Estimating an Option’s Expiration Probability with the Black–Scholes Model

Summary

The document explains how the Black–Scholes quantity N(d2) can approximate the probability that an option finishes in the money at expiration. N(d1), by contrast, corresponds to the option’s delta. This gives a practical starting point for estimating an equity option’s exercise likelihood from spot price, strike, time to expiration, rates, dividends, and a volatility estimate.

The probability from N(d2) is risk-neutral because the model uses the risk-free rate as the underlying drift. Substituting an estimated real-world drift can provide a practical approximation to an actual probability, but it changes the interpretation. Both approaches rely on assumptions including normally distributed log returns and constant volatility. The discussion offers no empirical comparison of estimates and does not resolve how to choose a volatility input; it notes that volatility may be estimated historically or implied from current option prices.

Key ideas

• Under Black–Scholes, N(d2) approximates the risk-neutral probability that an option expires in the money.
• N(d1) is associated with option delta rather than the expiration probability.
• Using a real-world drift instead of the risk-free rate can produce an estimate closer to an actual probability.
• The estimate depends on assumptions of normal log returns and constant volatility.

Tags

# what is the best way to calculate the probability of an equity option ending in the money?


# what is the best way to calculate the probability of an equity option ending in the money?












Given historical implied volatility and all other know variables (stock price, option strike price, option expiration date, dividend rate, interest rate) what is the best way to calculate the probability of an option being in the money at expiration?

## Answer by David Harper (score 4)

https://quant.stackexchange.com/a/2840

N(d2) is near to the probability the option will expire in the money; I have a video showing how d2 is similar to distance to default in the Merton here on youtube.

N(d1) is the delta.

The technical issue is that N(d2) is a risk-neutral probability; the input in d2 is the riskfree rate, although the theory is more involved.

But, if you replace the riskfree rate with a realistic drift (mu) you have a reasonable estimate, however N(d2) of course assumes normally distributed log returns. So, as with BSM, your answer here still makes the limiting assumptions, namely normal log returns and constant volatility. (I don't know what "historical implied volaility" is: the input is a current, instantaneous volatility estimate, it can be historical or implied)

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