At first glance, the Black Scholes Model looks like a Black (sc)Hole of Math. Let’s try and simplify it and understand exactly why this not-so-simple formula is groundbreaking.
The Black Scholes Formula, what we shall now call BSF, is a formula used for determining the pricing of a Call Option. (Call options are financial contracts that give the buyer the right, but not an obligation, to buy an asset or instrument at a specified price within a specific period.) It was developed by economists Fischer Black and Myron Scholes. Robert Merton further developed its understanding, which is why this is often also called as the Black-Scholes-Merton (BSM) model.
This formula is used only to price European options, as they can be exercised only on the expiration date.
BSF determines the price of an option by calculating the return an investor gets and deducting the amount that the investor has to pay, using log-normal distribution probabilities to account for volatility in the underlying asset. The formula has helped legitimize options trading, changing the narrative to make it seem more like a science, and less like a gamble. Now, the BSF is used by traders and investors alike as a strategy of hedging to control the risks associated with volatility in the assets that underlie the option.
This formula consists of complex mathematics, that considers five input variables, and the application of normal distribution. I have attempted to simplify this by making this flowchart.
Let’s tackle each of these, one by one.
The first is measuring volatility. Volatility is the measure of how fast and vastly the price of an asset changes. Stocks with higher volatility have a higher standard deviation and are also considered more valuable.
The next is S which stands for Current Stock Price, which is the current market value of the stock. This variable is used twice. Once when it is multiplied with the Normal Distribution of d1 and also to calculate d1.
The K stands for Strike Price, which is the price at which the Call Option can be exercised. This too is applied twice. Once, it is multiplied with the continuously compounded rate into the normal distribution of d2, and also to calculate d1 .
The r stands for Risk-Free Interest Rate, which is the theoretical rate of return on a risk-free investment. This too is used twice. Once to calculate the continuously compounded rate and to calculate d1.
The t, sometimes written as (T-t) stands for Time to Maturity, measures the amount of time between the present and when the bond matures. This is used to calculate d1 and d2. It is also used to calculate the continuously compounded rate.
Although the math seems extremely complicated (and it is!) these calculations can easily be done using online financial calculators, so that we can all harness the power that comes from BSF.