An alternative specification assumes that the call option function is homogeneous of
degree one in and
(as in the Black-Scholes formula) so that:
Combining the assumptions of (8.7) and (8.8) the
call pricing function can be further reduced to a function of
three variables
.
Another approach is to use a semiparametric specification based
on the Black-Scholes implied volatility. Here, the implied
volatility is modelled as a nonparametric function,
:
Empirically the implied volatility function mostly depends on two
parameters: the time to maturity and the moneyness
. Almost equivalently, one can set
where
and
is the
present value of the dividends to be paid before the expiration.
Actually, in the case of a dividend yield
, we have
. If the dividends are discrete, then
where
is the dividend payment date of the
dividend and
is its maturity.
Therefore, the dimension of the implied volatility function can be
reduced to
. In this case the call
option function is:
Once a smooth estimate of
is obtained,
estimates of
,
,
, and
can be calculated.
The previous section proposed a semiparametric estimator of the call pricing function and the necessary steps to recover the SPD. In this section the dimension is reduced further using the suggestion of Rookley (1997). Rookley uses intraday data for one maturity and estimates an implied volatility surface where the dimension are the intraday time and the moneyness of the options.
Here, a slightly different method is used which relies on all
settlement prices of options of one trading day for different
maturities to estimate the implied volatility surface
. In the second step, these estimates
are used for a given time to maturity which may not necessarily
correspond to the maturity of a series of options. This method
allows one to compare the SPD at different dates because of the
fixed maturity provided by the first step. This is interesting if
one wants to study the dynamics and the stability of these
densities.
Fixing the maturity also allows us to eliminate from the
specification of the implied volatility function. In the following
part, for convenience, the definition of the moneyness is
and we denote by
the implied
volatility. The notation
denotes the partial
derivative of
with respect to
and
the total derivative of
with respect to
.
Moreover, we use the following rescaled call option function:
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The rescaled call option function can be expressed as:
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The functional form of
is:
The quantities in (8.14) and (8.15) are a
function of the following first derivatives:
For the remainder of this chapter, we define:
The quantities in (8.14) and (8.15) also depend
on the following second derivative functions:
Consider the following data generating process for the implied volatilities:
This expansion suggests an approximation by local polynomial
fitting, Fan and Gijbels (1996). Hence, to estimate the implied
volatility at the target point
from
observations
, we minimize the
following expression:
For convenience use the following matrix definitions:
Hence, the weighted least squares problem (8.20) can be
written as
A nice feature of the local polynomial method is that it provides
the estimated implied volatility and its first two derivatives in
one step. Indeed, one has from (8.19) and
(8.20):
One of the concerns regarding this estimation method is the
dependence on the bandwidth which governs how much weight the
kernel function should place on an observed point for the
estimation at a target point. Moreover, as the call options are
not always symmetrically and equally distributed around the ATM
point, the choice of the bandwidth is a key issue, especially for
estimation at the border of the implied volatility surface. The
bandwidth can be chosen global or locally dependent on
. There are methods providing "optimal" bandwidths
which rely on plug-in rules or on data-based selectors.
In the case of the volatility surface, it is vital to determine one bandwidth for the maturity and one for the moneyness directions. An algorithm called Empirical-Bias Bandwidth Selector (EBBS) for finding local bandwidths is suggested by Ruppert (1997) and Ruppert et al. (1997). The basic idea of this method is to minimize the estimate of the local mean square error at each target point, without relying on asymptotic result. The variance and the bias term are in this algorithm estimated empirically.
Using the local polynomial estimations, the empirical SPD can be calculated with the following quantlet:
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The arguments for this quantlet are the moneyness ,
,
,
,
underlying price
corrected for future dividends,
risk-free interest rate
, and the time to maturity
. The output consist of the local polynomial SPD
(lpspd.fstar),
(lpspd.delta), and the
(lpspd.gamma) of the call-options.