Description 
Consider a European call option on a stock that has a current spot price of $85. The call option has a strike price of $85, matures on 1 March 2003 and has a market value of $15.75. The riskfree interest rate (on an actual/365 basis) is 6.0%. What is the implied volatility of this option as at 1 March 2002? 


Function Specification 
=oBS_IV(1, 15.75, "1/3/02", "1/3/03", 85, 85, 0.06) 


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 = 0.3414, S = X = 85, r = 0.0583, and T = 1 (365/365 days), d_{1} = 0.3414 and d_{2} = 0.0000.


As iPC = 1 (call), N(d_{1}) is 0.6336 and N(d_{2}) is 0.5000 (see oCumNorm( ) function), the oBS( ) equation gives the following solution: 



Since $13.7608 is below the market value of the option, $15.75, the volatility of 34.14% is too low. The oBS( ) value is therefore computed at a higher 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 40.356%, the oBS( ) equation returns $15.7489. As this value is slightly 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 40.3593%. 


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