In the year 1973 Fischer Black and Myron Scholes published their work as a paper “The pricing of options and corporate liabilities” in the Journal of Political Economy. Robert C. Merton published another paper expanding the mathematical understanding of the option pricing model and called it “Black-Scholes Options Pricing Model”. Fischer Black passed away in the year 1995. Later, Merton and Scholes received the 1997 Nobel Memorial prize in Economic Sciences for their work.

This Nobel Prize winning formula is mathematically challenging for the average options trader, but the good news is that they never have to use it. Almost all the modern trading platforms have built-in calculators that calculate the theoretical value of an option price.

In mathematical finance, Equation (1) is the famous Black-Scholes partial differential equation (PDE). It is a second order, non-linear PDE to the first degree. It is a Cauchy-Euler equation (for those who are mathematically inclined) which can be transformed into an unsteady (transient) heat diffusion equation in mechanical engineering by introducing the change of variable transformation. 1(a) and 1(b) are the boundary conditions and 1(c) is the terminal condition which becomes an initial condition when transformed. The standard method of solving the mechanical engineering unsteady heat diffusion equation subject to the transformed boundary and initial conditions and reverting the transformed variables to its original set of variables, t, T, C, S, X, sigma and r, gives the famous Black-Scholes formula, 2(a) and 2(b). This is definitely overwhelming for non-engineers or non-mathematicians.

The Black-Scholes Model assumes the price of the underlying is varying in a continuous manner with no jump discontinuity. Hence the limiting distribution of the price of the underlying asset becomes a standard normal distribution.

The version of the model presented by Black and Scholes was designed to value European options which were dividend-protected. European options (RUT, NDX) can only be exercised at expiration whereas the American style options (MSFT, AAPL) can be exercised anytime on or before the date of expiration.

The model is essentially divided into two parts: the first part, SN(d1), multiplies the price by the change in the call premium in relation to a change in the underlying price. This part of the formula shows the expected benefit of purchasing the underlying outright. The second part, N(d2)Xe^(-rt), provides the current value of paying the exercise price upon expiration (remember, the Black-Scholes model applies to European options that are exercisable only on expiration day). The value of the option is calculated by taking the difference between the two parts, as shown in the equation.

The spreadsheet shows the calculated values for a call option and a put option using the Black-Scholes formula. As you can see, the call option is in the money and the put option is out of the money.