Get Rich or Buy Trying - Part 1 [Solution]

Solution for the week of 10/6/24

Our problem from 10/6/24 was

We will consider two different types of behavior of this stock:

1. The stock has equal probability of going up or down each minute by p0/1000 where p0 is the initial value of the stock.

2. The stock has equal probability of going up to a value of 1.001*p_t-1 or down to a value .999*p_t-1 each minute (where p_t-1 is the value of the stock for the previous minute).

Your job for this first puzzle it to determine the distribution of the value of the stock after 1 day. Assume the stock is traded for 7 hours a day (a bit off, I know).

Big shout out to Seth Cohen from Concord, NH for being the first person to post their solution to this blog! I hope you’ll be joined by many others in the coming weeks and months. The solutions are posted here. Seth also generated the two distributions correctly.

These two problems ended up being exercises in applying transformations to the well-known binomial distribution. In both parts, we’re essentially just scaling the x-axis. For part one, we are performing a simpler linear transformation (scaling by a factor and translating) while for part two, we’re performing a non-linear transformation, which explains why the distribution is non-symmetric. In both cases, the only term in the binomial distribution that needed to be adjusted was the second argument of the binomial coefficient, which effectively scales the x-axis. I provide a bit more explanation here but the distributions ultimately look like:

\(P(X = x) = \binom{420}{\frac{1000(1-x) + 420}{2}} \left(\frac{1}{2}\right)^{420} \text{ where } x = 1 - \frac{1}{1000} (2k-n) \text{ for } k \in [0,1, \cdots , 420]\)

\( P(X = x) = \binom{420}{500 \cdot \ln(x \cdot e^{\frac{n}{1000}})} \left(\frac{1}{2}\right)^{420} \text{ where } x = 1.001^k \cdot 0.999^{n-k} \text{ for } k \in [0,1, \cdots , 420] \)

If you plotted these two distributions, you could be forgiven for thinking they might be the same—they are exceedingly similar, but they are different and we can see that if we modify the parameters a bit. The plot below is for 40 trials and a change of 10% on each trial.

I hope you found this problem fun! As always, feel free to leave comments, thoughts, etc. in the comments. Also, feel free to DM with any ideas you have for a puzzle! Hope to see you for the next one!