7.5 Implied Volatility
Mathematical Ideas
Historical volatility
Estimates ofhistorical volatilityof security prices use statistical estima-
tors, usually one of the estimators of variance. A main problem for historical
volatility is to select the sample size, or window of observations, used to es-
timateσ^2. Different time-windows usually give different volatility estimates.
Furthermore, for a lot of customized “over the counter” derivatives, the nec-
essary price data may not exist.
Another problem with historical volatility is that it assumes future market
performance is the same as past market data. Although this is a natural
scientific assumption, it does not take into account historical anomalies such
as the October 1987 stock market drop, which may be unusual. That is,
computing historical volatility has the usual statistical difficulty of how to
handle outliers. The assumption that future market performance is the same
as past performance also does not take into account underlying changes in
the market such as economic conditions.
To estimate the volatility of a security price empirically, observe the secu-
rity price at regular intervals, such as every day, every week, or every month.
Define:
- the number of observationsn+ 1
- Si,i= 0, 1 , 2 , 3 ,...,nis the security price at the end of theith interval,
- τ is the length of each of the time intervals (say in years),
and let
ui= ln(Si)−ln(Si− 1 ) = ln(
Si
Si− 1)
fori= 1, 2 , 3 ,...be the increment of the logarithms of the security prices.
We are modeling the security price as a Geometric Brownian Motion, so that
ln(Si)−ln(Si− 1 )∼N(rτ,σ^2 τ).
SinceSi=Si− 1 eui,uiis the continuously compounded return, (not an-
nualized) in theith interval. Then the usual estimates, of the standard
deviation of theui’s is
s=√√
√
√^1
n− 1∑ni=1(ui−u ̄)^2