In the functional data framework, the objects are usually modelled as realizations of a stochastic process
,
where
is a bounded interval in
. Thus, the set of functions
For the functional sample we can define the sample-counterparts of
,
and
in a straightforward way:
In practice, we observe the function values
;
only on a discrete grid
,
where
is the number of grid points for the
th observation.
One may
estimate the functions
via standard nonparametric regression methods,
Härdle (1990). Another popular way is to use a truncated functional basis expansion.
More precisely, let us denote a functional basis on the interval
by
and assume that the functions
are approximated by
the first
basis functions
,
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An example for a functional basis is the Fourier basis defined on by:
Our aim is to
estimate the IV-functions for fixed 1 month (1M) and 2 months (2M)
from the daily-specific grid of the maturities.
We estimate the Fourier coefficients on the moneyness-range
for maturities observed on particular day
. For
1M, 2M we calculate
by linear interpolation of the closest observable IV string with
,
and
,
:
The choice of delivers a good tradeoff between flexibility and smoothness of the strings.
At this moment we exclude from our analysis those days, where this procedure cannot be performed due to the
complete absence of the needed maturities,
and strings with poor performance of estimated coefficients,
due to the small number of contracts in a particular string or presence of strong outliers.
Using this procedure we obtain 77 ``functional" observations
, for the 1M-maturity and 66 observations
, for the 2M-maturity, as
displayed in Figure 5.3.