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oGK( ) Example 1 - Equity Call Option

Description

Consider a European option to buy GBP / sell USD. The current exchange rate is 1.56 USD per GBP, and the rate volatility is 12%. The 'domestic' risk-free rate in the USA is 6% while the 'foreign' risk-free rate in the UK is 8% (both expressed on an actual/365 basis). The option has a strike rate of 1.60 USD per GBP and matures on 1 May 2003. What is the value of the option as at 1 November 2002?

 

 

Function Specification

=oGK(1, "1/11/02", "1/5/03", 1.56, 1.60, 0.12, 0.06, 0.08, 0)

 

 

Solution

This option could be equivalently valued as either a call option or a put option. The following solution treats the option as a call on the GBP.

The continuous equivalent of the actual/365 risk-free interest rates are calculated as follows:

Equation Template
Equation Template

Referring to the equations for d1 and d2 (see model definition), if S = 1.56, X = 1.60, r = 0.0583, rf = 0.077, vol = 0.12, and T = 0.4959 (181/365 days), d1 = -0.3670 and d2 = -0.4516.

 

 

As iPC = 1 (call), N(d1) is 0.3568 and N(d2) is 0.3258 (see oCumNorm( ) function), the Garman Kohlhagen equation becomes:

 

Equation Template

 

Greeks

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

Delta

Given the above parameters, the delta equation becomes:

Equation Template

Gamma

As n(d1) = 0.3730, the Gamma equation becomes:

Equation Template

Theta

The equation for Theta becomes:

 

Equation Template

Vega

The equation for Vega becomes:

 

Equation Template

Rho

As r <> rf, the equation for Rho becomes:

 

Equation Template

Phi

The equation for Phi becomes:

 

Equation Template

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