Estimating Real-World Stock Price Probabilities with a Lognormal Model
Summary
The document shows how to estimate the real-world probability that a stock will finish within a specified price interval using a geometric Brownian motion model. It treats the stated required annual return as the drift and, for illustration, uses implied volatility as the volatility input. The resulting lognormal distribution is converted to standard normal probabilities at the upper and lower price bounds, then subtracted to obtain the interval probability.
The example starts with a stock price of 61.5, a 0.4-year horizon, 120% volatility, and an assumed 30% annual return, but it leaves the final probability calculation to the reader. The author emphasizes that implied volatility is typically risk-neutral, while a real-world forecast should ideally use historically estimated volatility. More broadly, model choice and investor beliefs affect real-world probabilities, so this calculation is illustrative rather than a reliable forecast. A second answer suggests forecasting volatility with methods such as GARCH or EWMA instead of expecting precise price forecasts.
Key ideas
- Under geometric Brownian motion, the stock price at a future date has a lognormal distribution.
- The interval probability is the cumulative probability at the upper bound minus that at the lower bound.
- A real-world calculation needs assumptions for drift and volatility; implied volatility is used only for illustration here.
- The answer cautions that model choice and investor beliefs limit the interpretation of a real-world probability estimate.
- Volatility forecasting methods such as GARCH and EWMA are offered as alternatives to direct price forecasting.
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Full text
# Probability of a stock price using implied volatility
# Probability of a stock price using implied volatility
I have attempted to use the fact of having implied volatility, but have not been able to come up with a viable way to calculate the probability, any ideas? Suppose that a stock $S_t$ follows a lognormal model and that on May 29,2019 the closing price of the stock was $S_0$ was 61.5 and the implied volatility of the options with maturity T=0.4 was 120%. If we assume that the annual return required by the investors is 30% what is the probability $P(40\le S_t \le 55)$
## Answer by Jan Stuller (score 3, accepted)
https://quant.stackexchange.com/a/54820
I assume you want to real-world probability, because the risk-neutral probability is not a probability in the 'likelihood' sense.
Under the real-world measure, we model the stock under the B-S model as:
$$X(t)=X(0)+\int^{t}_{0}\mu X(h)dh+\int^{t}_{0}\sigma X(h)dW(h)$$
If the market demands a 30% annual return, I will take that as the real-world rate $\mu$. Strictly speaking, we should also take the volatility estimated from a historical time series if we deal with the real-world measure, but I will just take your implied vol here for simplicity:
$$X(t)=61.5+\int^{t=0.4}_{0}0.3 X(h)dh+\int^{t=0.4}_{0} 1.2 X(h)dW(h) = \\ = 61.5exp \left( \left[ 0.3 - 0.5* 1.2^2 \right] 0.4 + 1.2 * \sqrt(0.4) Z \right) = \\ = 61.5exp\left( -0.456+0.759Z\right)$$
Therefore:
$$\mathbb{P}\left( 40<X_t<55\right)=\mathbb{P}\left(X_t<55\right)- \mathbb{P}\left( X_t<40\right)$$
Now:
$$\mathbb{P}\left(X_t<55\right)=\mathbb{P}\left(61.5exp\left( -0.456+0.759Z\right)<55\right) = \\= \mathbb{P}\left(ln(61.5) +\left( -0.456+0.759Z\right)<ln(55)\right) = \\= \mathbb{P} \left( Z< \frac{ln(\frac{55}{61.5})+0.456)}{0.759} \right) $$
You can do the same for $\mathbb{P}\left( X_t<40\right)$, work out the numbers yourself and you should get the answer.
Important Note: The above was just to demonstrate how real world probability could be calculated by blindly plugging numbers into the B-S model. However, pls note that if you want an actual real probability of a stock ending up within a specific range, the B-S model framework is not really suitable for that. Every market agent will have his or her (Bayesian) view of the state of the world and every market agent will view the probabilities differently. Even the choice of model that you will use to compute the probability is a Bayesian choice in itself. It's a really interesting problem, but its more of an "existentialistic" problem, rather than "practical" problem. High-frequency algo traders try to estimate probabilities all the time. They all use different models, different input data, etc...
## Answer by user28909 (score 1)
https://quant.stackexchange.com/a/54819
Asset prices follow a random walk, so assuming probabilities and forecasting stock prices are not that accurate. Hence, investors try to project volatility rather than asset prices (i.e. implied vol) using GARCH, EWMA, or other vol forecasting models.
The optimal portfolio is to invest long term in a globally diversified portfolio with a focus on uncorrelated asset styles (growth assets, real assets, and hedge assets) or asset classes.
If you figure out a way to project asset prices, with somewhat strong accuracy. Please let me know so that we can set up an investment management firm 😗. Joking aside, I wish you the best of luck on figuring this out and I look forward for your findings.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.