Trailing stops - the math says this is a complex decision

The math for trailing stops states that it may make good sense, but traders have known that for decades. However, the implementation of stop-loss is as difficult as the development of any trading rule. The core issue is determining how to optimally adjust those stops without losing potentially successful trades. Take three examples:

1. Fixed stops - Do not change the stop at all. Only question is determining where to put the initial stop. The probability of being stopped can be determined by the drift and volatility of the price series. The trader is willing to take more risk as the price moves away from the stop.

2. Variable stops - Change the stop with any gain in price. Question of keeping the probability of being stopped the same through time as reset by the price increase. Traders will still risk the same amount on price increases.

3. Trailing stops - Adjust the stop on a lagged basis as a percentage of the maximum value with some lag. More money will be risked as the trade increases in value.

This are not the only ways to describe stops, but it illustrates that the issue can be complex. It becomes even more complex if you think about the point of exit for a trade as an additional wrinkle to the problem. If you have a stop on the downside, then you should think about a stop or exit on the upside. The next higher level of complexity is determining when to enter in the first place. You don't need a stop if you don't put on a position, or more importantly, a stop is a form of error correction. The model has made a mistake. Hence, there is a need to stop-exit a position. Risk management cannot be detached from entry and exit decisions; however, many models perform worse once a stop-loss is added. Added to the stop-loss decision is the complex issue of when to re-enter a trade once a stop-loss is triggered.

This problem has been tackled by Tim Leung through his paper "Optimal Trading with a Trailing Stop" which is highly mathematical and in no way an easy read. I am still digesting the impact of his work, but I find that he has been able to provide structure to these higher levels of complexity.