9.5 Skewness Trades

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 $ K/(S_te^{(T-t)r})$:

Table 9.1: Definitions of moneyness regions.
    Moneyness(FOTM Put) $ <$ $ 0.90$
$ 0.90$ $ \leq$ Moneyness(NOTM Put) $ <$ $ 0.95$
$ 0.95$ $ \leq$ Moneyness(ATM Put) $ <$ $ 1.00$
$ 1.00$ $ \leq$ Moneyness(ATM Call) $ <$ $ 1.05$
$ 1.05$ $ \leq$ Moneyness(NOTM Call) $ <$ $ 1.10$
$ 1.10$ $ \leq$ Moneyness(FOTM Call)    


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 $ f^*$ is less skewed (for example more negatively skewed) than the time series SPD $ g^*$, i.e. skew($ f^*$) $ <$ skew($ g^*$), we sell the whole range of strikes of OTM puts and buy the whole range of strikes of OTM calls (S1 trade). Conversely, if the implied SPD is more skewed, i.e. skew($ f^*$) $ >$ skew($ g^*$), we initiate the S$ 2$ 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 $ s$ is a measure of asymmetry of a probability distribution. While for a distribution symmetric around its mean $ s=0$, for an asymmetric distribution $ s>0$ 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):

$\displaystyle C(S_t,K,r,T-t)$ $\displaystyle =$ $\displaystyle e^{-r(T-t)}\int_0^\infty$   max$\displaystyle (S_T-K,0)f^*(S_T)dS_T,$ (9.6)

where $ f^*$ is the implied SPD, we see that when the two SPD's are such that skew($ f^*$) $ <$ skew($ g^*$), agents apparently assign a lower probability to high outcomes of the underlying than would be justified by the time series density, see Figure 7.13. Since for call options only the right `tail' of the support determines the theoretical price, the latter is smaller than the price implied by equation (9.6) using the time series density. That is, we buy underpriced calls. The same reasoning applies to European put options. Looking at the pricing equation for such an option:
$\displaystyle P(S_t,K,r,T-t)$ $\displaystyle =$ $\displaystyle e^{-r(T-t)}\int_0^\infty$   max$\displaystyle (K-S_T,0)f^*(S_T)dS_T,$ (9.7)

we conclude that prices implied by this pricing equation using $ f^*$ are higher than the prices using the time series density. That is, we sell puts.

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 $ 0.95$ and one long position in a call option with moneyness of $ 1.05$. To further simplify, we assume that the future price $ F$ is equal to $ 100$ EUR. Thus, the portfolio has a payoff which is increasing in $ S_T$, the price of the underlying at maturity. For $ S_T<95$ EUR the payoff is negative and for $ S_T>105$ 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: S$ 1$ trade payoff at maturity of portfolio detailed in Table 9.2.
\includegraphics[width=1.4\defpicwidth]{SkewnessTrade1Payoff2PS.ps}

Figure 9.5 shows the payoff of a portfolio of $ 10$ short puts with strikes ranging from $ 86$ EUR to $ 95$ EUR and of $ 10$ long calls striking at $ 105$ EUR to $ 114$ EUR, the future price is still assumed to be $ 100$ EUR. The payoff is still increasing in $ S_T$ 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 $ S_T=106$ 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 $ S_T=107$ 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$ 2$ strategy payoff behaves in the opposite way. The same reasoning can be applied to explain its payoff profile. In contradiction to the S$ 1$ trade the S$ 2$ trade is favorable in a falling market.


Table 9.2: Portfolios of skewness trades.
  S$ 1$ OTM-S$ 1$
  Moneyness Moneyness
 
short put $ 0.95$ $ 0.86 - 0.95$
long call $ 1.05$ $ 1.05 - 1.14$



9.5.1 Performance

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 $ t=0$ and at expiration in $ t=T$. We ignore any interest rate between these two dates. Using EUREX settlement prices of $ 3$ month DAX put and calls we initiated the S$ 1$ strategy at the Monday immediately following the $ 3$rd Friday of each month, beginning in April $ 1997$ and ending in September $ 1999$. January, February, March $ 1997$ drop out due to the time series density estimation for the $ 3$rd Friday of April $ 1997$. October, November and December $ 1999$ drop out since we look $ 3$ 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 $ 5$% transaction costs on prices of bought and sold options. Since all options are kept in the portfolio until maturity (time to expiration is approximately $ 3$ months, more precisely $ \tau=$TTM$ /360$) the cash flow in $ t=T$ is composed of the sum of the inner values of the options in the portfolio.

Figure: Performance of S$ 1$ trade with $ 5$% transaction costs. The first (red), second (magenta) and the third bar (blue) show for each period the cash flow in $ t=0$, in $ t=T$ and the net cash flow respectively. Cash flows are measured in EUR. 19552 XFGSpdTradeSkew.xpl
\includegraphics[width=1.5\defpicwidth]{SkewnessTrade1PerformancePS.ps}

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 ($ 9855.50$ EUR). Note that the net cash flow (blue bar) is always positive except for the portfolios initiated in June $ 1998$ and in September $ 1998$ where we incur heavy losses compared to the gains in the other periods. In other words, this strategy would have procured $ 28$ 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 $ 1997$ and June $ 1998$ where large payoffs at expiration (positive and negative) occur. Indeed, the period of November and December $ 1997$ was a turning point of the DAX and the beginning of an $ 8$ month bull market, explaining the large payoff in March $ 1998$ of the portfolio initiated in December $ 1997$. The same arguing explains the large negative payoff of the portfolio set up in June $ 1998$ expiring in September $ 1998$ (refer to Figure 9.11). Another point to note is that there is a zero cash flow at expiration in $ 24$ 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 $ 1999$), when the implied SPD is more skewed than the time series SPD a comparison of the S$ 1$ 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 $ 1999$. But to what extend this may be significant is uncertain. For the same reason the S$ 2$ trade has no great informational content. Applied to real data it would have procured a negative total net cash flow. Actually, only in June $ 1999$ a portfolio would have been set up. While the S$ 1$ trade performance was independent of the knowledge of the implied and the time series SPD's the S$ 2$ 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.