9.7 A Word of Caution

Interpreting the implied SPD as the SPD used by investors to price options, the historical density as the `real' underlyings' SPD and assuming that no agent but one know the underlyings' SPD one should expect this agent to make higher profits than all others due to its superior knowledge. That is why, exploiting deviations of implied and historical density appears to be very promising at a first glance. Of course, if all market agents knew the underlyings' SPD, both $ f^*$ would be equal to $ g^*$ . In view of the high net cash flows generated by both skewness and kurtosis trades of type $ 1$, it seems that not all agents are aware of discrepancies in the third and fourth moment of both densities. However, the strategies seem to be exposed to a substantial directional risk. Even if the dataset contained bearish and bullish market phases, both trades have to be tested on more extensive data. Considering the current political and economic developments, it is not clear how these trades will perform being exposed to `peso risks'. Given that profits stem from highly positive cash flows at portfolio initiation, i.e. profits result from possibly mispriced options, who knows how the pricing behavior of agents changes, how do agents assign probabilities to future values of the underlying?

We measured performance in net EUR cash flows. This approach does not take risk into account as, for example the Sharpe ratio which is a measure of the risk adjusted return of an investment. But to compute a return an initial investment has to be done. However, in the simulation above, some portfolios generated positive payoffs both at initiation and at maturity. It is a challenge for future research to find a way how to adjust for risk in such situations.

The SPD comparison yielded the same result for each period but one. The implied SPD $ f^*$ was in all but one period more negatively skewed than the time series SPD $ g^*$. While $ g^*$ was in all periods platykurtic, $ f^*$ was in all but one period leptokurtic. In this period the kurtosis of $ g^*$ was slightly greater than that of $ f^*$. Therefore, there was no alternating use of type $ 1$ and type $ 2$ trades. But in more turbulent market environments such an approach might prove useful. The procedure could be extended and fine tuned by applying a density distance measure as in Ait-Sahalia, Wang and Yared (2000) to give a signal when to set up a portfolio either of type $ 1$ of type $ 2$. Furthermore, it is tempting to modify the time series density estimation method such that the monte carlo paths be simulated drawing random numbers not from a normal distribution but from the distribution of the residuals resulting from the nonparametric estimation of $ \sigma_{FZ}(\bullet)$, Härdle and Yatchew (2001).