Black Scholes formula

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In this below, we explain Black-Scholes formula.

Functions to load: Load Bl() and BS()
Usage: BS(K, T, S, vol, r, delta)
Arguments:
K: strike
T: maturity \§: spot price of underlying stock
vol: volatility
r: continuously compounding risk free interest rate
delta: dividend yield
Value: 2-D vector of call and put price.

BS model usually assumes the lognormal distribution for the stock price at time ${T}$ , that is,

$\displaystyle S_{T} \sim S_{0} e^{(r-\delta)T + Y},$

where

${r}$ is short rate, ${\sigma_{t}}$ is volatility, ${\delta}$ is dividend yield,
${Y \sim \mathcal{N}(-\frac 1 2 \nu^{2}, \nu^{2})}$ for ${\nu = (\int_{0}^{T} \sigma_{t}^{2} dt)^{1/2}.}$ In particular, ${\nu = \sigma \sqrt T}$ when ${\sigma_{t} = \sigma}$ is constant volatility.

R-code is written according to the following fact:

Proposition 1 Let ${C_{0}}$ and ${P_{0}}$ be the call and put premium with maturity ${T}$ and strike ${K}$ . Then,

$\displaystyle C_{0} = e^{-rT} \mathbb [((S_{T} - K))^{+}] = e^{-\delta T}Bl(Ke^{(\delta-r)T}, S_{0}, \nu, +1),$

and

$\displaystyle P_{0} = e^{-rT} \mathbb [((-(S_{T} - K))^{+}] = e^{-\delta T}Bl(Ke^{(\delta-r)T}, S_{0}, \nu, -1),$

Ex. Suppose stock spot price ${S = 10}$ , volatility ${\sigma = .4}$ , interest rate is ${r = .0427}$ , and no dividend ( ${\delta = 0}$ ). For strike ${K = 10}$ and maturity ${T = 10}$ years, we can compute call and put prices as follows.