The LP method introduced in the previous section will now be
applied under the assumption of a nonparametric autoregressive
model of the form (13.1) to estimate the volatility
function of the process
based on the observations
.
The conditional volatility and the conditional variance
respectively at time
is defined by
Referring back to the representation (13.8) of
the conditional variance we search for an estimator
for
with the form
In order to define these two estimators with the LP Method, after applying the steps discussed in the previous section we have to solve both of the following minimization problems:
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(14.11) |
This estimate is a direct modification of the estimator from the local polynomial, nonparametric regression in Tsybakov (1986).
To illustrate the estimator we consider an example.
Figure 13.1 above displays the DEM/USD exchange rate
from October 1, 1992 to September 30, 1993 in 20 minute intervals
(volatility time scale). There are observations. We
have calculated the returns of this series (see
Figure 13.1 below) and applied the
estimator (13.12) to the time series of the returns. Under
the model for the geometric Brownian motion for the price, which
is based on the Black-Scholes method (see Section 6.1),
the returns must follow an ordinary Brownian motion. Their
volatilities
are thus constant and independent of
. The
estimated conditional variance functions (see Figure
13.2) show a U shaped structure, which is called a
``smiling face'' structure or smile. The
estimated volatility functions
appear to be qualitatively analogous. This means that the expected
risk of the returns is significantly higher when extreme values
were observed the period before.
As an alternative to equation (13.9) it is also possible to first determine the sample residuals