Black–Scholes formula

Submitted by admin on Mon, 01/03/2011 - 23:22

The Black Scholes formula calculates the price of European put and call options. It can be obtained by solving the Black–Scholes partial differential equation.

The value of a call option for a non-dividend paying underlying stock in terms of the Black–Scholes parameters is:

$C(S,t)=N(d_{1})~S-N(d_{2})~K e^{-r(T-t)}\,$

$d_{1}=\frac{\ln(\frac{S}{K})+(r+\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}$

$d_{2}=\frac{\ln(\frac{S}{K})+(r-\frac{\sigma^{2}}{2})(T-t)}{\sigma\sqrt{T-t}}$

The price of a corresponding put option based on put-call parity is:

$\begin{array}[b]{rcl}<br /> P(S,t) &= &Ke^{-r(T-t)}-S+C(S,t)\\<br /> &= &(1-N(d_{2}))~K e^{-r(T-t)}-(1-N(d_{1}))~S\\<br /> \end{array}.\,$

For both, as above:

$N(\cdot)$ is the cumulative distribution function of the standard normal distribution
$T - t$ is the time to maturity
$S$ is the spot price of the underlying asset
$K(\cdot)$ is the strike price
$r$ is the risk free rate (annual rate, expressed in terms of continuous compounding)
$σ$ is the volatility of returns of the underlying asset

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Black–Scholes formula