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oGBS( ) Example 1 - Equity Call Option

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:

Equation Template

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:

 

Equation Template

 

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, Equation Template equals 1.0.

Gamma

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

Equation Template

 

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:

 

Equation Template

Vega

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

 

Equation Template

Rho

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

 

Equation Template

Phi

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

Equation Template

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