Description

Consider a European put option on a stock that pays no dividends and has a current spot price of $100.00. The call option has a strike price of $90.00, matures on 1 March 2003 and has a market value of $2.30. The risk-free interest rate (on an actual/365 basis) is 6.0%. What is the implied volatility of this option as at 1 June 2002?

Function Specification

=oGBS_IV(2, 2.3, "1/6/02", "1/3/03", 100, 90, 0.06, 0.06)

Solution

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

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 and T parameter values):

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:

Referring to the equations for d₁ and d₂ (see model definition), if vol = 0.6311, S = 100, X = 90, b = r = 0.0583 (see special cases), and T =0.7479 (273/365 days), d₁ = 0.5458 and d₂ = 0.0000.

As iPC = -1 (put), N(d₁) is 0.2926 and N(d₂) is 0.5000 (see oCumNorm( ) function), the oGBS( ) equation gives the following solution:

Since $13.82 is above the market value of the option, $2.30, the volatility of 63.11% is too high. The oGBS( ) value is therefore computed at a lower volatility, i.e., x₁ < x₀. Referring to the Newton-Raphson iteration procedure, x₁ is determined as:

Using the same parameter values as above with a new volatility estimate of 24.36%, the oGBS( ) equation returns $2.7515. As this value is above the market value of the option the next volatility trial is:

This process continues until the convergence criteria is met, which for this example occurs on the 6th iteration at a volatility of 22.50%.