Description 
Consider a European call option on a forward contract that has a volatility of 20%. The call option has a strike price of $100.00, matures on 1 December 2002, and has a market value of $1.22. The riskfree interest rate (on an actual/365 basis) is 6.0%. What is the implied forward price of the underlying asset as at 1 April 2002? 


Function Specification 
=oBLACK_IF(1, 1.22, "1/4/02", "1/12/02", 100, 0.2, 0.06) 


Solution 
As there is no closed form solution for implied forward prices, the NewtonRaphson iteration procedure is used to solve for F.


When calculating implied forward prices, the NewtonRaphson iteration procedure uses the strike rate as the initial estimate of the forward price, i.e., x_{0} = $100. The procedure will iterate using more and more precise estimates of the forward price until the difference between the option value derived from the forward price estimate and the given market option value is less than the desired accuracy level (see NewtonRaphson). In this example the desired accuracy level is eight 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 = 0.2, F = X = 100, r = 0.0583, and T =0.6685 (244/365 days), d_{1} = 0.0818 and d_{2} = 0.0818.


As iPC = 1 (call), N(d_{1}) is 0.5326 and N(d_{2}) is 0.4674 (see oCumNorm( ) function), the oBLACK( ) equation gives the following solution: 



Since $6.2674 is above the market value of the option, $1.22, the estimated forward price of $100 is too high. The oBLACK( ) value is therefore computed at a lower forward price, 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 forward price estimate of 89.9541, d_{1} = 0.5657 and d_{2} = 0.7292. As iPC = 1 (call), N(d_{1}) is 0.2858 and N(d_{2}) is 0.2329 (see oCumNorm( ) function), the oBLACK( ) equation gives the following solution: 



Since $2.3232 is above the market value of the option, $1.22, the estimated forward price of $89.9541 is too high. The next forward trial value is:


This process continues until the convergence criteria is met, which for this example occurs on the 11th iteration at an implied forward price of $85.0018. 


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