Probability That Realized Volatility Exceeds Implied Volatility in Black–Scholes

Summary

The document examines the probability that realized volatility exceeds at-the-money implied volatility under Black–Scholes assumptions. One response treats realized volatility as the absolute value of a single log return: because a standard normal observation falls within one standard deviation of its mean about 68% of the time, the chance its absolute value exceeds one standard deviation is about 32%. This is a single-period interpretation, not a general probability for volatility over a horizon.

A second response considers realized variance estimated from many sampled returns. In the idealized model, scaled realized variance follows a chi-square distribution with degrees of freedom tied to the number of observations, so the probability of exceeding the model variance is a chi-square tail probability that can be computed numerically. The answers therefore depend on what “realized volatility” means and how it is measured. The discussion also distinguishes risk-neutral implied variance from realized outcomes under the physical measure; it does not provide market data or account for estimation error, jumps, or departures from Black–Scholes.

Key ideas

• A single absolute Gaussian return exceeds one standard deviation with probability of roughly 32%.
• That single-return probability is different from the probability for a multi-observation realized variance estimate.
• Under the stated sampling assumptions, scaled realized variance follows a chi-square distribution.
• The multi-observation exceedance probability is obtained from the corresponding chi-square tail.
• The interpretation relies on Black–Scholes assumptions and a clear definition of realized volatility.

Tags

# Probability that realized volatility is larger than implied volatility


# Probability that realized volatility is larger than implied volatility












I did a test about quantitative finance. One of the question was :

> What is the probability, in the Black-Scholes world, that the realized volatility is larger the implied volatility ? And why ?

I couldn’t find any answer on the internet. Could you help me please ?

Thank you very much for your help

## Answer by unknownhuman (score 1)

https://quant.stackexchange.com/a/25182

I succeeded to have the expected answer, and I was on the right way :

The implied volatility is the ATM IV The realised volatility is defined as the absolute value of the log return

We know that a gaussian random variable is in the range +- 1*sigma with a probability of 68%. So, the correct answer is approximately 32%

Should you have any comment I am interested

## Answer by Phun (score 0)

https://quant.stackexchange.com/a/25201

It's clear that over time $t\rightarrow t+dt$ the (expected) implied variance in the BS world (under the risk-neutral measure $\mathbb{Q}$) is $\sigma^2 dt$.

The realized variance of an arbitrary single log stock path (under the physical measure $\mathbb{P}$) is $$\sum_{i=1}^\infty \sigma^2(W_{i+1}-W_i)^2 \approx \sum_{i=1}^n \sigma^2(W^{(n)}_{i+1}-W^{(n)}_i)^2 \sim \sigma^2 \frac{dt}{n}\chi_n$$ for large $n$ and $W^{(n)}_{i+1}-W^{(n)}_i \sim \mathcal{N}(0,\frac{dt}{n})$ i.i.d.. Now $$ \text{Prob}(\sigma^2 \frac{dt}{n}\chi_n > \sigma^2 dt) = \text{Prob}(\chi_n > n) = \text{Prob}(\gamma(\frac{n}{2},\frac{1}{2}) > n),$$ where $\gamma(\frac{n}{2},\frac{1}{2})$ is the gamma distribution. The last expression can be solved numerically.

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