While the previous section was dedicated to finding a
proxy for used by investors to price options, this section
approximates the historical underlyings' density
for date
using all the information available at date
. Of
course, if the process governing the underlying asset dynamics
were common knowledge and if agents had perfect foresight, then by
no arbitrage arguments both SPDs should be equal. Following
Ait-Sahalia, Wang and Yared (2000), the density extracted from the observed
underlyings' data is not comparable to the density implied by
observed option data without assumptions on investor's
preferences. As in Härdle and Tsybakov (1995), they apply an
estimation method which uses the observed asset prices to infer
indirectly the time series SPD. First, we will explain the
estimation method for the underlyings' SPD. Second, we apply it to
DAX data.
Assuming the underlying to follow an Îto diffusion process
driven by a Brownian motion
:
Let
denote the
conditional density of
given
generated by the dynamics
defined in equation (9.1) and
denote the
conditional density generated by equation (9.2)
then
can only be compared to the risk neutral density
and not to
.
A crucial feature of this method is that the diffusion functions
are identical under both the actual and the risk neutral dynamics
(which follows from Girsanov's theorem). Therefore, it is not
necessary to observe the risk neutral path of the DAX index
. The function
is estimated using
observed index values
and applying
Florens-Zmirou's (1993) (FZ) nonparametric version of the minimum
contrast estimators:
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0 | and |
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Once
is estimated, the time series SPD
can
be computed by Monte Carlo integration. Applying the Milstein
scheme (Kloeden, Platen and Schurz (1994)), we simulate
paths of the
diffusion process:
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(9.4) |
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In the absence of arbitrage, the futures price is the expected
future value of the spot price under the risk neutral measure.
Therefore the time series distribution is translated such that its
mean matches the implied future price. Then the bandwidth
is chosen to best match the variance of the IBT implied
distribution. In order to avoid over- or undersmoothing of
,
is constrained to be within
to
times the
optimal bandwidth implied by Silverman's rule of thumb. This
procedure allows us to focus the density comparison on the
skewness and kurtosis of the two densities.
Using the DAX index data from
M
D
*BASE
we estimate the diffusion
function
from equation (9.2) by
means of past index prices and simulate (forward)
paths
to obtain the time series density,
.
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To be more precise, we explain the methodology for the first
period in more detail. First, note that Friday, April ,
, is the
rd Friday of April
. Thus, on Monday,
April
,
, we use
months of DAX index prices from
Monday, January
,
, to Friday, April
,
, to
estimate
. Following, on the same Monday, we start the
months `forward' Monte Carlo simulation. The bandwidth
is determined by Cross Validation applying the
XploRe
quantlet
regxbwcrit
which determines the optimal bandwidth
from a range of bandwidths by using the resubstitution estimator
with the penalty function 'Generalized Cross Validation'.
Knowing the diffusion function it is now possible to Monte Carlo simulate the index
evolution. The Milstein scheme applied to equation (9.2) is given by:
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With these ingredients we start the simulation with index value
(Monday, April
,
) and time to maturity
and
. The expiration date is Friday, July
,
. From these simulated index values we calculate
annualized log-returns which we take as input of the
nonparametric density estimation (see equation
(9.5)). The
XploRe
quantlet
denxest
accomplishes the estimation of the time series density by means of
the Gaussian kernel function:
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First of all, we calculate the optimum bandwidth given
the vector of
simulated index values. Then we search the
bandwidth
'
which implies a variance of
to
be closest to the variance of
(but to be still within
to
times
). We stop the search if var(
) is within
a range of
of var(
). Following, we translate
such
that its mean matches the futures price F. Finally, we transform
this density over DAX index values
into a density
' over log-returns
. Since
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