Description

Consider a European call option on a stock that has a current spot price of $100, a volatility of 30% and pays no dividends. The option has a strike price of $100 and matures on 1 September 2003. The risk-free interest rate (on an actual/365 basis) is 6%. What is the value of this option as at 1 September 2002?

Function Specification

=oGBS(1, "1/9/02", "1/9/03", 100, 100, 0.3, 0.06, 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 S = 100, X = 100, b = r = 0.0583 (see special cases), vol = 0.30, and T = 1 (365/365 days), d₁ = 0.3442 and d₂ = 0.0442.

As iPC = 1 (call), N(d₁) is 0.6347 and N(d₂) is 0.5176 (see oCumNorm( ) function), the Generalized Black Scholes equation becomes:

Greeks

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

Delta

As the underlying asset is a non-dividend paying stock, delta is equal to N(d₁) or 0.5699. This can be confirmed by referring to the Delta equation, where the first part of the equation, equals 1.0.

Gamma

As b = r and T = 1, the Gamma equation simplifies to:

The n(d₁) equation gives 0.3760, and since S = 100 and vol= 0.30, Gamma is 0.0125.

Theta

As b = r and 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:

Phi

As b = r = and iPC = 1, the equation for Phi becomes: