Black Scholes formula

This function is available in R-fiddle (here). See also its upper level page (here).

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.

\displaystyle [C, P] = BS(K, T, S, \sigma, r, \delta) = [5.834, 2.358741].

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s