Volatility is implicitly defined as the standard deviation of
the yield of an asset. The value of an option increases with
volatility. The higher the volatility the greater the option's chance
during its life to be "in the money" - convertible to the underlying
asset at a handsome profit.
Without delving too deeply into the model, this mathematical expression
works well during trends and fails miserably when the markets change
sign.
There is disagreement among scholars and traders whether one should
better use historical data or current market prices - which include
expectations - to estimate volatility and to price options correctly.
From "The Econometrics of Financial Markets" by John Campbell, Andrew
Lo, and Craig MacKinlay, Princeton University Press, 1997:
"Consider the argument that implied volatilities are better forecasts
of future volatility because changing market conditions cause
volatilities (to) vary through time stochastically, and historical
volatilities cannot adjust to changing market conditions as rapidly.
The folly of this argument lies in the fact that stochastic volatility
contradicts the assumption required by the B-S model - if volatilities
do change stochastically through time, the Black-Scholes formula is no
longer the correct pricing formula and an implied volatility derived
from the Black-Scholes formula provides no new information.
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