4.3.3 Black-Scholes formula

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4.3.3 Black-Scholes formula $\diamondsuit$

The material in this section is intended for students at a more advanced level than your profile

In the case of plain vanilla call and put options, the price can be evaluated in terms of the cumulative normal distribution

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

denotes the (spot) price of an underlying share that pays a dividend

and has a historical volatility $\sigma$ ,

is the strike price of the option evolving in time $t\in[0;T]$ from the present to the expiry date and

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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