Why Black-Scholes Delta Is Not Usually the Risk-Neutral ITM Probability
Summary
The document disentangles the common description of call delta as the probability that the option finishes in the money. Under Black-Scholes, the risk-neutral probability of finishing above the strike is represented by N(d2), while the call's delta, assuming no dividends, is N(d1). These quantities have distinct interpretations and are not generally equal.
The answer explains that N(d2) relates to a digital option's value after accounting for the relevant discounting, whereas delta can also be interpreted as an in-the-money probability under a different equivalent martingale measure, using the asset as numéraire. Delta may be approximated by N(d2) when maturity is very short or volatility is low, under the stated no-dividend assumptions. The discussion is specific to the Black-Scholes framework and its measure choices; it does not establish delta as a universal real-world probability forecast.
Key ideas
- In Black-Scholes, N(d2) gives the risk-neutral probability of finishing above the strike under the money-market numéraire.
- For a call without dividends, delta is N(d1), not generally N(d2).
- Delta can have a probability interpretation under a measure using the asset as numéraire.
- The delta and N(d2) values can be close at short maturities or low volatility.
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Full text
# Option delta - Conditional probability definition? # Option delta - Conditional probability definition? Can someone help me interpret this definition of delta? > Delta is a conditional probability of terminal value (St) being greater than the Strike (X) given that St > X for a call option. Is the second part of the statement not completely redundant? This may be completely obvious, but I don't see why this is conditional. The source of the definition is: Initially found here, where three definitions are given: > https://financetrainingcourse.com/education/2012/09/sales-trading-interview-guide-understanding-greeks-option-delta-and-gamma/ Also in: > Farid (2015) - An Option Greeks Primer: Building Intuition with Delta Hedging and Monte Carlo Simulation in Excel. ## Answer by Quantuple (score 4, accepted) https://quant.stackexchange.com/a/25790 IMHO the 'definition' you mention is not a mathematical definition per se, but rather an approximation used by some practitioners. Mathematically, it is $N(d_2)$ in the BS formula which figures the conditional probability that the terminal asset price $S_T$ will finish above the strike level $X$ given the information we possess today (represented by $S_t$), and that, under a risk-neutral measure which uses the risk-free money market account as numéraire. Omitting discounting, $N(d_2)$ is indeed the price of a binary or digital option assuming GBM. The delta (omitting dividend payments) is on the other hand given by $N(d_1)$ not $N(d_2)$. Only for extremely short time to maturity or low volatility can $d_1$ be approximated by $d_2$ and hence $\Delta$ be approximated by $N(d_2)$ (again assuming no divs). Still, we could also say that $\Delta$ corresponds to a probability of finishing in the money, but in an equivalent martingale measure which uses the asset itself as numéraire. It therefore depends on the probability measure in which you are working/interpreting your results. In the sell-side we like to think in terms of risk-neutral probabilities (or binary options). See the definition of $d_1$ and $d_2$ here for instance.
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