Description 
Consider a European option to sell GBP / buy USD. The current exchange rate is 1.45 USD per GBP. The 'domestic' riskfree rate in the United States is 6% while the 'foreign' riskfree rate in Great Britain is 7% (both expressed on an actual/365 basis). The option has a strike rate of 1.40 USD per GBP, matures on 1 December 2002, and has a market value of $0.0310 USD per GBP. What is the implied volatility as at 1 April 2002? 


Function Specification 
=oGK_IV(2, 0.031, "1/4/02", "1/12/02", 1.45, 1.4, 0.06, 0.07) 


Solution 
This option is treated as a put on the GBP. This option is treated as a call on the USD. As there is no closed form solution for implied volatility, the NewtonRaphson iteration procedure is used to solve for volatility.


When calculating implied volatilities, the NewtonRaphson iteration procedure uses the Manaster and Koehler seed value as the initial estimate of the volatility. This is calculated as follows (see below for r and T parameter values): The procedure will iterate using more and more precise estimates of volatility until the difference between the option value derived from the volatility estimate and the given market option value is less than the desired accuracy level (see NewtonRaphson). In this example the desired accuracy level is 11 decimal places.
The continuous equivalent of the actual/365 riskfree interest rates are calculated as follows: Referring to the equations for d_{1} and d_{2} (see model definition), if vol = 0.4707, S = 1.45, X = 1.4, r = 0.0583, r_{f} = 0.0677, and T =0.6685 (244/365 days), d_{1} = 0.2673 and d_{2} = 0.1175.


As iPC = 1 (put), N(d_{1}) is 0.3946 and N(d_{2}) is 0.5468 (see oCumNorm( ) function), the oGK( ) equation gives the following solution: 



Since $0.18935 is above the market value of the option, $0.0310, the volatility of 47.07% is too high. The oGK( ) value is therefore computed at a lower volatility, i.e., x_{1} < x_{0}. Referring to the NewtonRaphson iteration procedure, x_{1} is determined as:


Using the same parameter values as above with a new volatility estimate of 10.76%, the oGK( ) equation returns $0.0308. As this value is below the market value of the option the next volatility trial is:


This process continues until the convergence criteria is met, which for this example occurs on the 4th iteration at a volatility of 10.8077%. 
Copyright 2013 Hedgebook Ltd.