In the previous section we learned that the implied and the time series SPD's reveal differences in skewness and kurtosis. In the following two sections, we investigate how to profit from this knowledge. In general, we are interested in what option to buy or to sell at the day at which both densities were estimated. We consider exclusively European call or put options.
According to Ait-Sahalia, Wang and Yared (2000), all strategies are designed
such that we do not change the resulting portfolio until maturity,
i.e. we keep all options until they expire. We use the following
terms for moneyness which we define as
:
|
where FOTM, NOTM, ATM stand for far out-of-the-money, near out-of-the-money and at-the-money respectively.
A skewness trading strategy is supposed to exploit differences in skewness of two
distributions by buying options in the range of strike prices where they are underpriced
and selling options in the range of strike prices where they are overpriced. More
specifically, if the implied SPD is less skewed (for example more negatively
skewed) than the time series SPD
, i.e. skew(
)
skew(
), we sell the
whole range of strikes of OTM puts and buy the whole range of strikes of OTM calls (S1
trade). Conversely, if the implied SPD is more skewed, i.e. skew(
)
skew(
),
we initiate the S
trade by buying the whole range of strikes of OTM puts and selling
the whole range of strikes of OTM calls. In both cases we keep the options until
expiration.
Skewness is a measure of asymmetry of a probability
distribution. While for a distribution symmetric around its mean
, for an asymmetric distribution
indicates more weight
to the left of the mean. Recalling from option pricing theory the
pricing equation for a European call option, Franke, Härdle and Hafner (2001):
Since we hold all options until expiration and due to the fact
that options for all strikes are not always available in markets
we are going to investigate the payoff profile at expiration of
this strategy for two compositions of the portfolio. To get an
idea about the exposure at maturity let us begin with a simplified
portfolio consisting of one short position in a put option with
moneyness of and one long position in a call option with
moneyness of
. To further simplify, we assume that the
future price
is equal to
EUR. Thus, the portfolio has a
payoff which is increasing in
, the price of the underlying
at maturity. For
EUR the payoff is negative and for
EUR it is positive.
However, in the application we encounter portfolios containing several long/short calls/puts with increasing/decreasing strikes as indicated in Table 9.2.
Figure 9.5 shows the payoff of a portfolio of
short puts with strikes ranging from
EUR to
EUR and of
long calls striking at
EUR to
EUR, the future
price is still assumed to be
EUR. The payoff is still
increasing in
but it is concave in the left tail and convex
in the right tail. This is due to the fact that our portfolio
contains, for example, at
EUR two call options which are
in the money instead of only one compared to the portfolio
considered above. These options generate a payoff which is twice
as much. At
EUR the payoff is influenced by three ITM
calls procuring a payoff which is three times higher as in the
situation before etc. In a similar way we can explain the slower
increase in the left tail. Just to sum up, we can state that this
trading rule has a favorable payoff profile in a bull market where
the underlying is increasing. But in bear markets it possibly
generates negative cash flows. Buying (selling) two or more calls
(puts) at the same strike would change the payoff profile in a
similar way leading to a faster increase (slower decrease) with
every call (put) bought (sold).
The S strategy payoff behaves in the opposite way. The same
reasoning can be applied to explain its payoff profile. In
contradiction to the S
trade the S
trade is favorable in a
falling market.
Given the skewness values for the implied SPD and the time series SPD we now have a look
on the performance of the skewness trades. Performance is measured in net EUR cash flows
which is the sum of the cash flows generated at initiation in and at expiration in
. We ignore any interest rate between these two dates. Using EUREX settlement
prices of
month DAX put and calls we initiated the S
strategy at the Monday
immediately following the
rd Friday of each month, beginning in April
and
ending in September
. January, February, March
drop out due to the time
series density estimation for the
rd Friday of April
. October, November and
December
drop out since we look
months forward. The cash flow at initiation
stems from the inflow generated by the written options and the outflow generated by the
bought options and hypothetical
% transaction costs on prices of bought and sold
options. Since all options are kept in the portfolio until maturity (time to expiration
is approximately
months, more precisely
TTM
) the cash flow in
is composed of the sum of the inner values of the options in the portfolio.
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Figure 9.6 shows the EUR cash flows at
initiation, at expiration and the resulting net cash flow for each
portfolio. The sum of all cash flows, the total net cash flow, is
strongly positive ( EUR). Note that the net cash flow
(blue bar) is always positive except for the portfolios initiated
in June
and in September
where we incur heavy losses
compared to the gains in the other periods. In other words, this
strategy would have procured
times moderate gains and two
times large negative cash flows. As Figure 9.5
suggests this strategy is exposed to a directional risk, a feature
that appears in December
and June
where large
payoffs at expiration (positive and negative) occur. Indeed, the
period of November and December
was a turning point of the
DAX and the beginning of an
month bull market, explaining the
large payoff in March
of the portfolio initiated in
December
. The same arguing explains the large negative
payoff of the portfolio set up in June
expiring in
September
(refer to Figure 9.11). Another point to
note is that there is a zero cash flow at expiration in
periods. Periods with a zero cash flow at initiation and at
expiration are due to the fact that there was not set up any
portfolio (there was no OTM option in the database).
Since there is only one period (June ), when the implied SPD
is more skewed than the time series SPD a comparison of the S
trade with knowledge of the latter SPD's and without this
knowledge is not useful. A comparison of the skewness measures
would have filtered out exactly one positive net cash flow, more
precisely the cash flow generated by a portfolio set up in June
. But to what extend this may be significant is uncertain.
For the same reason the S
trade has no great informational
content. Applied to real data it would have procured a negative
total net cash flow. Actually, only in June
a portfolio
would have been set up. While the S
trade performance was
independent of the knowledge of the implied and the time series
SPD's the S
trade performance changed significantly as it was
applied in each period (without knowing both SPD's). The cash flow
profile seemed to be the inverse of Figure 9.6
indicating that should there be an options mispricing it would
probably be in the sense that the implied SPD is more negatively
skewed than the time series SPD.