7.5. IMPLIED VOLATILITY 255
C = 1.7647, which is too low. SinceC is a increasing function ofσ, this
suggests we try a value ofσ= 0.30. This givesC = 2.1010, too high, so
we bisect the interval [0. 20 , 0 .30] and tryσ = 0.25. This value ofσgives
a value ofC = 1.9268, still too high. Bisect the interval [0. 20 , 0 .25] and
try a value ofσ= 0.225, which yieldsC = 1.8438, slightly too low. Try
σ= 0.2375, givingC= 1.8849. Finally tryσ= 0.23125 givingC= 1.8642.
To 2 significant digits, the significance of the data,σ= 0.23, with a predicted
value ofC= 1.86.
A faster procedure is to use Newton’s method which is iterative. Essen-
tially we are trying to solve
f(σ,S,K,r,T−t)−C= 0,so from an initial guessσ 0 , we form the Newton iterates
σi+1=σi−f(σi)/(df(σi)/dσ).This means one has to differentiate the Black-Scholes formula with respect
toσ. This derivative is one of the “Greeks” known asvegawhich we will look
at more extensively in the next section. A formula for vega for a European
call option is
df
dσ
=S
√
T−tΦ′(d 1 ) exp(−r(T−t)).A natural way to do the iteration is with a computer program rather than
by hand.
Implied volatility is a “forward-looking” estimation technique, in contrast
to the “backward-looking” historical volatility. That is, it incorporates the
market’s expectations about the prices of securities and their derivatives, or
more concisely, market expectations about risk. More sophisticated com-
binations and weighted averages combining estimates from several different
derivative claims can be developed.
Sources
This section is adapted from:Quantitative modeling of Derivative Securities
by Marco Avellaneda, and Peter Laurence, Chapman and Hall, Boca Raton,
2000, page 66; andOptions, Futures, and other Derivative Securitiessecond
edition, by John C. Hull, Prentice Hall, 1993, pages 229–230.