III. Quantitative Finance Applications
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08. September 2023

# The intuition behind the reason why delta is not the probability of an option expiring in the money

**Why delta is not the probability of an option expiring in the money in layman’s terms…**

The connection between the Delta of an option and the probability of that option expiring in-the-money under the assumption mu = 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 mu = 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 mu = 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 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 mu = 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 (mu) 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.

#BlackScholesModel #RiskNeutralValuation #DeltaMisconceptions#OptionPricing

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