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The Black and Scholes option pricing formula exhibits certain
biases on several parameters used in the model. It has been
observed that the implied volatilities are high for `in-the-money'
options and low for `out-of-the-money' options indicating
that the Black-Scholes model underprices `in-the-money'
options and overprices `out-of-the-money' options. Further,
implied volatility also varies with maturity. In addition
to the problems of changing implied volatility across moneyness
and maturity, Nifty options also suffer from `cost-of-carry'
bias, as future prices of Nifty options are usually less
than Nifty spot prices plus interest element. Since the
inception of Nifty future trading in India, Nifty future
even traded below the Nifty spot value. These deformities
obviously cause difference between the actual price of Nifty
options and the prices calculated using Black-Scholes formula.
Black tried to address this problem of negative cost of
carry by using forward prices in the option pricing model
instead of spot prices. He argued that actual forward prices
not only incorporate cost of carry but also capture various
irregularities faced by market forces. In his model, he
replaced the spot price term (S) by the discounted value
of future price (F.e-rt) in the original Black-Scholes Formula.
On the otherhand black's model is widely used for valuing
options on physical commodities as the discounted value
of a quoted future price is found to be a better proxy of
the current spot prices as an input to Black-Scholes formula.
In this study, the theoretical options prices of Nifty options
are calculated using both the Black formula and Black-Scholes
formula, and these theoretical values are compared with
the actual quoted prices in the market. It is found that
the Black formula provides better result in comparison to
Black-Scholes formula for Nifty options.
Black-Scholes (1973) call option pricing formula was a
landmark in the history of financial modeling and continues
to be the preferred model for theoretical valuation of option
prices. However, the deviation of observed market prices
for options from the theoretical results given by the Black-Scholes
formula has produced both academics and practitioners, alike.
Black (1975) himself was one of the first to observe biases
in the Black-Scholes option pricing formula. Latane and
Rendleman (1976) observed that out-of-money put options
are generally overpriced in the market. MacBeth and Merville
(1979) found that the implied volatilities are high for
`in-the-money' options and low for `out-of-the-money' options
indicating that the Black-Scholes model underprices `in-the-money'
options and overprices `out-of-the-money' options.
One of the possible reasons for mispricing of the options
on volatility measures can be attributed to the fact, that,
actual stock price movement do not follow lognormal distribution
as assumed in the model. Mandelbrot (1963), while trying
to model cotton prices, observed that the asset price returns
were highly leptokurtic, which he termed as `fat tails'.
The actual returns from the market showed extreme moves
more likely than that can be explained by the lognormal
distribution. This is one of the reasons for higher implied
volatilities of `deep in-the-money' and `deep out-the-money'
options than that of `at-the-money' options.
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