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oGBS_IX( ) Example - Equity Call Option

Description

Consider a European call option on a stock that pays no dividends, has a current spot price of $100.00 and a volatility of 35%. The call option matures on 1 April 2003 and has a market value of $5.25. The risk-free interest rate (on an actual/365 basis) is 6.5%. What is the implied strike rate of this option at 1 September 2002?

 

 

Function Specification

=oGBS_IX(1, 5.25, "1/9/02", "1/4/03", 100, 0.35, 0.065, 0.065)

 

 

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 = $100. 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 8 decimal places.

 

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

Equation Template

Referring to the equations for d1 and d2 (see model definition), if S = X = 100, vol = 0.35, b = r = 0.0630 (see special cases), and T =0.5808 (212/365 days), d1 = 0.2705 and d2 = 0.0038.

 

 

As iPC = 1 (call), N(d1) is 0.6066 and N(d2) is 0.5015 (see oCumNorm( ) function), the oGBS( ) equation gives the following solution:

 

Equation Template

Since $12.3124 is above the market value of the option, $5.25, the strike rate of $100 is too low. The oGBS( ) value is therefore computed at a higher 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 114.673, the oGBS( ) equation returns $6.6877. 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 7th iteration at a strike rate of of 120.00.

Equation Template

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