Nvidia Stock Crash Prediction

Summary

To keep things computationally simple, we are going to use a binomial model for the price of the underlying Nvidia stock. We don’t know the daily volatility, so we’ll keep that as a variable we call \(\sigma\). We will pretend that each day, the Nvidia stock price can either grow with a factor of \(e^\sigma\) or shrink with a factor of \(e^{-\sigma}\).5 This is a geometric binomial walk. We could transform everything in the reasoning below with the logarithm and get an additive walk in log-returns. Thus, on day zero, the Nvidia stock trades for $184. On day one, it can take one of two values: \(184e^\sigma\) because it went up, or \(184e^{-\sigma}\) because it went down. On day two, it can have one of three values: \(184e^{2\sigma}\) (went up both in the first and second day), \(184e^{\sigma - \sigma} = 184\) (went up and then down, or vice versa), or \(184e^{-2\sigma}\) (went down both days). If it’s easier, we can visualise this as a tree. Each day, the stock price branches into two possibilities, one where it rises, and one where it goes down. In the graph below, each column of bubbles represents the closing value for a day. This looks like a very crude approximation, but it actually works if the time steps are fine-grained enough. The uncertainties involved in some of the other estimations we’ll do dwarf the inaccuracies introduced by this model.6 Even for fairly serious use, I wouldn’t be unhappy with daily time steps when the analysis goes a year out. It is important to keep in mind that the specific numbers in the bubbles depend on which number we selected for the daily volatility \(\sigma\). Any conclusion we draw from this tree is a function of the specific \(\sigma\) chosen to construct the tree. When we have chosen an initial \(\sigma\) and constructed this tree, we can price an option using it. Maybe we have a call option expiring on day three, with a strike price of $180. On day four, the last day, the option has expired, so it is worth nothing. We’ll p...