Consider put or
options on a given
underlier. They have
different strikes but the same
expiration. If we obtain market prices for
those options, we can apply the
Black-Scholes (1973) model to back-out
implied volatilities. Intuitively, we might expect the implied
volatilities to be identical. In practice, it is likely that they will not
Coffee options trade on New York's Coffee, Sugar and Cocoa
Exchange (CSCE). Exhibit 1 indicates implied volatilities at various
strikes for the May 2001 calls based upon their March 12, 2001 settlement
CSCE May 2001 coffee call option implied
volatilities as of March 12, 2001. Implied volatilities were
calculated from settlement prices using
option pricing model. The pattern of implied volatilities form a
"smile" shape, which is called a volatility smile.
The pattern of implied volatilities forms a "smile" shape,
which is called a volatility smile.
Such a smile persists over time in the coffee options markets with
out-of-the-money volatilities generally higher than
Most derivatives markets exhibit persistent patterns of
volatilities varying by strike. In some markets, those patterns form a
smile. In others, such as equity index options markets, it is more of a
skewed curve. This has motivated the name
volatility skew. In practice, either the term "volatility smile" or
"volatility skew" (or simply skew) may be used to
refer to the general phenomena of volatilities varying by strike. Indeed, you may even hear of
smirks" or "volatility sneers", but such names are often as much whimsical as
they are descriptive of any particular volatility pattern.
Volatility smile (left) is compared to
volatility skew (right).
There are various explanations for why volatilities
exhibit skew. Different explanations may apply in different markets. In
most cases, multiple explanations may play a role. Some explanations
relate to the idealized assumptions of the Black-Scholes approach to
valuing options. Almost every one of those assumptions—lognormally
distributed returns, return
homoskedasticity, etc.—could play a role. For example, in
most markets, returns appear more
leptokurtic than is assumed by a lognormal distribution. Market
leptokurtosis would make way out-of-the-money or way in-the-money options
more expensive than would be assumed by the Black-Scholes formulation. By
increasing prices for such options, volatility smile could be the markets'
indirect way of achieving such higher prices within the imperfect
framework of the Black-Scholes model. Other explanations relate to
relative supply and demand for options. In
equity markets, volatility skew
could reflect investors' fear of market crashes—which would cause them to
bid up the prices of options at strikes below current market levels. In
electricity markets, utilities and other purchasers are concerned about
price spikes. Not surprisingly, electricity volatilities exhibit the
opposite skew—with volatilities elevated for higher strikes.
Another dimension to the problem of volatility skew is
that of volatilities varying by expiration. This is illustrated for CSCE
coffee options in Exhibit 3. It indicates what is known as a
three-dimensional graph indicating implied volatilities by both strike and
Volatility surface for CSCE cofee options
on March 12, 2001.
For early expirations, the graph exhibits a volatility
smile. Volatilities dip and then rise, taking on a skew for the September
contract before falling and flattening out. This is a pattern that recurs
every year for coffee volatilities. September is the harvest month for
Brazilian coffees. Brazil is a major exporter, and every year, the harvest
is threatened by frost. Market participants watch the harvest, knowing
that a frost will drive world coffee prices sharply higher. This explains
both the rise in volatilities as well as the skew for September. For that
month, traders are concerned about a spike in prices, and the volatility
surface reflects this.
If we take a cross section of a volatility surface at a
particular strike, we obtain a curve that describes implied volatilities
as a function of expiration for that strike. This is called a
volatility term structure.
Traders aren't interested only in static volatility surfaces.
They also want to know how skew will respond to the passage of time and
changes in the underlier's value. While the dynamics of a volatility
surface are complicated, there are two simple models that are useful for
describing an aspect of those dynamics. Introduced by Derman (1999),
these are called the sticky strike and
sticky delta models.
If a skew is behaving according to the sticky strike
model, its implied volatilities are associated with specific strikes. The
curve of the skew will not shift with the value of the underlier.
With the sticky delta model, implied volatilities are
associated with specific deltas. The entire curve of volatilities moves
with the underlier. For example, suppose a strike 75 call has a delta of
.65 and an implied volatility of 14%. If the value of the underlier rises
so that the strike 80 option becomes the delta .65 option, then the 14%
implied volatility will migrate to that option.
Obviously, both models are simplifications. No market
exhibits one behavior or the other at all times, and actual volatility
dynamics often blend the two.
Volatility skew complicates the tasks of pricing and
hedging options. Consider the task of calculating an option's
delta. If we assume the
sticky delta model, this will affect how we calculate the option's delta.
Changes in implied volatilities that are expected to accompany changes in
the value of the underlier will impact the option's value. Deltas need to
be adjusted to reflect this. This is more than a theoretical
consideration. If a trader is
dynamically hedging an options position and
fails to incorporate skew into her delta calculations, her hedge ratio
will be off.
Skew poses even greater challenges for
They need to adopt dynamic models for entire volatility surfaces. These
are far more sophisticated than the simple sticky strike and sticky delta
models. Standard models include
stochastic volatility models of Hull and
White (1988) and
local volatility models of Dupire (1994),
Derman and Kani (1994),
Rubinstein (1994) and Andersen and
mixed distribution models such as Brigo and
A jump-diffusion model
adds random jumps to the geometric
Brownian motion that Black-Scholes (1973)
assumes for the underlier. Among other things, this has the effect of
giving the underlier's value a leptokurtic distribution.
volatility model models both the underlier's value and its
volatility as stochastic processes. As with a jump-diffusion model, this
has the effect of giving the underlier's value a leptokurtic distribution.
Heston's (1993) is a popular
stochastic volatility model.
models have various names. They have been called
volatility function models or
implied tree models (implied binomial tree or implied trinomial
tree models, depending on the type of tree). They assume that future
volatilities will be a deterministic function of the underlier value and
time. This function is implied by the current volatility skew and can be
fully reflected in a suitably calibrated binomial or trinomial tree.
models model the underlier's value with a mixture of
distributions. This has the effect of giving the underlier a leptokurtic
distribution. Brigo and Mercurio (2000)
use a mixture of lognormals.
Other models are hybrids, combining aspects of those
described above. Bates' (1996)
stochastic volatility jump-diffusion model is an example. The
universal volatility models
of Dupire (1996) and Britten-Jones and
among others, combine elements of local volatility, jump-diffusion and
stochastic volatility models. These are rich models, but they can be
difficult to implement.
ARCH A category of conditionally heteroskedastic
derivative instrument An instrument
which derives its value from the value of other financial
instruments. Article includes a list of vanilla and exotic derivatives.
Greeks A set of
factor sensitivities used to measure risk exposures related to
options or other derivatives.
condition where a stochastic process has non-constant second
kurtosis A parameter describing the peakedness and tails of a
option pricing theory The
body of financial theory used by financial engineers to value options and other
spreads Positions combining one or more options in a single
A formula that relates the price of a put to the price of a
A category of conditionally heteroskedastic
time series and
stochastic processes An introductory article.
The two components that comprise an option's market value.
A metric of variability in a stochastic process.
is an excellent book for experienced options traders. Taleb (1996)
is even more sophisticated. See Lipton (2003)
and Rebonato (2004)
for discussions of financial engineering models for skew.
Ads by Contingency Analysis
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