ResolutionPro is an analytical toolkit for Excel for derivatives valuation and risk management.
Includes;
Free demo version is available.
Current Articles | RSS Feed
Consider a European call option on a stock that has a current spot price of $50 and a volatility of 25%. The option has a strike price of $60 and matures on 1 April 2012. The risk-free interest rate (on an actual/365 basis) is 7%. What is the value of this option as at 1 April 2011?
The Black Scholes option pricing model is defined as follows:
OV = option value.
S = spot price of the underlying asset.
X = exercise price (strike).
r = risk-free interest rate, expressed with continuous compounding.
vol = volatility of the relative price change of the underlying asset.
T = time to maturity measured in years (actual/365 basis).
N(.) = cumulative normal distribution of (.).
iPC = 1 for call / -1 for put.
Solution
The continuous equivalent of the actual/365 risk-free interest rate is calculated as follows:
Referring to the equations for d1 and d2 (see model definition), if S = 50, X = 60, r= 0.0677, vol = 0.25, and T = 1 (365/365 days), d1 = -0.3334 and d2 -0.5837.
As iPC = 1 (call), N(d1) is 0.3693 and N(d2) is 0.2797 (see oCumNorm( ) function), the Black Scholes equation becomes:
Greeks
As iPC = 1 (call), the delta equation simplifies to:
As T = 1, the Gamma equation simplifies to:
The n(d1) equation gives 0.3773, and since S = 50 and vol = 0.25, Gamma is 0.0302
As iPC = T = 1, the equation for Theta becomes:
As T = 1, the equation for Vega becomes:
As iPC = T = 1, the equation for Rho becomes:
Allowed tags: <a> link, <b> bold, <i> italics
