Assumptions of
the BSM Approach
Before
proceeding with the description of the fundamental assumptions of the BSM model
on option pricing, it is necessary to define what an option actually is. As
Black and Scholes (1973) assert, an option is a financial contract that gives
an investor the right to buy or sell an asset within a specified time window
and under determined conditions. There are two main kinds of options: the
European option and the American option; the former differs from the latter as
it gives the owner the right to sell the shares at a prearranged price, the
exercise price (or strike price), at the prearranged date: the expiration
date. On the other hand, the American option can be exercised at any date
before maturity (Merton, 1977). Furthermore, option types can be divided
according to flexibility in terms of execution: the owner of a call or a put
option – namely options that can be bought or sold on or before a specific date
 has the right to exercise it for a price said to be the option premium. Both
of these two options can be either naked – when the owner does not own the
underlying stock  or covered (Fortune, 1996). Nevertheless, the emphasis of
the BSM is put on the Europeanstyle options.
The assumptions
that make the BSM valid are presented below, amongst the standard assumptions
^{
}
(Bailey, 2005; Merton, 1998; Black
and Scholes, 1971) for general option pricing models:

Markets
are frictionless; there are no transaction costs in buying or selling an
underlying asset, no penalties are imposed for short selling and no
institutional restrictions are enforced on trading.

It is
possible to lend and borrow in unlimited amounts, if a constant and
riskfree interest rate is guaranteed. The assets are perfectly divisible,
so that each part can be sold or lent independently.

The asset
pays no dividends or other distributions during the life of the option,
which is protected against possible stock splits. Additionally, the
options considered are only the European type, meaning they can only be
exercised at the prearranged maturity date.

There are
no riskless arbitrage opportunities: this can be viewed more like an
implication from the other assumptions rather than an assumption itself.

Investors’
preferences are assumed to be such that they would prefer to hold more
rather than less.

Logarithmic
returns of an underlying stock are normally distributed, and follow a
continuous random walk according to a Brownian motion
^{
[1]
}
.
Moreover, markets are continuously open so that trading can happen
anytime.
With the last
assumption, Black, Scholes and Merton made a significant contribution to the
standard assumptions already existing in financial models such as the ones
described by Sharpe
(1964)
and Lintner (
1965)
for the Capital Asset Pricing Model. The set of assumptions helps give credibility
to the predictions. Bearing in mind the abovementioned theorizations, it is
now possible to outlay the formulae for the option premium (
Hull, 2011)
regarding
European call (1) and put options (2).
1.
c=S
_{
0
}
N(d
_{
1
}
)Ke
^{
rT
}
N(d
_{
2
}
)
2. p=Ke ^{ rT } N(d _{ 2 } ) S _{ 0 } N(d _{ 1 } ) _{ }
where
3.
d
_{
1
}
=
ln(S
_{
0
}
/K)+(r+σ
^{
2
}
/2)T
σ√T
4.
d
_{
2
}
=
ln(S
_{
0
}
/K)+(r+σ
^{
2
}
/2)T
= d
_{
1
}
σ√T
σ√T
In particular,
c
and
p
are the European call and put option respectively;
S
_{
0
}
represents the initial
stock price at time
t
=0 while
K
equals the strike price.
R
is the risk free rate, which is
continuously compounded;
T
is the
exercise or maturity date, and
σ
is
the volatility of the stock price.
N(x)
represents the cumulative probability distribution function for a N(0,1)
distribution; thus the probability that a variable that has a standard normal
distribution
^{
[2]
}
will be less than
x
.
The empirical
estimation given by Black and Scholes (1971) aimed at determining the
equilibrium value of an option shows ambiguous results; the difficulty in
estimating the variance caused the model to predict overpriced options on high
variance stocks, and underpriced options on low variance stocks. The study
also sheds light on the presence of nonconstant variance, and the difficulty
in completely ruling out the transaction costs that seem to be quite high. As
can be seen from (1) and (2), the only variable that is impossible to observe
before the option price is computed is the (explicit) volatility
σ
. Moreover, it is clear that the
price of a call or put option does not depend on the expected rate of return,
suggesting that even if investors do not agree on the return of some asset, the
model will still hold. It is also independent of investors’ beliefs and
preferences, but all investors must agree on the value of the volatility for
the model to be true.
Merits and
Critiques of the Model
Although the
model was constructed to be true only for the Europeanstyle options, Merton
(1973) describes how if there are no dividend payments during the time up to
the maturity date of an option  and hence there is no profit in exercising the
option before its exercise date  then the BSM model applies also to American
call options, that do not pay a dividend prior to the expiration date. In his
paper, Merton underlines the versatility of the model and also expands on the
arbitrage principle and dynamic hedging.
In particular,
he explains how it is possible to minimise or totally eliminate the risk
associated when an option is issued by constructing a replicating portfolio.
Specifically, the financial intermediary creates a hedging portfolio made of
risky and riskfree assets; if the payoffs of an option are exactly replicated
by the portfolio, this is called a “replicating portfolio” (Merton, 1998, p.
329). When this happens, the replicating portfolio only consists of the
underlying asset, which the call option was sold on.
Nevertheless,
the dynamic hedging strategies are only possible theoretically; but when the
hedging portfolio is well diversified, the risk involved purely concerns market
risk  that cannot be hedged. This means the elimination of nonsystematic
risk, and only the market risk affects the return of the hedging portfolio. The
adaptability of the model makes it possible for it to still be employed today
to compute prices of put and call options; its simplicity is a crucial element
that determined its success, alongside the fact that it provides a benchmark to
assess competing models. It can also be used to price various elements of a
firm capital's structure; in other words, the total value of the firm can be
employed as an individual security thus replacing the concept of option under
which the whole model is based (Merton, 1973).
However, the
BSM approach has been shown to have several limitations too.
Recalling the
Merton (1973) development of the dynamic hedging portfolio, making use of this
strategy cannot always be possible due to discontinuous trading and transaction
costs that violate assumption (1); additionally dynamic hedging could imply
risk and is associated with costs, hence the riskfree implication does not
hold.
Another of the
assumptions can be subject to criticism; the perfect divisibility of
assets
(assumption 2, which allows for
the assets to be split and sold or lent separately) is violated as often,
financial intermediaries construct a bundle of option contracts to be traded.
Moreover, assumption (6) can be argued: markets cannot allow continuous trading
as there will always be execution lags – although these lags have been
considerably reduced by the advent of new technologies that allow trading to
happen within nanoseconds. Additionally, the formulae to get the price of a
call and put option also lack an important element: Garleanu et al. (2009) show
how the price of the option is determined not only by the factors shown in the
BSM model, but also by its demand.
Fortune (1996)
empirically confutes parts of the model assumptions, specifically the one
regarding the normal logarithmic distribution of the underlying stock price, by
observing that the relative frequency distribution is not normally distributed,
is very leptokurtic
^{
}
(
distribution whose points form a higher peak than in the normal
distribution),
and
shows skewness.
Nonetheless,
amongst the shortcomings of the model, particular importance must be given to
the role of volatility. Although Black and Scholes (1973) were well aware of
the difficulties that could be encountered when trying to estimate volatility,
a wrong calculation could lead to completely wrong predictions of the option
price. In order to give a wellrounded analysis it is necessary to first make a
distinction between explicit (realised) and implicit (or implied) volatility.
As Bailey (2005) states, the former is calculated as the standard deviation of
previous data on asset price, with the following formula:
s
^{
2
}
=
(g
_{
1
}
g)
^{
2
}
+(g
_{
2
}
g)
^{
2
}
+(g
_{
3
}
g)
^{
2
}
+…+(g
_{
N
}
g)
N1
Where
g
is the sample average,
g
_{
1, 2, 3
}
_{
}
etc. are
the rates of return for each date. It is assumed that the volatility remains
constant within time and does not depend on investor sentiment, although there
is no consensus about how it should be modelled if it were nonconstant.
Interestingly,
implicit volatility is the market’s assessment of the underlying asset’s
volatility – the market’s estimate of the constant volatility parameter
(Mayhew, 1995)  and the value that satisfies the BSM formulae. In order to
calculate it, one would simply substitute the price of the option into the
formula and use the volatility as the unknown to be found. One useful trait of
implicit volatility is that it can be defined as forward looking, meaning that
it is founded on the future expectations of the investors. Moreover, implicit
volatility can be computed for different option contracts within the same
underlying asset: when they are not equal, there is evidence of nonconstant
volatility and the BSM formulae do not hold
(Bailey, 2005)
.
In fact, implied volatility can graphically assume different shapes; as Mayhew
(1995) underlines, across the strike price value they take the form of “smiles”
or “skews”. Considering the defects of volatility, modelfree estimates of the
implicit volatility have been implemented; one of the most significant is the
Chicago Board Options Exchange VIX index. Looking at the historical background,
in 1993 the CBOE
^{
[3]
}
introduced the VIX index that was originally constructed to measure the
market’s expectations on the S&P100’s (
Standard and Poor’s)
30day implicit volatility and consequently, became
the benchmark for the U.S. volatility. It is founded on the assumption of
arbitrage absence and lack of discrete jumps. A derivation of the formula goes
beyond the scope of this paper, but it takes the following form:
Figure
1
Derivation of Chicago Board Options Exchange VIX index
Where:
s
^{
2
}
= (VIX/100);
T
is the time to expiration;
F
is the
forward index level desired;
K
_{
o
}
_{
}
equals the first strike below
F
;
K
_{
1
}
_{
}
is the
strike price of the
ith
out of the
money option;
ΔK
_{
i
}
represents the interval between strike prices;
R
equals the riskfree interest rate;
Q(K
_{
i
}
)
is the “midpoint” of the bidask spread for each
option with strike
K
_{
i
}
. The
VIX index can be considered as a market forecast of variability, because it
reflects the beliefs and option preferences of the investors.
It can also be
employed for other scopes: it constructs the underlying asset for options
traded in the CBOE; it prices variance swaps – a type of forward contract; it
estimates variance risk premia.
With these
concepts about volatility in mind, Bollerslev and Zhou (2007) construct a
different model to account for time changing volatility to explain price
variation; they focus on the difference between implied and realised variances
finding more accurate predictions than the ones of the BSM model, suggesting
that the approach can be further implemented.
Alternative
Applications
Although the
BlackScholesMerton model was initially developed to fit the analysis of
option prices, it can also be applied to other contingent claims and extended
to the analysis of other types of derivatives. In particular, the analysis can
be applied to any security whose payoff is dependent on the returns of another
asset (
Bailey, 2005)
.
Merton (1977,
1998) proceeds on extending the analysis and applies the BSM model in special
cases, for example to loan guarantees and deposit insurance. Specifically, he
claims that a contract that insures against the losses caused by the default of
an underlying asset is equivalent to a put option whose exercise price is equal
to the defaultfree contract value. Thus, a purchase of a debt instrument that
involves possible defaults can be seen as an issue of a loan guarantee to
insure the buyer against losses. Similarly, deposit insurance guarantees the
depositor a refund (full or partial) in case of a bank run. Merton applies the
analysis to the U.S. financial system and government guarantees; the contrast
with the put option is made clear if we consider the deposit as an act of
protection from the buyer against default, and the loan guarantee as a
condition in which the bank holds the deposit and will only exercise if the
borrower does not make the prearranged payments.
Additionally,
Merton (1977) extends the analysis to nonfinancial instruments: the socalled
“realoptions”. These options regard real estate investments or development
decisions, or, in general, every instrument that involves future uncertainty.
The BSM model
can also be applied to revolving credit agreements, for which a company is
obliged to give funds to another company in case this is in need of credit. The
price of this decision is studied by Hawkins (1982) and depends on various
elements such as the cost a bank might impose for borrowing (that could be less
or more than the one imposed by the company lending the funds), and the rate at
which funds are lent. Finally, Lauterbach and Schultz (1990) make use of the
BSM (adjusted for dilution) to analyse price warrants rather than options,
although their findings might suggest that the model performs rather poorly in
this case.
Conclusion
The power of
the BlackScholesMerton approach comes from the fact that it is a simple and
straightforward way to calculate option prices, by using both explicit and
implicit information in the formulae described earlier in the paper. Its
versatility makes it possible for it to be applied to other contingent claims
that are not strictly put and call options, as Merton (1977), Hawkins (1982)
and Lauterbach and Schultz (1990) describe. For these reasons, the model is
still widely used in the financial world today.
Nevertheless,
the assumptions of the model often prove to be unrealistic; the fact that the
volatility is not observable and the violations of other assumptions, for
example the ones concerning continuous trading or perfect divisibility of an
asset, prevent the model from being an accurate prediction of an option’s
price. Thus, although the model is extremely effective in some aspects, it
shall not be viewed as a flawless way of evaluating options’ and other
securitylike instruments’ prices; caution should be exercised when making
empirical use of the approach.
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statement
©
Eleanora
Cambone.
This article is licensed under a Creative Commons Attribution 4.0
International Licence (CC BY).