Properties of time averages in a risk management simulation

This paper investigates a simple risk management problem where an investor is forced to hold a risky asset and then allowed to trade put options on the asset. I simulate the distribution of returns for different quantities of options and investigate statistics from the distribution. In the first section of the paper, I compare two types of averages: the ensemble and the time average. These two statistics are motivated by research that uses ideas from ergodic theory and tools from statistical mechanics to provide new insight into decision making under uncertainty. In a large sample setting, I find that the ensemble average leads an investor to buy zero put options and the time average leads them to buy a positive quantity of options; these results are in agreement with stylized facts from the literature. In the second section, I investigate the stability of the optimal quantity under small sample sizes. This is a standard resampling exercise that shows large variability in the optimal quantity associated with the time average of returns. In the third section, I conclude with a brief discussion of higher moments from the distribution of returns. I show that higher moments change substantially with different quantities of options and suggest that these higher moments deserve further attention in relation to the time average.