Get Rich or Buy Trying - Part 3 [Solution]
Solution for the week of 10/20/24
Our problem from 10/20/24 was
Consider the following stock behavior:
The stock’s value will change by a factor of (1+δ) from it’s pervious value each hour where δ is uniformly distributed from [-1/100, 1/100]. For example, after two hours, the value would be p_0 * δ_1 * δ_2. Note: these values are slightly different from Part 2.
The question for this week is, what’s the average difference in performance between an oracle that can predict whether the stock will go up each hour vs the oracle that can predict whether the stock will go up each day? The oracles are to perform over 5 days, 7 hours each day.
This problem ended up being tricky more because of some involved algebraic bookkeeping than any particularly challenging intuitive leaps. The hourly oracle’s returns were relatively easy to get. We just needed to compute the expectation of
Where X is a uniformly distributed random variable on [0, 1/100], and Y_h is a binomially distributed random variable with p = .5 and N = 35. Then with that we can use the definition of the expectation to compute the exact value. The full writeup is provided here. The ends up with an expected return of 9.40%. Interestingly, my posted solution doesn’t quite match the simulated solution (I’m assuming the simulation to be correct) and is off by about .1%. I suspect I made an algebra error somewhere in my solution but haven’t been able to find it yet. This was emblematic of the challenge in this problem—keeping all the constants and terms straight.
Computing the returns for the daily oracle is definitely the more complicated part. For this we need to first determine the distribution of the stock price over the course of the day. For that we can use our result from Part 2 coupled with some rescaling to limit the distribution to values above 1. From there we do the same thing and compute the expectation. This time around, the integral for the expectation is quite complex and I resorted to software to help compute it. I still haven’t gotten my analytic solution to match up with simulated solution so my posted solution probably isn’t correct but I think the overall approach is ok. I’ll continue working on it and update this solution post once its correct. In the meantime, a simple simulation will show that the expected returns for the daily oracle are 3.11%. A pretty big difference.
Let me know if you had any better luck with this problem or approached it differently. As always, thanks for reading!