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oBLACK_IX( ) Example - Put Option on a Forward Contract

Description

Consider a European put option on a forward contract with a current forward price of $105.00 and a volatility of 20%. The put option matures on 1 February 2003 and has a market value of $1.65. The risk-free interest rate (on an actual/365 basis) is 7%. What is the implied strike rate of this option as at 1 May 2002?

 

 

Function Specification

=oBLACK_IX(2, 1.65, "1/5/02", "1/2/03", 105, 0.2, 0.07)

 

 

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 forward rate as the initial estimate of the strike price, i.e., x0 = $105. The procedure will iterate using more and more precise estimates of the forward price until the difference between the option value derived from the forward 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 X = S = 105, vol = 0.2, r = 0.0677, and T =0.7562 (276/365 days), d1 = 0.0870 and d2 = -0.0870.

 

 

As iPC = -1 (put), N(d1) is 0.4654 and N(d2) is 0.5346 (see oCumNorm( ) function), the oBLACK( ) equation gives the following solution:

 

Equation Template

 

Since $6.9131 is above the market value of the option, $1.65, the strike rate of $105 is too high. The oBLACK( ) 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 94.6392, d1 = 0.6843 and d2 = 0.5104. As iPC = -1 (put), N(d1) is 0.2469 and N(d2) is 0.3049 (see oCumNorm( ) function), the oBLACK( ) equation gives the following solution:

 

Equation Template

 

Since $2.7847 is above the market value of the option, $1.65, the strike rate of 94.6392 is too high. 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 an implied strike price of $90.00.

Equation Template

 

 

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