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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
 

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 one-dimensional diffusion process:
(with ÁB representing the cost-of-carry of the ETF), returns the same prices
for European-style index options as the n-dimensional 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 well-known correspondence between the sto-
chastic volatility of a pricing model and its corresponding (Dupire-type)
local volatility (see Derman, Kani & Kamal, 1997, Britten-Jones & Neu-
berger, 2000, Gatheral, 2001, and Lim, 2002).
The problem, of course, is that the conditional expectation is difficult

  

Source: Avellaneda, Marco - Department of Mathematics, Courant Institute of Mathematical Sciences, New York University

 

Collections: Mathematics