# Black Scholes Formula - Explained

What is the Black Scholes Formula?

# What is the Black-Scholes Formula?

The Black-Scholes is a formula also known as Black-Scholes-Merton formula. The economists used it the first time for option pricing. It basically estimates a theoretical value of options in European-style with the help of current stock prices, the options strike price, expected dividends, expected interest rates, expected volatility and expiry time.

## How Does the Black Scholes Method Work?

There were 3 economists who introduced this formula, namely Robert Merton, Fischer Black and Myron Scholes. It is widely used and most popular model of options pricing. In 1973, their research paper published with the title The Pricing of Options & Corporate Liabilities. They introduced it in this publication of the Journal of Political Economy. Black left the world 2 years before Merton and Scholes got the Nobel Prize of 1997 in the subject of Economics for their outstanding work. They brought forth a new way to find the derivatives value. The Nobel Prize is generally not awarded after death. However, Blacks contribution was highly acknowledged by the Nobel Committee regarding the Black Scholes Formula. There are specific assumptions for the Black Scholes Model, i.e.

• The option is only European. It can just be exercised at expiry.
• During the lifespan of the option, no dividend will be paid out.
• There is market efficiency, means the movement of the markets is unpredictable.
• To buy an option, transaction costs will not be applied at all.
• The rate is risk-free. This rate and the volatility are constant and known.
• There is a normal distribution of the returns of the model.

Note: The original model of Black Scholes did not take into consideration the impacts of dividends paid in the lifespan of the option. The company adopts the model frequently for dividends to account by finding the date value of the ex-dividend for the underlying stock.

Black-Scholes Formula

The Black Scholes Model considers multiple variables, i.e.

• the present underlying price
• Price of options strike
• Time till expiry, which is shown being a yearly percentage.
• Implied volatility
• Interest rates which are risk-free.

The model has 2 part. The 1st part is SN(d1) multiplies the price with call premium variation in response to the underlying price variation. So, this part explains the expected benefit of buying the underlying outright.

The 2nd part N(d2) Ke-rt gives the present value of making payment of the exercise price in expiry. The point to be noted is that application of the BS model (Black Scholes) is only on the European options which we can exercise on expiry day only. To calculate the value of the option, subtract both parts as given in the equation. The maths part of the formula is tricky and may be intimidating.

Luckily, one does not have to be an expert in mathematics for using this model ones own strategy. Options investors can access many of the options calculators online. Several commercial platforms present analysis instruments for robust options, e.g. spreadsheets and indicators that make the calculations and give the price values of the options as output.

The formula of the Black Scholes Merton Model estimates only European call options, mainly for equity (exercising on expiry date only). It integrates factors, including price volatility of the underlying stock, the relation in present price and the exercise price of the option, expected interest rates, expiry time of the option and expected dividends.