Why Option Delta Is Not the Probability of Finishing In the Money
Summary
The document distinguishes option delta from the probability that an option expires in the money. In the Black-Scholes model for a European call, delta measures sensitivity of the option price to the underlying price and includes a carry adjustment. Because it can exceed one in some cases, it cannot generally be interpreted as a probability.
The discussion separates this sensitivity from risk-neutral probabilities associated with particular payoff events. The term involving the second Black-Scholes parameter corresponds to a risk-neutral probability of finishing above the strike, while the first parameter is tied to a probability under a different numéraire. Neither quantity predicts the real-world outcome distribution. The formulas and payoff interpretations explain why delta may sometimes look like a probability proxy, but closeness between the two parameters is only an approximation in some settings. These statements are framed within the Black-Scholes model and should not be confused with a forecast of realized exercise frequency.
Key ideas
- Call delta is the derivative of option value with respect to the underlying price.
- Delta includes carry effects and is not generally bounded like a probability.
- A risk-neutral probability associated with finishing above the strike uses the second Black-Scholes parameter.
- The first parameter corresponds to a probability under a different numéraire.
- Risk-neutral probabilities and delta do not predict the real-world price outcome.
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Full text
# What is delta of an option signaling?
# What is delta of an option signaling?
In an interview I was once asked what the delta of an option was and my answer started from the fact that it is the first derivative of the option with respect to the price, and then I concluded saying that it is practically used as probability of the option to end In The Money at Maturity. The interviewer, very bother by this conclusion, replied:
> That is absolutely not true from a mathematical point of view
Why is that so?
## Answer by Kevin (score 5)
https://quant.stackexchange.com/a/67960
The comments already give links to many top answers and articles outlining the answer. Here's the summary:
The Black-Scholes formula for European-style call options is $$C = Se^{-qT}\Phi(d_1)-Ke^{-rT}\Phi(d_2).$$ The option delta (sensitivity to changes in the stock price) is $$\Delta=\frac{\partial C}{\partial S} =e^{-qT}\Phi(d_1).$$
Firstly, the delta of an option cannot be the probability of anything: it can exceed one, depending on the cost of carry $q$ (think of long-dated deep ITM currency options).
You can show that $\Phi(d_2)$ is the risk-neutral probability of the event $\{S_T\geq K\}$. Thus, a few people call $\Phi(d_2)$ the probability of ending up in the money. It couldn't be further from the truth. This number doesn't tell you where the asset will likely be at maturity. Risk-neutral valuation is a beautiful and very convenient pricing tool, but it makes no predictions about the future distribution of stock prices.
You can show that $\Phi(d_1)$ is the (risk-neutral) probability of the event $\{S_T\geq K\}$ associated to a different numéraire. That's even more technical and even less related to where the underlying is going to end up in the real world.
Because $\Delta$ is easily observable on any trading platform, and because $d_1=d_2+\sigma\sqrt{T}\overset{?!}{\approx} d_2$, some people may suggest that delta proxies the probability of exercise. As you now know, this is just wrong and bad. As @Jan said this is a "red flag in a quant interview".
What you can do: You can interpret $Se^{-qT}\Phi(d_1)$ as price of an asset-or-nothing and $e^{-rT}\Phi(d_2)$ as price of a cash-or-nothing option, or as the aforementioned risk-neutral probabilities.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.