 # The Black Scholes Model The Black Scholes model is a price disparity formula used to determine European call options price. Additionally, it is referred to as the Black-Scholes-Merton model. This model offers investors an alternative to investing in securities which helps them earn a risk-free interest rate.
With this formula, one can use to acknowledge that an option cost is a function of stock-price volatility. The increased volatility equals a higher premium value on an option. It is essential to recognize that Black-Scholes sees call options as forward contracts that deliver securities at contractual prices. This is also referred to as strike price.

## Background

Fischer and Myron created the Black-Scholes formula, but it was further improved by R. Merton. Keep in mind that Fischer designed the formula over a long period of time. He first developed a valuation formula for asset warrants and formulated this principle after.

Scholes joined Fischer during his studies. The new team discovered a pricing equation that is still used today. While both researchers take credit for creating this principle, A. James Boness went on to improve it to the formula we know today. Boness included it in his Ph.D. dissertation while studying at the University of Chicago.

## How Black Scholes Model Works

The Black-Scholes equation is an excellent formula to calculate fair prices for assets. This formula has five inputs are that are required. These include:

1. Current asset price
2. Time-to-expiration

iii. Risk-free rate

1. Volatility
2. Strike price

This equation assumes that all asset prices are positive since they will never be negative. It also assumes that transaction costs or taxes are not incurred. This also applies to risk-free interest rates, which must remain constant for all maturities.

It’s important to note that it isn’t acceptable to use proceeds when short selling assets. This also means that riskless arbitrage opportunities are unavailable.

To calculate the formula, multiply an asset’s cost by a cumulative-standard normal probability distribution function, which can be named as X1. Next, multiply the net present value of a strike cost by a cumulative-standard normal distribution, represented as X2. Subtract X2 from X1 to get the final value.

Here is the mathematical notation:

C=S*N (d1)-KE^ (-r*T)*N (d2)

Calculate the value of put-option as follows:

P=Ke^ (-r*T)*N (-d2)-S*N (-d1)

Representation:

S = asset price

K = the strike price

r = the risk-free interest rate

T = time-to-maturity

Assumptions of the Model

1. Dividends are not paid out from stocks
2. Prediction of market movements is impossible
3. Being European and are exercised at expiration
4. Risk-free rate and volatility are constant
5. No transaction costs are incurred during the purchase of an option

## Model Limitations

1. Asset prices are continuous
2. Asset volatility can only be estimated but not observed
3. High dividend securities are mispriced
4. It overvalues deep out-of-the-money calls and undervalues in-the-money calls

## Wrapping it all up:

Thanks to financial experts, we have more ways of processing limitations. One method to do this is called Autoregressive Conditional Heteroskedasticity (ARCH). This equation can be used to replace constant volatility with schostatic volatility. And even despite this, traders and investors still prefer the classic Black-Scholes equation above other methods. 