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oGK_IS( ) Example - Equity Put Option

Description

Consider a European option to sell NZD / buy AUD. The current rate volatility is 18%. The 'domestic' risk-free rate in Australia is 7% while the 'foreign' risk-free rate in New Zealand is 6% (both expressed on an actual/365 basis). The option has a strike rate of 0.88 AUD per NZD and matures on 1 February 2003, and has a market value of $0.0150 AUD per NZD. What is the implied spot exchange rate as at 1 October 2002?

 

 

Function Specification

=oGK_IS(2, 0.015, "1/10/02", "1/2/03",.88, 0.15, 0.07, 0.06)

 

 

Solution

This option is treated as a put on the NZD. As there is no closed form solution for implied spot prices, the Newton-Raphson iteration procedure is used to solve for S.

 

When calculating implied spot rates, the Newton-Raphson iteration procedure uses the strike rate as the initial estimate of the spot exchange rate, i.e., x0 = 0.88. The procedure will iterate using more and more precise estimates of the spot rate until the difference between the option value derived from the spot rate 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 11 decimal places.

 

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 = X = 0.88, vol 0.15, r = 0.0677, rf = 0.0583, and T =0.3370 (123/365 days), d1 = 0.0799 and d2 = -0.0072.

 

 

As iPC = -1 (put), N(d1) is 0.4682 and N(d2) is 0.5029 (see oCumNorm( ) function), the oGK( ) equation gives the following solution:

 

Equation Template

 

Since $0.028575 is above the market value of the option, $0.015, the exchange rate of 0.88 is too low. The oGK( ) value is therefore computed at a higher spot rate, 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 spot estimate of 0.91074, the oGK( ) equation returns $0.016787. As this value is above the market value of the option the next exchange rate trial is:

Equation Template

 

 

This process continues until the convergence criteria is met, which for this example occurs on the 9th iteration at an implied spot exchange rate of 0.91672.

Equation Template

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