The Black–Scholes model (see Black and Scholes, 1973) gives options
prices as a function of volatility. If an option price is given by the
market we can invert this relationship to get the implied volatility.
If the model were perfect, this implied value would be the same
for all option market prices, but reality shows this is not the case.
Implied Black–Scholes volatilities strongly depend on the maturity
and the strike of the European option under scrutiny. If the implied
volatilities of at-the-money (ATM) options on the Nikkei 225 index
are 20% for a maturity of six months and 18% for a maturity of one
year, we are in the uncomfortable position of assuming that the
Nikkei oscillates with a constant volatility of 20% for six months
but also oscillates with a constant volatility of 18% for one year.
It is easy to solve this paradox by allowing volatility to be time-
dependent, as Merton did (see Merton, 1973). The Nikkei would
first exhibit an instantaneous volatility of 20% and subsequently a
lower one, computed by a forward relationship to accommodate
the one-year volatility. We now have a single process, compatible
with the two option prices. From the term structure of implied
volatilities we can infer a time-dependent instantaneous volatility,
because the former is the quadratic mean of the latter. The spot
process S is then governed by the following stochastic differential
Pricing with a Smile
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DERIVATIVES PRICING: THE CLASSIC COLLECTION
where r(t) is the instantaneous forward rate of maturity t implied
from the yield curve.
Some Wall Street houses incorporate this temporal information
in their discretisation schemes to price American or path-
However, the dependence of implied volatility on the strike, for
a given maturity (known as the smile effect) is trickier. Researchers
have attempted to enrich the