A kurtosis trading strategy is supposed to exploit
differences in kurtosis 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 has more kurtosis than
the time series SPD
, i.e. kurt(
)
kurt(
), we
sell the whole range of strikes of FOTM puts, buy the whole range
of strikes of NOTM puts, sell the whole range of strikes of ATM
puts and calls, buy the whole range of strikes of NOTM calls and
sell the whole range of strikes of FOTM calls (K
trade).
Conversely, if the implied SPD has less kurtosis than the time
series density
, i.e. kurt(
)
kurt(
), we
initiate the K
trade by buying the whole range of strikes of
FOTM puts, selling the whole range of strikes of NOTM puts, buying
the whole range of strikes of ATM puts and calls, selling the
whole range of strikes of NOTM calls and buying the whole range of
strikes of FOTM calls. In both cases we keep the options until
expiration.
Kurtosis measures the fatness of the tails of a
distribution. For a normal distribution we have
. A
distribution with
is said to be leptokurtic and has
fatter tails than the normal distribution. In general, the bigger
is, the fatter the tails are. Again we consider the
option pricing formulae (9.6) and
(9.7) and reason as above using the
probability mass to determine the moneyness regions where we buy
or sell options. Look at Figure 7.14 for a situation in
which the implied density has more kurtosis than the time series
density triggering a K
trade.
To form an idea of the K strategy's exposure at maturity we
start once again with a simplified portfolio containing two short
puts with moneyness
and
, one long put with moneyness
, two short calls with moneyness
and
and one
long call with moneyness
. Figure 9.7 reveals
that this portfolio inevitably leads to a negative payoff at
maturity regardless the movement of the underlying.
Should we be able to buy the whole range of strikes as the K
trading rule suggests, the portfolio is given in Table
9.3, FOTM-NOTM-ATM-K
, we get a payoff
profile (Figure 9.8) which is quite similar to the
one from Figure 9.7. In fact, the payoff function
looks like the `smooth' version of Figure 9.7.
Changing the number of long puts and calls in the NOTM regions can
produce a positive payoff. Setting up the portfolio given in Table
9.3, NOTM-K, results in a payoff function
shown in Figure 9.9. It is quite intuitive that the
more long positions the portfolio contains the more positive the
payoff will be. Conversely, if we added to that portfolio FOTM
short puts and calls the payoff would decrease in the FOTM
regions.
As a conclusion we can state that the payoff function can have
quite different shapes depending heavily on the specific options
in the portfolio. If it is possible to implement the K trading
rule as proposed the payoff is negative. But it may happen that
the payoff function is positive in case that more NOTM options
(long positions) are available than FOTM or ATM (short positions)
options.
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To investigate the performance of the kurtosis trades, K and K
, we proceed in the
same way as for the skewness trade. The total net EUR cash flow of the K
trade,
applied when kurt(
)
kurt(
), is strongly positive (
EUR).
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What would have happened if we had implemented the K trade
without knowing both SPD's? Again, the answer to this question can
only be indicated due to the rare occurences of periods in which
kurt(
)
kurt(
). Contrarily to the S
trade, the
density comparison would have filtered out a strongly negative net
cash flow that would have been generated by a portfolio set up in
July
. But the significance of this feature is again
uncertain.
About the K trade can only be said that without a SPD
comparison it would have procured heavy losses. The K
trade
applied as proposed can not be evaluated completely since there
was only one period in which kurt(
)
kurt(
).