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oGK_IX( ) Example - Equity Call Option

Description

Consider a European option to buy USD / sell CND. The current exchange rate is 1.46 CND per USD and the annual rate volatility is 13.5%. The 'domestic' risk-free rate in Canada is 6% while the 'foreign' risk-free rate in the USA is 6.5% (both expressed on an actual/365 basis). The option matures on 1 January 2003, and has a market value of $0.0200 CND per USD. What is the implied strike rate as at 1 June 2002?

 

 

Function Specification

=oGK_IX(1, 0.02, "1/6/02", "1/1/03", 1.46, 0.135, 0.06, 0.065)

 

 

Solution

This option is treated as a call on the USD. As there is no closed form solution for implied strike prices, the Newton-Raphson iteration procedure is used to solve for X.

 

 

When calculating implied strike prices, the Newton-Raphson iteration procedure uses the spot rate as the initial estimate of the strike price, i.e., x0 = 1.46. The procedure will iterate using more and more precise estimates of the spot price until the difference between the option value derived from the spot price estimate and the given market option value is less than the desired accuracy level (see Newton-Raphson). In this example the desired accuracy level is 10 decimal places.

 

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

Equation Template
Equation Template

Referring to the equations for d1 and d2 (see model definition), if S = X = 1.46, vol = 0.135, r = 0.0583, rf = 0.063, and T =0.5863 (214/365 days), d1 = 0.0250 and d2 = -0.07838.

 

 

As iPC = 1 (call), N(d1) is 0.5010 and N(d2) is 0.4688 (see oCumNorm( ) function), the oGK( ) equation gives the following solution:

 

Equation Template

Since $0.05616 is above the market value of the option, $0.0200, the strike rate of $1.46 is too low. The oGK( ) value is therefore computed at a higher strike price, i.e., x1 > x0. Referring to the Newton-Raphson iteration procedure, x1 is determined as:

Equation Template

 

Using the same parameter values as above with a new strike rate estimate of 1.5398, the oGK( ) equation returns $0.02769. As this value is above the market value of the option the next strike trial is:

Equation Template

 

This process continues until the convergence criteria is met, which for this example occurs on the 6th iteration at a strike exchange rate of of 1.5725.

Equation Template

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