The Anatomy of the Black-Scholes-Merton formula

This article gives you a background to the famous option pricing formula

The Black-Sholes-Merton formula for a call option reads as follows:

where C stands for the price of a call option, S(0) is the initial price of the option’s underlying asset, K is the option’s Strike Price, r is the risk-free rate, T is the time to maturity. N(d1) and N(d2) are cumulative probability distribution functions for a variable with a standard Normal distribution, which are specified as follows:

where σ^2 is the underlying asset’s variance.

Let us now take a practical example of how the Black-Scholes-Merton formula can be used to calculate the price of the Option A in our presentation. The Option A is European Long Call and has the following characteristics:

Asset price (S0)
Strike Price (K)
Time-to-maturity in years (T)
Interest Rate (e^(-rT)
Volatility (σ)
30 %

As a first step, let us proceed to calculate eqs. (2) and (3):

Finally, we have that eq. (1) takes the following form:
Thus, the price of the Option A is approximately 20.

Further reading

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