# Statistics - Black-Scholes model

The Black Scholes model is a mathematical model to check price variation over time of financial instruments such as stocks which can be used to compute the price of a European call option. This model assumes that the price of assets which are heavily traded follows a geometric Brownian motion having a constant drift and volatility. In case of stock option, Black Scholes model incorporates the constant price variation of the underlying stock, the time value of money, strike price of the option and its time to expiry.

The Black Scholes Model was developed in 1973 by Fisher Black, Robert Merton and Myron Scholes and is still widely used in euporian financial markets. It provides one of the best way to determine fair prices of options.

## Inputs

The Black Scholes model requires five inputs.

• Strike price of an option

• Current stock price

• Time to expiry

• Risk-free rate

• Volatility

## Assumptions

The Black Scholes model assumes following points.

• Stock prices follow a lognormal distribution.

• Asset prices cannot be negative.

• No transaction cost or tax.

• Risk-free interest rate is constant for all maturities.

• Short selling of securities with use of proceeds is permitted.

• No riskless arbitrage opportunity present.

## Formula

${ C = SN(d_1) - Ke^{-rT}Nd_2 \\[7pt] \, P = Ke^{-rT}N(-d_2) - SN(-d_1) \\[7pt] \, where \\[7pt] \, d_1 = \frac{1}{{\sigma \sqrt T}} [ln(\frac{S}{K}) + (r + \frac{\sigma^2}{2}T)] \\[7pt] \, d_2 = d_1 - \sigma \sqrt T }$

Where −

• ${C}$ = Value of Call Option.

• ${P}$ = Value of Put Option.

• ${S}$ = Stock Price.

• ${K}$ = Strike Price.

• ${r}$ = Risk free interest rate.

• ${T}$ = Time to maturity.

• ${\sigma}$ = Annualized volatility.

## Limitations

The Black Scholes model have following limitations.

• Only applicable to European options as American options could be exercised before their expiry.

• Constant dividend and constant risk free rates may not be relistic.

• Volatility may fluctuate with the level of supply and demand of option thus being constant may not be true.