Description 
Consider a European call option on a forward contract with a current forward price of $100.00. The call option has a strike price of $120.00, matures on 1 March 2003 and has a market value of $2.15. The riskfree interest rate (on an actual/365 basis) is 8.0%. What is the implied volatility of this option as at 1 July 2002? 


Function Specification 
=oBLACK_IV(1, 2.15, "1/7/02", "1/3/03", 100, 120, 0.08) 


Solution 
As there is no closed form solution for implied volatility, the NewtonRaphson iteration procedure is used to solve for vol.


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 rate is calculated as follows: Referring to the equations for d_{1} and d_{2} (see model definition), if vol = 0.6275, F = 100, X = 120, r = 0.077, and T =0.6658 (243/365 days), d_{1} = 0.1001 and d_{2} = 0.6121.


As iPC = 1 (call), N(d_{1}) is 0.4601 and N(d_{2}) is 0.2702 (see oCumNorm( ) function), the oBLACK( ) equation gives the following solution: 



Since $12.9073 is above the market value of the option, $2.15, the estimated volatility of 62.75% is too high. The oBLACK( ) 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 27.79%, d_{1} = 0.6906 and d_{2} = 0.9174. As iPC = 1 (call), N(d_{1}) is 0.2449 and N(d_{2}) is 0.1795 (see oCumNorm( ) function), the oBLACK( ) equation gives the following solution: 



Since $2.8066 is above the market value of the option, $2.15, the estimated volatility of 27.79% is too high. The next volatility trail is:


This process continues until the convergence criteria is met, which for this example occurs on the 5th iteration at a volatility of 25.00%. 


Copyright 2013 Hedgebook Ltd.