MacBeth and Merville's comparison between the actual
option price and the Cox and Black and Scholes model price
shows that the Cox model price fits the market data better than
the Black and Scholes model price for every money position.
Merton's simulation results show that the Black and Scholes
model has discrepancies for the short time-to-maturity of
But our results shoe that we cannot say consistently that
any model explains the observed market option price better than
other models for every money position and time-to-maturity. It
depends on the time-to-maturity and money position : In
general, the CEV model and the Black and Scholes model show
opposite bias direction. The Black and Scholes model underprices
in-the-money and deep-on-the-money options and overprices
deep-out-of-the-money and out-of-the-money. The CEV model
underprices out-of-the-money and deep-out-of-the-money
options, but overprices deep-in and in-the-money.
The simplified Jump model shows bias pattern similar to the
Black and Scholes model.. With regard to the bias magnitude,
each model showed a different magnitude of bias according to
the money position and time to maturity. In general, for the
deep-out-of-the-money and out-of-the-money options, the Black
and Scholes model has smaller bias magnitude than other
models. For in-the-money and deep-in-the-money of
long-tine-to-maturity, the Simplified Jump model the least bias
magnitude, but the CEV model shows the least magnitude for
deep-in-the-money and short time to maturity.
Even though the CEV model is excellent in reducing bias
for deep-in-the-money and short-time-to-maturity in our sample
data, the extra effort to estimate the CEV parameters can ma다
practitioners hesitate to use that model.
Since one formula does not seem to explain all of our
observations across time, perhaps, we need to build a composite
model. Other additional analysis mighty help us : perhaps, we
can suspect the synchronization of markets. If markets are
nonsynchronized, the above irregular bias pattern can be
observed across the models. we can examine this problem by
testing boundary condition. Also explanation of exercise price
bias or time-to-maturity bias will help us to understand the
above model comparison results.