Arc Sine laws and stock prices
In a graduate course in probability we give the three arc sine laws for Brownian motion
(B(t), 0 < t < 1), which say that the following three random variables have arc sine
So the usual random walk/Brownian motion models for stock market prices,
looked at over a short period, predict that the analogous statistics should follow the arc sine law.
Ideally one should look at intra-day prices for each of say 1,000 days.
This data is not so easily available; instead we show plots (made by Gah-Yi Vahn in May 2009)
giving daily closing S&P500 values for non-overlapping 20-day blocks
(there were 746 blocks).
length of time t that B(t) > 0
- last time t that B(t) = 0
- time t that B(t) attains its maximum.
Because the model is really a 20-step random walk, not Brownian motion, there are discretization effects.
A better comparison of the data above is with a simulation of the random walk, shown below.
Ignoring the beta plots, for the 3 statistics the data fits the model simulations pretty well.
Note that the discretization effect (deviation from beta) is most noticable for the statistic
and in retrospect this is easy to explain.
If B(19/20) is close to 0 then it's very likely that T > 19/20 but only has
chance about 1/2 that B(1) > 0, whch is needed for the discretization of T to be 1.
So the height of the rightmost bar in the histogram for T should be only about 1/2 of
the height of the leftmost bar, and that's what we see in data and the model.
- T = last time t that B(t) = 0
Why this example?
These arc sine laws are both counter-intuitive and non-elementary -- they are not just
simple consequences of "fair bet" or "IID sums" -- and there's no way you would predict this behavior except using the specific probability model.