# GBM

If you want to make money trading, you're going to need a way to identify when an asset is likely to be cheap and when it is likely to be expensive. You want to be a net buyer of the cheap stuff and a net seller of the expensive stuff. Thanks, Capitain Obvious. You're welcome. How does this relate to equity options? If we take the (liquid) US Equity options market as an example then there are an absolute ton of options contracts you could be trading. 95% of them are sufficiently fairly valued that you won't make much money trading them once you've paid all the costs to buy and sell them and hedge your risk. The remaining 5% are worth looking for. Options have a positive dependency on volatility. In looking for "cheap" or "expensive" options, we're really looking for cheap or expensive "volatility". So we ask the following questions: When does the forward volatility "implied" by options prices tend to be lower than the volatility that realises in the subsequent stock price process? We would look to buy...

There are 2 good reasons to buy put options: because you think they are cheap because you want downside protection. In the latter case, you are looking to use the skewed payoff profile of the put option to protect a portfolio against large downside moves without capping your upside too much. The first requires a pricing model. Or, at the least, an understanding of when and under what conditions put options tend to be cheap. The second doesn't necessarily. We'll assume that we're going to have to pay a premium to protect our portfolio - and that not losing a large amount of money is more important than the exact price we pay for it. Let's run through an example… We have a portfolio comprised entirely of 100 shares of SPY. About $29k worth. We can plot a payoff profile for our whole portfolio. This is going to show the dollar P&L from our portfolio at various SPY prices. At the time of writing, SPY closed at $287.05 if (!require("pacman")) install.packages("pacman") pacman::p_load(tidyverse, rvest, slider, tidyquant, alphavantager, kableExtra) SPYprice <- 287.05...

We've been working on visualisation tools to make option pricing models intuitive to the non-mathematician. Fundamental to such an exercise is a way to model the random nature of asset price processes. The Geometric Brownian Motion (GBM) model is a ubiquitous way to do this. We can represent the price of an asset at time [latex] t [/latex] as the state [latex] x(t) [/latex] of a GBM process. [latex] x(t) [/latex] satisfies an Ito differential equation [latex display="true"] dx(t) = \mu x(t) dt + \sigma x(t) dw(t) [/latex] where [latex] w(t) [/latex] follows a wiener process with drift [latex] \mu [/latex] and volatility [latex] \sigma [/latex]. The probability distribution of future prices follows a log-normal distribution [latex display="true"] [(\mu - \frac{\sigma^2}{2}) + \log{x_0}, \sigma\sqrt{t}] [/latex] OK, nerd, but how do I get the probability distribution of future prices from the starting price of the asset, and assumptions about the return distribution? I couldn't work that out quickly, so I asked Wolfram Mathematica that question. PDF[GeometricBrownianMotionProcess[\[Mu], \[Sigma], Subscript[x, 0]][t], x] [latex display="true"] \frac{exp{\frac{(-t(\mu - \frac{\sigma^2}{2}) + \log{x} - log{x_0})^2}{2t\sigma^2}}}{\sqrt{2\pi}\sqrt{t}x\sigma}, x > 0...