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oBSdd_IV( ) Example 1 - Equity Call Option

Description

Consider a European call option on a stock that has a current spot price of $100 and pays a dividend of $2.00 on 1 August 2002. The call option has a strike price of $100, matures on 1 August 2002, and has a market value of $3.68. The risk-free interest rate is 6% (on an actual/365 basis). What is the implied spot at 1 June 2002?

 

 

Function Specification

=oBSdd_IV(1, 3.68, "1/6/02", "1/8/02",100, 100, 0.06, D5:E5)

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

 

 

Solution

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

 

 

When calculating implied volatilities, the Newton-Raphson 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, T, and S* parameter values):

Equation Template

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 Newton-Raphson). In this example the desired accuracy level is 11 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* = 100 - 1.98 = 98.02

Referring to the equations for d1 and d2 (see model definition), if vol = 0.3505, S = 98.02, X = 100, r = 0.0583 and T = 0.1671 (61/365 days), d1 = 0.0000 and d2 = -0.1433.

 

 

As iPC = 1 (call), N(d1) is 0.5000 and N(d2) is 0.4430 (see oCumNorm( ) function), the oBSdd( ) equation gives the following solution:

 

Equation Template

 

Since $5.1362 is above the market value of the option, $3.68, the volatility of 35.05% is too high. The oBSdd( ) value is therefore computed at a lower volatility, 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 volatility estimate of 25.942%, the oBSdd( ) equation returns $3.6804. As this value is slightly above the market value of the option the next volatility trial is:

Equation Template

 

 

This process continues until the convergence criteria is met, which for this example occurs on the 4th iteration at a volatility of 25.9404%.

Equation Template

 

 

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