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oBSdd_IX( ) Example - Equity Call Option with Discrete Dividends

Description

Consider a European call option on a stock that has a current spot price of $50.00, a volatility of 20% and pays two dividends of $3.00 on 1 December 2002 and 1 June 2003. The call option matures on 1 September 2003 and has a market value of $11.40. The risk-free interest rate is 6% (on an actual/365 basis). What is the implied spot as at 1 September 2002?

 

 

Function Specification

=oBSdd_IX(1, 11.40, "1/9/02", "1/9/03", 50, 0.20, 0.06, D5:E5)

It is assumed that the cell references for the dividend schedule contain the appropriate input values.

 

 

Solution

As there is no closed form solution for implied strike prices, the Newton-Raphson iteration procedure is used to solve for X.

 

 

When calculating implied strike prices, the Newton-Raphson iteration procedure uses the spot rate as the initial estimate of the strike price, i.e., x0 = S* = $44.17 (see below). 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 Newton-Raphson). In this example, the desired accuracy level is 9 decimal places.

 

The continuous equivalent of the actual/365 risk-free interest rate is calculated as follows:

Equation Template

S*, the spot price less the present value of the future dividends paid during the life of the option, is calculated as follows:

S* = 50 - 5.83 = 44.17

Referring to the equations for d1 and d2 (see model definition), if S* = X = 44.17, vol = 0.20, r = 0.0583, and T =1 (365/365 days), d1 = 0.3913 and d2 = 0.1913.

 

 

As iPC = 1 (call), N(d1) is 0.6522 and N(d2) is 0.5759 (see oCumNorm( ) function), the oBSdd( ) equation gives the following solution:

 

Equation Template

Since $4.8126 is below the market value of the option, $11.40, the strike rate of $44.17 is too high. The oBSdd( ) value is therefore computed at a lower strike price, i.e., x1 < x0. Referring to the Newton-Raphson iteration procedure, x1 is determined as:

Equation Template

 

Using the same parameter values as above with a new strike rate estimate of 32.0458, the oBSdd( ) equation returns $14.0206. As this value is above the market value of the option the next strike trial is:Equation Template

 

This process continues until the convergence criteria is met, which for this example occurs on the 6th iteration at a strike rate of of 35.00.

Equation Template

 

 

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