Black Scholes model is a price disparity formula used to determine European call options price. Also referred to as Black-Scholes-Merton model, it takes into account that investors have an alternative to investing in securities earning risk free interest rate.
Basically, this formula is used to acknowledge that an option cost is a function of stock-price volatility. Which means increased volatility equals to a higher premium value on an option. It is important to understand that Black-Scholes treats call options as forward contracts designed to deliver securities at contractual prices also referred to as strike price.
Black-Scholes was formulated by Fischer and Myron before being improved further by R. Merton What you need to know is that Fischer did not design this formula overnight. He began by developing a valuation formula for asset warrants before formulating this principle.
During his studies, Fischer was joined by Scholes. As a team, both went on to discover a pricing equation that is used even today. Although both researchers take credit for creating this principle, it was further improved by A. James Boness. Boness made it part of his Ph.D dissertation while studying at the University of Chicago.
How Black Scholes Model Works
The Black-Scholes equation is a great formula for calculating fair prices for assets. What you need to know is that five inputs are required. They include:
i. Current asset price
iii. Risk free rate
v. Strike price
The equation does assume that asset prices are positive since they can never be negative. Furthermore, it is assumed that no transaction costs or taxes are incurred. This is also true for risk free interest rate which must remain constant for all maturities.
When it comes to short selling of assets, use of proceeds is not accepted. Furthermore, no riskless arbitrage opportunities are available.
The formula is calculated by multiplying an asset’s cost by cumulative-standard normal probability distribution function which we can name as X1. This should be followed by multiplication of net present value of a strike cost by cumulative-standard normal distribution which can be represented by X2. Subtracting X2 from X1 gives you the final value.
The mathematical notation used is:
C=S*N (d1)-KE^ (-r*T)*N (d2)
The value of put-option can be calculated as follows:
P=Ke^ (-r*T)*N (-d2)-S*N (-d1)
Here is what they stand for:
S represents asset price
K represents strike price
r represents risk free interest rate
T represents time-to-maturity
Assumption of this Model
i. No dividends are paid out from stocks
ii. They are European and are exercised at expiration
iii. Prediction of market movements is impossible
iv. No transaction costs are incurred during purchase of an option
v. Risk free rate and volatility are constant
Limitations Of This Model
i. Asset prices are continuous
ii. Asset volatility can be estimated and not observed
iii. High dividend securities are mispriced
iv. This equation overvalues deep out-of-the-money calls and undervalues in-the-money calls
Financial experts have found more ways of dealing with limitations. One method is called Autoregressive Conditional Heteroskedasticity (ARCH). This equation is used to replace constant volatility with schostatic volatility. Despite this, traders and investors still prefer classic Black-Scholes equation.