Description 
Consider a stock that has 1 million outstanding shares, a current spot price of $25, a volatility of 30%, and pays a dividend of $2.00 on 1 December 2002. There are 100,000 warrants on issue each entitling the holder to 10 shares in the stock. The warrants have a strike of $35 and mature on 1 June 2003, while the riskfree interest rate is 6% (on an actual/365 basis). What is the value of each warrant as at 1 August 2002? 




Function Specification 
=oBSw("1/8/02", "1/6/03", 25, 35, 0.3, 0.06, 1000000, 100000, 10, A1:B1, 0) It is assumed the cell references for the dividend schedule contain the appropriate input values. 




Solution 
When calculating warrant values, the iteration procedure uses zero as the initial estimate of the warrant price. The procedure will iterate using more and more precise estimates of the warrant price until the outputted warrant value is within 15 decimal places of the inputted warrant value (see model definition). 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 S = 25, X = 35, r = 0.0583, vol = 0.3, t_{i} = 0.4986 (182/365 days) and T = 0.8329 (304/365 days), d_{1} = 0.9148 and d_{2} 1.1886.



As N(d_{1}) is 0.1801 and N(d_{2}) is 0.1173 (see oCumNorm( ) function), the Black Scholes warrant equation becomes: 




Since the outputted warrant value ($1.2086) is above the inputted warrant price ($0.00), the valuation is rerun with W = $1.5411. This gives a warrant price of 1.6296.



This process continues until the convergence criteria is met, which for this example occurs on the 14th iteration at a warrant price of $1.6351. 










Greeks 
The following Greeks are computed using a discrete approximation of the partial derivative (see oBSw() Model Greeks): 

Delta 
0.6243511 

Gamma 
0.1858957 

Theta 
4.7362376 

Vega 
22.17412 

Rho 
10.61864 



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