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oBS_IS( ) Example - Equity Put Option

Description

Consider a European put option on a stock which has a volatility of 40%. The option matures on 1 March 2003, has an exercise rate of $70.00, and a market value of $4.55. The risk-free interest rate (on an actual/365 basis) is 7%. What is the implied spot of the stock as at 1 March 2002?

 

Function Specification

=oBS_IS(2, 4.55, "1/3/02", "1/3/03", 70, 0.4, 0.07)

 

 

Solution

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

 

 

When calculating implied spot prices, the Newton-Raphson iteration procedure uses the strike rate as the initial estimate of the spot price, i.e., x0 = $70. 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

Referring to the equations for d1 and d2 (see model definition),if S = X = 70, vol = 0.4, r = 0.0677, T =1 (365/356 days), and using an initial spot price of 70, d1 = 0.3691 and d2 = -0.0309.

 

 

As iPC = -1 (put), N(d1) is 0.3560 and N(d2) is 0.5123 (see oCumNorm( ) function), the oBS( ) equation gives the following solution:

 

Equation Template

 

 

Since $8.5948 is above the market value of the option, $4.55, the estimated spot price of $70 is too low. The oBS( ) value is therefore computed at a higher spot 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 spot estimate of 81.3614, d1 = 0.7452 and d2 = 0.3452. As iPC = -1 (put), N(d1) is 0.2281 and N(d2) is 0.3650 (see oCumNorm( ) function), the oBS( ) equation gives the following solution:

 

Equation Template

 

Since $5.3202 is above the market value of the option, $4.55, the estimated spot price of $81.3614 is too high. The next spot trial value is:

Equation Template

 

This process continues until the convergence criteria is met, which for this example occurs on the 6th iteration at an implied spot price of $85.00.

Equation Template

 

 

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