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Black-Scholes model
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 volatipty. 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
Volatipty
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 selpng 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.
${sigma}$ = Annuapzed volatipty.
Limitations
The Black Scholes model have following pmitations.
Only apppcable to European options as American options could be exercised before their expiry.
Constant spanidend and constant risk free rates may not be repstic.
Volatipty may fluctuate with the level of supply and demand of option thus being constant may not be true.