Black Scholes Model Explanation and Formula
Black Scholes Model – An option pricing model that determines the value of a call or put option on the basis of volatility, type of option, underlying stock price, time, strike price, and risk-free rate.
The Black Scholes model has two important features. First, the parameters of model except share price volatility are contained in the agreement that takes place between buyer and seller. Secondly, even having impractical assumptions, the model is able to determine the value of options quite well.
Black Scholes Model Assumptions
- The value of underlying share and the risk free rate remains constant throughout the life of the option.
- There are no transaction cost or taxes.
- Market is efficient.
- The rates of return on a share are log normally distributed.
Black Scholes Model Formula
C0 = S0 N(d1) – Ee^-rft N(d2)
C0 = the current value of the option
S0 = current market value of the share
E = exercise power
e = 2.7183, the exponential constant
rf = the risk free rate of interest
t = the time to expiration
N(d1) = cumulative normal probability function
d1 = ln(S0/E) + [rf+σ^2/2]t/σ½
d2 = d1 – σ½
Here, ln = natural logarithm, σ = standard deviation and σ² = variance of the continuously compounded annual return on the share.
Value of put option (P0) = C0 – S0 + Ee^-rft
This model also takes into consideration Hedge ratio which is also known as option’s delta. Hedge ratio is a tool that enables the investor to summarize the overall exposure of portfolios of options with various exercise prices and maturity periods. A call option has positive hedge ratio and put option has negative hedge ratio.