# Black Scholes Model

**Definition:** The **Black-Scholes Model** is the options pricing model developed by Fischer Black, Myron Scholes, and Robert Merton, wherein the formula is used to calculate the theoretical price of the European call and put option based on five determinants: Stock price, strike price, volatility, expiration date and the risk-free interest rate.

The formula to calculate the price of the call options is:

**C = SN(d1) – N(d2)Ke-rt**

**d1= [ln (S/K) + (r+****∂**^{2}**/2)t]/ ****∂√t**

**d2= d1- ****∂√t **

**Where,
**C = Call Premium

S = Current Stock Price

t= time until expiration

K = Option strike price

r = Risk-free Rate

N = Normal distribution

e = exponential

∂ = Standard Deviation

ln = log

The formula given above is divided into two parts, the first part, i.e. **SN(d1)**, represents the expected benefits of purchasing the underlying asset. It shows the change in the call premium due to the change in the price of an underlying asset. The second part, Viz. **N(d2)Ke ^{-rt}** shows the current value of the exercise price that needs to be paid when the option is exercised. The difference between these helps to determine the option’s price.

Before the application of the above formula, the assumptions associated with the Black Scholes model must be understood clearly. **These assumptions are:**

- The options are only European, that can be exercised only at the date of expiration.
- No dividends are paid throughout the life of the options.
- There are no commissions.
- There is an efficient market, which means the market movements cannot be predicted.
- The risk-free rate and the volatility of change in the price of the underlying asset are known and constant.
- The returns on the underlying asset are normally distributed
**.**

The formula for calculating the option price is quite complex, but the investors need not have to go through the lengthy calculations and can use the Black-Scholes option calculator, which is easily available online.