The d₁ term in Black and Scholes model explained in Layman’s terms…

The Black-Scholes model calculates the theoretical value of European-style options, assuming stock prices follow a log-normal distribution. The model is known for constant volatility, no dividend payments, and its innovative use of stochastic calculus, significantly influencing both theoretical finance and practical trading.

The formula for a European call option is:
C = S₀ * N(d₁) - X * e^(-rT) * N(d₂)

- C is the call option price.
- S₀ is the current stock price.
- X is the strike price.
- T is the time to expiration (years).
- r is the risk-free interest rate.
- σ is the stock return volatility.
- N(·) is the standard normal cumulative distribution function.
- d₁ and d₂ are defined as:

d₁ = (ln(S₀/X) + (r + σ²/2)T) / (σ√T)
d₂ = d₁ - σ√T

d₁ = (ln(S₀/X) + (r + σ²/2)T) / (σ√T)

- ln(S₀/X) is the natural logarithm of the stock-to-strike price ratio.
- r is the risk-free interest rate.
- σ² is the squared volatility of the stock.
- T is the option's time to expiration.

The term σ² / 2 corrects for the higher mean of a log-normally distributed variable compared to its normally distributed logarithm, addressing the right skewness of the log-normal distribution.

Standardization in statistics means adjusting data so that it has a mean of 0 and a standard deviation of 1. In the Black-Scholes formula, dividing by σ√T standardizes d₁ and d₂. This process scales the impact of stock price, strike price, volatility, and time, making the model applicable across different scenarios.

This standardization aligns d₁ and d₂ with the properties of a standard normal distribution, facilitating the use of N(d₁) and N(d₂) to estimate probabilities relevant to the option’s value.

The Black-Scholes model, with its components σ² / 2 and σ√T, provides a robust framework for option pricing. The σ² / 2 term addresses log-normal distribution characteristics, while σ√T standardizes the model's calculations.


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