Why delta is not the probability of an option expiring in the money simply explained

The connection between the Delta of an option and the probability of that option expiring in-the-money under the assumption μ = r is rooted in the Black-Scholes option pricing model and the concept of risk-neutral valuation.

In risk-neutral valuation, all assets are assumed to grow at the risk-free rate (r), which means μ = r. Under this assumption, we eliminate the risk premium associated with any asset. In a risk-neutral world, investors are indifferent to risk. 

The Delta of a European call option in the Black-Scholes model can be expressed as:

Delta = N(d1)

Where N(d1) is the cumulative distribution function of the standard normal distribution evaluated at d1. In the risk-neutral world (where μ = r), N(d1) can be thought of as the factor by which the stock price (contingent on exercise) exceeds the exercise price, adjusted by the risk-neutral probability. The term d1 in the Black-Scholes formula indeed incorporates elements such as the risk-free rate, the stock’s volatility, and the time to expiration. The model assumes that the expected growth rate of the underlying asset is equal to the risk-free rate. This assumption ensures that the model is consistent with risk-neutral valuation, allowing the calculation of option prices without needing to know the actual probability of various future stock prices.

However, while this risk-neutral framework allows for the derivation and use of the Black-Scholes formula, it’s crucial to remember that the real-world probability (where risks are essential, and the expected returns of risky assets might not equate to the risk-free rate) can be different from the risk-neutral probability. It’s important to note that the real-world probability (where risks matter and returns aren't always equal to the risk-free rate) can differ from the risk-neutral probability. The Delta, which aligns with risk-neutral probability when using the Black-Scholes model, doesn't directly give the real-world probability of the option expiring ITM. It merely gives an adjusted probability under the risk-neutral measure, which is a mathematical tool used for option pricing.

The idea that the Delta could be interpreted as the probability of an option expiring ITM is specifically under the assumption of risk-neutral valuation, which assumes μ = r. However, this doesn't reflect real-world scenarios where risks matter and expected returns on risky assets aren't equal to the risk-free rate. To accurately determine the probability of an option expiring in-the-money, we must consider μ rather than just relying on the risk-free rate (r). While the risk-neutral valuation provides a useful mathematical framework for pricing options, making real-world investment decisions requires us to incorporate the true expected return on the underlying asset.