Learning about probability distribution is pretty boring unless one knows how to relate these to at least some real life application. Well, that can be said about almost anything learnt in life .However in the case of probability distributions, it is strikingly more true.

There is a fair chance that a lot of people know about normal,log normal, student t,weibull,poisson,Cauchy, exponential, chi-square ,brownian, gbm, levy, etc etc.. Somehow I have never found merely learning about these distributions interesting. There are all sorts of distributions which have various parameters. Sometimes I feel I should least know half a dozen real life apps for each of these distributions.

It is always an aha moment for me when I see a application of these distributions in some practical way. One of the aha moments I had recently, was with the arc sine density function. Most of the people working in finance know about brownian motion becoz finance literature is filled with BM constructs. I remember reading about Zero Hitting time of a brownian motion which follows arc sine density function. I conveniently forgot about it. Honestly, who the hell cares whether it follows arc sine or arc cos density function. This was until a few days back when I came across a powerful way of looking at a residual trading strategy where author looks at a spread trade and uses Zero hitting time of a brownian motion to discard trading in that spread!! NOW THAT’s WHAT I LIKE. A true application of something from theory.

Well , the reasoning is elegant. The trader looks at a spread and compares the zero crossing frequency of the spread. If the spread is to be traded, it has to mean revert which means it has to cross 0 often, 0 often meaning, probability of hitting 0 given it hit 0 at some point in its history should be high. AND, brownian motion doesnt let that happen. One might think…..Its a random walk so the path hits 0 pretty regularly!! But that’s not the case. In any case, the brownian motion can thus be used to check the non stationarity of the zero hitting time.

ArcSin Density

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I don’t think I will ever forget this arc sine density function again becoz a viewpoint from a practical application is worth more than 1000 hours of studying about it.