Estimating One-Day Exercise Probability with the Black–Scholes Model

Summary

The document checks the probability that a stock priced at $27 will finish above a $28 strike one day later, assuming zero interest and 15% annual volatility. It applies the Black–Scholes d₂ expression and translates annual volatility into daily volatility by dividing by the square root of 252 trading days. The strike requires a move of roughly 3.7%, or about 3.9 daily standard deviations, which corresponds to an approximately 0.005% upper-tail probability under a normal return approximation.

This comparison shows why the resulting probability, about one in 17,000, is small but consistent with the stated assumptions. The explanation is an approximation tied to the model's distribution and inputs; it does not establish the real-world chance of exercise for other stocks, volatility estimates, or time horizons. It also treats finishing in the money as the relevant event, without discussing early exercise or other option-specific factors.

Key ideas

• Annual volatility scales to daily volatility by dividing by the square root of the number of trading days.
• The $28 strike is about 3.9 daily standard deviations above a $27 stock price under the given assumptions.
• A normal-tail calculation gives an upper-tail probability near 0.005%.
• The model result depends on the assumed volatility, time to expiry, rate, and return distribution.

Tags

# Am I reading this correctly? probability way too small with BS model


# Am I reading this correctly? probability way too small with BS model












For a stock trading at $27, $28 strike, 0% interest, 15% annual vol, and one day until expiration there is about a 1 in 17000 chance of it being exercised?

$d_2 = \frac{1}{.15\sqrt{1/252}}\left[\ln\left(\frac{27}{28}\right) + \left(0 - \frac{.15^2}{2}\right)(1/252)\right]$

$1/(0.5 (1 + Erf[(Log[27/28] - (1/252) (0.15^2/2))/(0.15 *2^{1/2} Sqrt[1/252])]))$

that seem way too small but it's the answer I got

## Answer by RRG (score 9, accepted)

https://quant.stackexchange.com/a/8481

With $15\%$ annual volatility we have $15\%/\sqrt{252}\approx0.94\%$ daily volatility. To go from $27$ to $28$ is a $1/27\approx 3.7\%$ move which is $3.7/0.94\approx 3.9$ standard deviations. For a normal distribution this is about $0.005\%$ probability which is in line with your result.

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