Deriving Positive-Return Probabilities from Drift and Volatility

Summary

The document derives the probability that an investment has a positive return over a chosen horizon under a geometric Brownian motion model. Starting with a price process whose drift is 15% and volatility is 10%, it applies Itô’s lemma to show that log returns are normally distributed. The expected log return uses the drift adjusted downward by half the variance, while the standard deviation scales with the square root of elapsed time.

Standardizing that distribution gives a cumulative normal probability for a positive return. The worked example reports about 92.65% for a one-year horizon and about 50.08% over one second, using a year of 365 days. It illustrates how a favorable annual probability can coexist with a near-even chance over a very short interval. These figures depend on the assumed constant drift and volatility, the geometric Brownian motion model, and the selected time convention; the calculation does not establish that those assumptions describe actual returns.

Key ideas

• Under geometric Brownian motion, log returns are normally distributed over a fixed horizon.
• The expected log return is the price drift minus half the variance rate.
• Return volatility over a horizon scales with the square root of time.
• The probability of a positive return follows from standardizing the log-return distribution.
• The example’s probabilities rely on constant parameters and a specific time convention.

Tags

# Probability of success given expected return and volatility


# Probability of success given expected return and volatility












I am reading Taleb "Fooled By Randomness", and the author says that a 15% return with 10% volatility translates to 93% success in a year and 50.02% success in any given second.

Could someone help me understand this calculation?

## Answer by Kevin (score 7)

https://quant.stackexchange.com/a/77610

It's probably a simple textbook example to illustrate Taleb's point. Suppose $\text{d}S_t=\mu S_t \text{d}t+\sigma S_t \text{d}W_t$ with $\mu=0.15$ and $\sigma=0.1$.

By Itô's lemma, the log return follows a normal distribution, $$R=\ln(S_T/S_t)\sim N\left(\left(\mu-\frac{1}{2}\sigma^2\right)(T-t),\sigma^2 (T-t)\right).$$

Then, your success probability is $$\mathbb{P}[R>0]=1-\mathbb{P}[R\leq0]=1-\Phi\left(-\frac{\left(\mu-\frac{1}{2}\sigma^2\right)(T-t)}{\sigma \sqrt{T-t}}\right)=\Phi\left(\frac{\mu-\frac{1}{2}\sigma^2}{\sigma}\sqrt{T-t}\right),$$ where $\Phi$ is the cdf of a standard normal random variable. Note that $\frac{0.15-\frac{1}{2}0.1^2}{0.1}=1.45$.

- The probability of earning money over one second is $\Phi\left(1.45\cdot \sqrt{\frac{1}{365\cdot24\cdot60}}\right)=50.08\%$.

- The probability of a positive return over one year is $\Phi(1.45\cdot\sqrt{1})=92.65\%$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.