Installment Warrant Valuation using Gaussian Quadrature Methods
March 8, 2011
Here is a very interesting and a possibly a very attractive retail product. Letís say thatDelhi Bombay bank issues installment put warrant on Hang Seng Index (HSI). The warrants have the following structure: the investor pays $3.00 on the purchase date. One year later, the investor may choose to investment another $3.00 and receive an American put option on the Hang Seng Index, making the total cost of the warrant as $6.00. Alternative, he may choose not to pay the $3.00 and let the option expire worthless.
How do we value this installment warrant (compound option)?
There are quite a few numerical routines to evaluate this product. For example, if the underlying option and the compound option are both of European style then Rubenstein has suggested a Newton Raphson method to solve a non-linear equation. However, if the compound option becomes American in style then such a method does not yield a robust result.
Gaussian Quadrature (GQ) methods can be used to estimate the value of the above American style compound option in a robust way. The value of the above option would be given by:
In the above, and are strike prices and is the underlying put option and is the normal density function.
Gaussian Quadrature methods are a robust way of estimating numerical integrals. Essentially, we approximate the integrable function by the zeros of the orthogonal polynomials which are integration points (the abscissas) on the plane and , the weighting function related to the orthogonal polynomials.
There are many kinds of Gaussian Quadrature (GQ) algorithms. Here are three important Gaussian quadrature algorithms.
- Gauss-Kronrod local quadrature method (an adaptive routine of Q1DA by Kahaner, Moler and Nash is based on Gauss-Konrod method and is suggested by IzzyNelken to value an installment warrant like the one mentioned above).
- Gauss-Legendre quadrature method (we explain and use this method to solve valuation problems in our CFE classes);
- Gauss-Laguerre quadrature method.
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