Risk Latte - The “Forgotten Models” of Warrant Pricing

The "Forgotten Models" of Warrant Pricing

Rahul Bhattacharya
January 13, 2007


Very few, if anyone, in the world of derivatives would remember the name of Sheen Kassouf, Guynemer Giguere or John Shelton. However, at one point in time, long ago before we were completely overwhelmed and captivated by Black-Scholes type derivatives pricing models these individuals had come up with econometric models of pricing options and warrants.

Of course Black-Scholes' analysis and their theoretical construct for option pricing was the most appropriate and very powerful so as to correctly price options and option like derivatives. And that is why we live in a Black-Scholes world today. But given the enormous fascination of the financial market players and investors with options and derivatives these days and the simply humongous volume of trading in these instruments it may be worthwhile to pause for breath and take a peek into the not too distant past and see what was the derivatives world like before Fisher Black, Myron Scholes and Robert Merton came along.

The formulas and analysis given below may not have any relevance any more and are relegated to the pages of history. But sometimes, it does not hurt to play the historian.

Shelton's Model of Warrant Pricing

Given the grounding of present day Black-Scholes type models on theory of probability and stochastic calculus it may be hard for us to imagine that before Black-Scholes many academicians were thinking of options and warrants in terms of econometric models. One such person was John Shelton, who came up with an econometric model to evaluate warrants but could very easily be extended to options. Shelton''s model for warrants was mathematically expressed as:

The biggest drawback of such an econometric model was that it did not take into account the interest rates of an economy and the volatility of the stock.

Kassouf's Model for Warrant Pricing

In 1965 Prof. Sheen Kassouf developed an econometric model for option and warrant pricing as his doctoral dissertation. Kassouf's model has been used - yes, it may surprise many of us today - over the years by many practitioners to value warrants (more of that a bit later). The Kassouf's model for warrant pricing is given by:

The two big improvements of this model over Shelton's model was that it takes into account the volatility of the stock and the model could be fitted to any curve because the stock price can be varied in .

Actually, Kassouf's formula could be reduced to a very simple warrant pricing formula if certain adjustments are made (and under certain assumptions). A simple version of Kassouf's warrant pricing formula is given by:

Giguere's Model for Warrant Pricing

Guynemer Giguere produced a very simple model for warrant valuation where the warrant price had a parabolic variation with the stock price. Giguere's valuation formula was given by:

Giguere's formula was quite different from Shelton''s and Kassouf's formula and will produce quite different results for the warrant price.


Reference: A Survey of Options, their Valuation, and their Use by Victor Nemchenok, Department of OR and Industrial Engineering, Cornell University, New York; Security Analysis, 5 th ed., by Sidney Cottle, Roger F. Murray and Frank E. Block, McGraw Hill, Financial Analysts Journal, May-June and July-August, 1967.

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