How to Calculate N in Black Scholes Formula

The Black-Scholes formula is a mathematical model used to determine the theoretical value of European-style options. At its core, the formula calculates the fair price of an option by considering factors like the underlying asset's price, strike price, time to expiration, risk-free interest rate, and volatility. One key component of this formula is the cumulative distribution function (N), which represents the probability that a standard normal random variable will be less than or equal to a given value.

What is N in Black-Scholes?

The N function in the Black-Scholes formula represents the cumulative distribution function (CDF) of the standard normal distribution. In practical terms, it calculates the probability that a random variable from a standard normal distribution (mean = 0, standard deviation = 1) will be less than or equal to a specific value.

In options pricing, N is used to determine the probability that the underlying asset's price will be above or below the strike price at expiration. This probability is then used to calculate the present value of the option's potential payoff.

N is often referred to as the "normal cumulative distribution function" or "standard normal CDF." It's a fundamental statistical concept that bridges the gap between the Black-Scholes formula and real-world probability calculations.

The N Function Formula

The standard normal cumulative distribution function is defined mathematically as:

N(x) = ∫_-∞^x (1/√(2π)) * e^-t²/2 dt

Where:

N(x) is the cumulative probability from -∞ to x
√(2π) is the normalization constant
e^-t²/2 is the probability density function of the standard normal distribution

In the Black-Scholes formula, N is typically used in its standardized form where the variable x is calculated as:

x = (ln(S/K) + (r + σ²/2)T) / (σ√T)

Where:

S = current stock price
K = strike price
r = risk-free interest rate
σ = volatility of the stock
T = time to expiration (in years)

Calculation Methods

There are several ways to calculate N in the Black-Scholes formula:

Excel Function: The NORM.S.DIST function in Excel can calculate the cumulative distribution function.
Statistical Tables: Standard normal distribution tables provide pre-calculated values for N(x).
Programming Libraries: Most programming languages have built-in functions for calculating the standard normal CDF.
Approximation Formulas: For quick calculations, you can use approximation formulas like the one developed by Abramowitz and Stegun.

For precise financial calculations, it's recommended to use built-in functions or specialized financial software rather than approximation formulas.

Worked Example

Let's calculate N for a call option with the following parameters:

Current stock price (S) = $50
Strike price (K) = $55
Risk-free rate (r) = 5% or 0.05
Volatility (σ) = 20% or 0.20
Time to expiration (T) = 0.5 years

First, we calculate the d1 term:

d1 = (ln(50/55) + (0.05 + 0.20²/2)*0.5) / (0.20*√0.5)

d1 ≈ (ln(0.909) + (0.05 + 0.02)*0.5) / (0.20*0.707)

d1 ≈ (-0.0953 + 0.055) / 0.1414

d1 ≈ (-0.0403) / 0.1414 ≈ -0.2847

Now we calculate N(d1):

N(-0.2847) ≈ 0.3870

This means there's approximately a 38.7% probability that the stock price will be above the strike price at expiration.

FAQ

What is the difference between N and n in Black-Scholes?

N represents the cumulative distribution function of the standard normal distribution, while n typically represents the standard normal probability density function. In the Black-Scholes formula, N is used for cumulative probabilities, while n is used for point probabilities.

Can I use the same N function for both call and put options?

Yes, the N function is the same for both call and put options. The difference in calculations comes from the d1 and d2 terms, which are adjusted based on whether you're pricing a call or put option.

What happens if the volatility is zero in the Black-Scholes formula?

If volatility (σ) is zero, the option price will be determined solely by the intrinsic value (the difference between the stock price and strike price). In this case, the N function will approach either 0 or 1 depending on whether the stock price is below or above the strike price.

Is the N function the same as the z-score?

No, the N function is the cumulative distribution function of the standard normal distribution, while the z-score is a measure of how many standard deviations a value is from the mean. The N function uses the z-score as its input to calculate probabilities.