Continuous Time Finance Notes, Spring 2004 – Section 9,
March 31, 2004
Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connec
tion with the NYU course Continuous Time Finance.
This section begins the third segment of the course: a discussion of the volatility skew/smile
– its sources and consequences. We start with a general discussion of the phenomenon, and
its relation to “fat tails,” following Hull’s Chapter 15. There are three main approaches to
modeling the volatility skew/smile quantitatively: (a) local vol models, (b) jumpdiffusion
models, and (c) stochastic vol models.
We have time for just a very brief introduction
to each; you’ll learn much more in the course Case Studies. Today we focus on local vol
models, explaining why it is “in principle” easy but in practice quite difficult to extract a
local volatility function from market data on calls. The heart of the matter is “Dupire’s
equation.”
*****************
The implied volatility skew/smile.
Consider call options on an underlying which earns no
dividends. We assume the interest rate
r
is constant. We write
C
(
S
0
, K, T
) = market price of a call option with strike
K
and maturity
T
where
S
0
is the spot price and the current time is
t
= 0. Now define
C
BS
(
S
0
, K, σ, T
) = BlackScholes value of the call, using constant volatility
σ
.
Then the
implied volatility
σ
I
(
S
0
, K, T
) is defined by the equation
C
(
S
0
, K, T
) =
C
BS
(
S
0
, K, σ
I
(
S
0
, K, T
)
, T
)
.
Since the BlackScholes value of a call is a monotone function of
σ
, the implied volatility
is welldefined. If the constantvol BlackScholes model were “correct,” i.e. if it gave the
actual market values of call options, then
σ
I
would be constant, independent of
S
0
,
K
, and
T
.
In fact however
σ
I
is not constant. The “volatility skew/smile” refers to its dependence on
K
. Typically, for equities,
σ
I
decreases as
K
increases. For foreign exchange the typical
behavior is different:
σ
I
is smallest when
K
≈
S
0
so its graph looks like a “smile.”
The definition of implied vol depends on the choice of payoff. But if we used puts rather
than calls we would get the
same
implied vols, by putcall parity. (We use here the fact
that putcall parity is modelindependent!).
Hull discusses at length how the skew/smile reflects “fat tails” in the riskneutral probability
distribution. No need to repeat his discussion here. But note what underlies it. Since prices
are linear, the value of any option with maturity
T
is a linear function of its payoff at time
T
. We recognize from the relation
option value =
e

rT
E
RN
[payoff]
1
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that (for a complete market model) this linear relation is expressed by integration against
the riskneutral probability density times
e

rT
. In other words, if the payoff is
f
(
S
T
) and
the riskneutral probability density of
S
T
at time
T
(given price
S
0
at time 0) is
p
(
ξ, T
;
S
0
)
then
option value =
e

rT
∞
∞
f
(
ξ
)
p
(
ξ, T
;
S
0
)
dξ.
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 Fall '11
 Bayou
 Finance, Volatility, Mathematical finance, market data, Σi, local volatility

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