 
Summary: Notice that this expression depends on calendar time and the individual
stock prices, and not just on the index level. A local volatility function for
the index (that is, one that depends only on the price of the index and
time) can be obtained by calculating the expectation of 2
B conditional on
the value of the index. More precisely, the function B, loc = B, loc(B, t),
defined as:
is such that the onedimensional diffusion process:
(with µB representing the costofcarry of the ETF), returns the same prices
for Europeanstyle index options as the ndimensional model based on the
dynamics for the entire basket.
To see this, we observe that B(S, t) can be viewed as a stochastic volatil
ity process that drives the index price B(t), with the vector of individual
stock prices S playing the role of ancillary risk factors. The above formu
la for 2
B, loc expresses a wellknown correspondence between the sto
chastic volatility of a pricing model and its corresponding (Dupiretype)
local volatility (see Derman, Kani & Kamal, 1997, BrittenJones & Neu
berger, 2000, Gatheral, 2001, and Lim, 2002).
The problem, of course, is that the conditional expectation is difficult
