Barle-Cakici's modification of Derman-Kani's Implied
Binomial Tree (IBT) yields a proxy for the option
implied SPD, , see Chapter 7.
XploRe
provides quantlets computing Derman-Kani's and Barle-Cakici's
IBT's. Since the latter proved to be slightly more robust than the
former, Jackwerth (1999), we decide to use
Barle-Cakici's IBT to compute the option implied SPD. In the
following subsection, we follow closely the notation used in
Chapter 7. That is,
denotes the number of
evenly spaced time steps of length
in which the tree is
divided into (so we have
levels).
is the forward price of the underlying,
, at
node
at, level
. Each level
corresponds to time
.
Using the DAX index data from
M
D
*BASE
, we estimate the month
option implied IBT SPD
by means of the
XploRe
quantlets
IBTbc
and
volsurf
and a two week cross section of
DAX index option prices for
periods beginning in April
and ending in September
. We measure time to maturity (TTM)
in days and annualize it using the factor
, giving the
annualized time to maturity
TTM
. For each
period, we assume a flat yield curve. We extract from
M
D
*BASE
the maturity consistent interest rate.
We describe the procedure in more detail for the first period.
First of all, we estimate the implied volatility surface given the
two week cross section of DAX option data and utilizing the
XploRe
quantlet
volsurf
which computes the
dimensional implied volatility surface (implied volatility over
time to maturity and moneyness) using a kernel smoothing
procedure. Friday, April
,
is the
rd Friday of April
. On Monday, April
,
, we estimate the volatility
surface, using two weeks of option data from Monday, April
,
, to Friday, April
,
. Following, we start the IBT
computation using the DAX price of this Monday, April
,
. The volatility surface is estimated for the moneyness
interval
and the time to maturity interval
. Following, the
XploRe
quantlet
IBTbc
takes
the volatility surface as input and computes the IBT using Barle
and Cakici's method. Note that the observed smile enters the IBT
via the analytical Black-Scholes pricing formula for a call
and for a put
which
are functions of
,
,
,
and
. We note, it may happen that
at the edge of the tree option prices, with associated strike
prices
and node prices
, have to be
computed for which the moneyness ratio
is
outside the intverall
on which the volatility surface
has been estimated. In these cases, we use the volatility at the
edge of the surface. Note, as well, that the mean of the IBT SPD
is equal to the futures price by construction of the IBT.
Finally, we transform the SPD over into a SPD over
log-returns
as follows:
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A crucial aspect using binomial trees is the choice of the number
of time steps in which the time interval
is divided.
In general one can state, the more time steps are used the better
is the discrete approximation of the continuous diffusion process
and of the SPD. Unfortunately, the bigger
, the more node
prices
possibly have to be overridden in the IBT
framework. Thereby we are effectively losing the information about
the smile at the corresponding nodes. Therefore, we computed IBT's
for different numbers of time steps. We found no hint for
convergence of the variables of interest, skewness and kurtosis.
Since both variables seemed to fluctuate around a mean, we compute
IBT's with time steps
and consider the average
of these ten values for skewness and kurtosis as the option
implied SPD skewness and kurtosis.
Applying this procedure for all periods, beginning in April
and ending in September
, we calculate the time
series of skewness and kurtosis of the
month implied SPD
shown in Figures 9.3 and 9.4. We
see that the implied SPD is clearly negatively skewed for all
periods but one. In September
it is slightly positively
skewed. The pattern is similar for the kurtosis of
which is
leptokurtic in all but one period. In October
the density
is platykurtic.