Calculating GBM Probabilities of Reaching a Price by a Fixed Time
Summary
The document explains how to calculate the model probability that a stock price will meet or exceed a specified level at a fixed future time under geometric Brownian motion. Applying Itô’s lemma to the log price gives a normally distributed expression for the terminal price; rearranging the threshold condition reduces the probability to a standard normal cumulative distribution, using the same d2 term as in the Black–Scholes call formula.
The answer distinguishes this terminal-price probability from the chance of touching a level at any point before the horizon, which is not derived here. It also stresses that the result is conditional on the model and its inputs, including drift, volatility, and dividends. A pricing model is designed to value and replicate options, so its probabilities should not automatically be treated as real-world forecasts; the quality of any forecast depends on how well the assumptions represent actual price behavior.
Key ideas
- Under geometric Brownian motion, the log of the terminal stock price is normally distributed.
- A fixed-time threshold probability can be reduced to a standard normal probability using the Black–Scholes d2 term.
- The model probability depends on assumptions about growth, volatility, and dividends.
- A probability of finishing above a threshold differs from the probability of touching it before the horizon.
- Pricing-model probabilities may not represent real-world forecasts.
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Full text
# Determining the probability of arriving at a price by a time T # Determining the probability of arriving at a price by a time T A useful calculation for ascertaining the risk of something might be determining the probability of a realization of a set of stock prices $X$ being greater than or equal to some future price $x$. I understand from my amateur knowledge of this field is that one model of the price of an asset is GBM. Given that prices are assumed to be lognormally distributed (and therefore, returns are normally distributed) it may be easier to determine the probability of a future return arriving at a price $x$. This leaves me with two questions: - For the GBM model of prices - I don't have the first clue of determining a probability of a stock reaching (or exceeding) a price. I would guess you would need to do this in a monte carlo fashion - but I have no idea. - For the return model since it is normally distributed you could determine the probability it will eventually reach a return by using $P(X <= x)$ to get the probability the price $X$ is less than or equal to some price $x$. Then taking $1 - P(X <= x)$ would give you the probability the return exceeds $x$. However there's no time component here. I'm not sure how to integrate it. More to the point, I dont think this really tells me anything about the price. For both of these I would really appreciate some direction - either directly or links to some resources that will help me implement them. Thanks! EDIT: Bad assumption on (2). Should be probability of returning x. ## Answer by Alex (score 2) https://quant.stackexchange.com/a/41066 Regarding the probability of a stock exceeding a certain level: let's assume for simplicity we are in a Black-Scholes world (but similar reasoning for other models) Assuming you believe a GBM is a good representation of the stock price, and assuming you can correctly estimate the "real" growth rate for the stock (and other parameters such as volatility to use and dividends if any), then you can calculate the probability of the stock S exceeding at future time T a certain level K (according to the model): using ito's lemma applied to ln(St), and after integrating, you obtain an expression of S(T) = S(0) * exp( (r-d-vol*vol/2)*T + vol * sqrt(T)*epsilon) You write down the inequality S(T) >= K then isolate epsilon (a gaussian variable) such that -epsilon <= ... Hence proba(S(T)>= K) is equal to proba( -epsilon <= d2 ) = N(d2) N(x) denotes the standard normal cumulative distribution function. d2 here is the same as in the B-S call formula. You can easily find details of the above calculations on the internet or in "Exotic Options and Hybrids" (Mohamed Bouzoubaa) for example. This is a probability according to a model. But if what you want is "real" probabilities, then the question is do you really believe that, in "real life", GBM is really a good representation of the stock price, do you believe that any pricing model is a good enough represention of what happens in real life. It probably is not as the goal of any pricing model is not to forecast anything (but just to tell you how to replicate an option and how much it costs to do so). This does not completely answer your question but hope it might help a little.
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