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4.3.3 Black-Scholes formula $ \diamondsuit $

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In the case of plain vanilla call and put options, the price can be evaluated in terms of the cumulative normal distribution $ N(x)$ and yields the well known Black-Scholes formula

$\displaystyle V_\mathrm{call}(S,t)=SN(d_1)-K\exp[-r(T-t)]N(d_2)$ (4.3.3#eq.1)

$\displaystyle d_{1,2}=\frac{\ln(S/K)+(r-D\pm\sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}$ (4.3.3#eq.2)

$\displaystyle V_\mathrm{put}(S,t)-V_\mathrm{call}(S,t)+S=K\exp[-r(T-t)]$ (4.3.3#eq.3)


Remember that $ S$ denotes the (spot) price of an underlying share that pays a dividend $ D$ and has a historical volatility $ \sigma$ , $ K$ is the strike price of the option evolving in time $ t\in[0;T]$ from the present to the expiry date and $ r$ the risk-free interest (spot) rate. Note that the last relation (4.3.3#eq.3) is nothing more than the put-call parity previously obtained in (2.1.3#eq.2), where the guaranteed payoff has been discounted back in time to achieve the risk free return of the spot rate. The cumulative normal distribution is related with the so-called error function $ N(x)=\frac{1}{2}(1+\mathrm{erf}(\frac{x}{\sqrt{2}}))$ , which is available in Matlab and can be approximated with 6 digits accuracy using the polynomial expansion [1]
$\displaystyle N(x)$ $\displaystyle \approx$ \begin{displaymath}\left\{ \begin{array}{lr} \displaystyle 1-N^\prime(x)\left(a_... ...ge 0\\ 1-N(-x) &\mathrm{when}\quad x < 0\\ \end{array}\right.\end{displaymath}  
   $\displaystyle \mathrm{where}$ $\displaystyle N^\prime(x)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^2}{2}\right) \qquad \mathrm{and}\qquad k=\frac{1}{1+\gamma x}$ (4.3.3#eq.4)


with the coefficients g=0.2316419, a1=0.319381530, a2=-0.356563782, a3=1.781477937, a4=-1.821255978, a5=1.330274429.

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