oBLACK( ) Example 1 - Call Option on a Forward Contract

Description

Consider a European call option on a forward contract with a forward price (as of maturity) of \$115 and a volatility of 20%. The option has a strike price of \$120 and matures on 1 June 2003. The risk-free interest rate (on an actual/365 basis) is 6%. What is the value of this option as at 1 June 2002?

Function Specification

=oBLACK(1, "1/6/02", "1/6/03", 115, 120, 0.2, 0.06, 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 F = 115, X = 120, r = 0.0583, vol = 0.2, and T = 1 (365/365 days), d1 = -0.1128 and d2 = -0.3128.

As iPC = 1 (call), N(d1) is 0.4551 and N(d2) is 0.3772 (see oCumNorm( ) function), the Black equation becomes: Greeks

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

Delta

As the option is a call and T = 1 the Delta equation simplifies to Gamma

As T = 1 and the n(d1) equation gives 0.3964, the Gamma equation simplifies to: Theta

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

As b = r and T = 1, the equation for Vega becomes: Rho

As b <> 0 and iPC = T = 1, the equation for Rho becomes:  