## Kurtosis Trades

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 f * has more kurtosis than the time series SPD g*, i.e. kurt(f*) > kurt(g*), 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 (K1 trade). Conversely, if the implied SPD has less kurtosis than the time series density g*, i.e. kurt(f*) < kurt(g*), we initiate the K2 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 k measures the fatness of the tails of a distribution. For a normal distribution we have k = 3. A distribution with k > 3 is said to be leptokurtic and has fatter tails than the normal distribution. In general, the bigger k 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 K1 trade.

To form an idea of the K1 strategy's exposure at maturity we start once again with a simplified portfolio containing two short puts with moneyness 0.90 and 1.00, one long put with moneyness 1.00, two short calls with moneyness 1.00 and 1.10 and one long call with moneyness 1.05. 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 K1 trading rule suggests, the portfolio is given in Table 9.3, FOTM-NOTM-ATM-K1, we get

Payoff of K1 Trade

Payoff of K1 Trade

Underlying

Figure 9.7. Kurtosis trade 1 payoff at maturity of portfolio detailed in Table 9.3.

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.

Payoff of K1 Trade : FOTM-NOTM-ATM

Payoff of K1 Trade : FOTM-NOTM-ATM

Underlying

Figure 9.8. K1 trade payoff at maturity of portfolio detailed in Table 9.3.

Underlying

Figure 9.8. K1 trade payoff at maturity of portfolio detailed in Table 9.3.

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-K1, 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 K1 trading rule as proposed the payoff is negative. But it may

Payoff of K1 Trade : NOTM

Payoff of K1 Trade : NOTM

Underlying

Figure 9.9. K1 trade payoff at maturity of portfolio detailed in Table 9.3.

Underlying

Figure 9.9. K1 trade payoff at maturity of portfolio detailed in Table 9.3.

happen that the payoff function is positive in case that more NOTM options (long positions) are available than FOTM or ATM (short positions) options.

K1 |
FOTM-NOTM-ATM-K1 |
NOTM-K1 | |

Moneyness |
Moneyness |
Moneyness | |

short put |
0.90 |
0.86 - 0.90 |
0.90 |

long put |
0.95 |
0.91 - 0.95 |
0.91 - 0.95 |

short put |
1.00 |
0.96 - 1.00 |
1.00 |

short call |
1.00 |
1.00 - 1.04 |
1.00 |

long call |
1.05 |
1.05 - 1.09 |
1.05 - 1.09 |

short call |
1.10 |
1.10 1.14 |
1.10 |

Table 9.3. Portfolios of kurtosis trades.

Table 9.3. Portfolios of kurtosis trades.

### 9.6.1 Performance

To investigate the performance of the kurtosis trades, K1 and K2, we proceed in the same way as for the skewness trade. The total net EUR cash flow of the K1 trade, applied when kurt(f*) > kurt(g*), is strongly positive (10, 915.77 EUR). As the payoff profiles from figures 9.7 and 9.8 already suggested, all portfolios generate negative cash flows at expiration (see magenta bar in Figure 9.10). In contrast to that, the cash flow at initiation in t = 0 is always positive. Given the positive total net cash flow, we can state that the K1 trade earns its profit in t = 0. Looking at the DAX evolution shown in Figure 9.11, we understand why

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