1
The Smile in Option Prices
When option prices are well described by the Black-Scholes
model, then when we examine the prices of a set of options on
the one asset with a common maturity date and different
exercise prices they will all have the same
implied volatility
.
• Suppose
S
= $65;
X
= $65;
t
= 1/4 year (3 months);
continuously compounded risk-free rate
r
= 9.53102% p.a.
• Suppose the option is trading at $5.285407.
⇒
If the market is using Black-Scholes, then its estimate of the
volatility of the stock is 35% p.a.; i.e.,
σ
= 0.35.
ln
(
)
(
).
.
.
2
rt
1
2
1
2
1
S
r
t
X
2
c
SN d
e
XN d
d
d
d
t
t
σ
σ
σ
−
+
+
=
−
≡
≡
−

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