Description

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 2003. The risk-free interest rate (on an actual/365 basis) is 7%. What is the value of this option as at 1 April 2002?

Function Specification

=oBS(1, "1/4/02", "1/4/03", 50, 60, 0.25, 0.07, 0)

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

The following Greeks are computed using the formulas specified in oBS() Model Greeks:

Delta

As iPC = 1 (call), the delta equation simplifies to:

Gamma

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

Theta

As iPC = T = 1, the equation for Theta becomes:

Vega

As T = 1, the equation for Vega becomes:

Rho

As iPC = T = 1, the equation for Rho becomes: