Description 
Consider a European put option on a stock that has a volatility of 35% and pays a dividend of $4.50 on 1 December 2002. The option has a strike price of $80, matures on 1 February 2003, and a current market value of $2.41. The zero curve is flat at 6% (on an actual/365 basis). What is the implied spot at 1 August 2002? 


Function Specification 
=oBSdd_IS(2, 2.41, "1/8/02", "1/2/03", 80, 0.35, D4:E4, D5:E5) It is assumed the cell references for the zero curve and dividend schedule contain the appropriate input values. 


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


When calculating implied spot prices, the NewtonRaphson iteration procedure uses the strike rate as the initial estimate of the spot price, i.e., x_{0} = $80. The procedure will iterate using more and more precise estimates of the spot price until the difference between the option value derived from the spot 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 9 decimal places. Note, the spot price that is referred to in this example is the spot price of the underlying instrument less the present value of any discrete dividend paid during the life of the option. The present value of the dividend in this example is $4.4133.
The continuous equivalent of each rate in the flat zero curve is calculated as follows: Referring to the equations for d_{1} and d_{2} (see model definition), if S* = X = 80, vol = 0.35, r = 0.0583 and T =0.5041 (184/356 days), d_{1} = 0.2425 and d_{2} = 0.0060.


As iPC = 1 (put), N(d_{1}) is 0.4042 and N(d_{2}) is 0.5024 (see oCumNorm( ) function), the oBSdd( ) equation gives the following solution: 



Since $6.6924 is above the market value of the option, $2.41, the spot price of $80 is too low. The oBSdd( ) value is therefore computed at a higher spot 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 spot price estimate of 90.5945, the oBSdd( ) equation returns $3.3966. As this value is above the market value of the option the next spot price trial is:


This process continues until the convergence criteria is met, which for this example occurs on the 7th iteration at an implied spot price of $95.5867: S, or the spot price with present value of the dividend included, would be $100.00. 


Copyright 2013 Hedgebook Ltd.