markets since 1970s. Nature of volatility in
different markets at different times are
discovered, which are indeed of great interest
for financial economists. Financial economists
are also interested about the causes and
variables behind the existence and nature as
well as the anomalies relating to market
volatility.In [1] where they used geometric
Brownian motion model to study the
behaviour of stock market price. In their study,
volatility was considered over a very long
period of time in such case, it is difficult for
investors to predict the behaviour of stock
price for a short period. Also[47] used Heston
model, where volatility was considered over a
short period. In this study, we reviewed the
model used by [1] and [47],where empirical
verification of Heston model is done by using
data of selected companies from Nigerian
Stock Exchange.
Page 3 of 17
Journal for Studies in Management and Planning
Available at http://internationaljournalofresearch.org/index.php/JSMaP
e-ISSN: 2395-0463
Volume 01 Issue 07
August 2015
Available online: http://internationaljournalofresearch.org/ P a g e | 111
2.1 THE HESTON MODEL
In this section, we show derivation of the
parabolic partial differential equation of the
model and show how the solution of the model
satisfied the derived parabolic partial
differential equation. The following
assumptions are made; the interest rate
is
constant,Stock price St follows a Black- Scholes type of stochastic process, but with a
stochastic variance Vt that follows a
Cox,Ingersoll and Ross(CIR) process. Then
the model is given as:
dSt= Stdt Vt
StdWt
dVt=x(
Vt
)dt +
Vt
dZt
(2.1)
dWtdZt= dt
The parameters above are defined as below:
is the drift coefficient of the stock price
is the long-term mean of variance
x is the rate of mean reversion
is the volatility of volatility
St and Vt are the stock price and volatility
process respectively
To take into account the leverage effect, stock
returns and implied volatility are negatively
correlated.Wt and Zt are correlated Wiener
process, and the correlation coefficients is
.We dropped the time index and write
S=St,V=Vt.
We explain how to derive the PDE from
the Heston model. This derivation is a special
case of a PDE for general stochastic volatility
models form of a portfolio consisting of one
option V = V (S,v, t),
units of the stock S,
and
units of another option U =U(S, v, t)
that is used to hedge the volatility. The
portfolio has value,
= V +
S +
U
where
=
t. Assuming the portfolio is self-financing, the change in portfolio
value is
d
= dV +
dS +
dU
Apply Itô’s Lemma to dV . We must differentiate with respect to the variables
t,S, and v. Hence
