Sept. 25, 2019

This **HTML file** deals with stopping times. It is a notion related to stochastic processes where we try to determine where and when certain values will be hit like a price target for instance.

The note argues that it is not the first stopping time that should be the main interest, but the last one where you might have no means to determine when and at what level it might be reached if at all. Yet, getting closer to that last stopping time might have more merit since it should tend to increase profits.

Added Sept. 27

For those wishing to learn more about **stopping times**, **hitting times** follow the links.

To go even further, look up **Riemann sums**, **Itô calculus**, and the **Newton-Cotes formulas**.

What you will find is that those formulas date, way back to the 1850s and '60s. Even **Bachelier** in his 1900 thesis (La Théorie de la Spéculation) used formulas from the 1870s. Since Bachelier's thesis we have that the variance is proportional to the square-root of time. Therefore, we should expect that the longer we hold some shares, the higher the variance will be. Over the short-term, this might not show up that much, but over the long term, it definitely will.

One mathematician I admire a lot is **Kiyosi Itô** for his work. This does not derail the remarkable contributions of his predecessors or successors. We all build on their shoulders anyway.

The extension of these time series formulas to applications in finance is not new either. Nor is the payoff matrix that I often use in my posts: Σ(H ∙ ΔP). It is even embedded in the **fundamental theorem of calculus**, and that dates way back. It is how we use these formulas that we can extend their applications to push the limits of what was considered as barriers for many many years. Things like the efficient frontier which in my research appears at most as a line in the sand. Where you can simply jump over it since it was a self-imposed theoretical quadratic upper limitation to portfolio management.

It is not by figuring out the partial sum of the parts (factors, weights or whatever) having an upper limit that you will exceed this limit. If 4 or 5 factors are sufficient to explain 95% of a price series, it is all it can do. It is the same for the efficient portfolio on the efficient frontier. If that is the target, you can aim for it, but you will not exceed it.

What you have to do is change the portfolio management mechanics, take what you find useful in all those theories and readymade formulas and make them do more.

Sept. 25, 2019, © Guy R. Fleury. All rights reserved.