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Minyan Mailbag: Black Scholes Valuation



Editor's Note: Our goal in Minyanville is to remove intimidation from the financial markets and encourage an interactive dialogue among the Minyanship. We share this next discussion with that very intent.

Prof. Succo,

This is about pricing, not the merits or lack thereof, of an individual security.

Options are not my regular fare. I was wondering if you could give me some really squishy guidance, (not advice), on how some warrants might be priced.

I own Yamana AUYWS warrants traded on the pink sheets (very thin, it has taken a lot of work to accumulate them using limits). The AUY common is at about $3.65 and the warrants are at approximately $2.50. The warrants are exchangeable between now and July 2008 with 1 warrant plus $1.50 Canadian for 1 common share. So, $2.50 & $1.20 US = $3.70.

The way I am playing it is as a fly-or-die junior mining company. I would rather have all the upside (the warrants trade almost right at intrinsic value) with less ultimate capital at risk on the downside. Any gains would be magnified with the smaller capital investment. The company is proposing to force conversion with a "sweetened" offer of 1.0356 shares per warrant & $1.20.

They say that they are offering more than a Black Scholes valuation implies. That doesn't seem right to me. With a 2008 expiration, where is the time premium? I am very unhappy about a forced conversion, but the question is...

What should a ballpark Black Scholes valuation look like?

Thanks in advance
Minyan Jeff


First of all we need to establish an appropriate volatility to the stock. Based only on past volatility (I do not know the stock at all, and normally I need to be familiar with the stock and the situation to "tweek" my estimate), I would assign a 53% implied volatility (I would sell 60% and buy 40%). The vega on the warrant you describe is $.002; if you disagree and think the stock is worth a 63% volatility, you add $.002 x 10 = $.02 more for the option than the value I describe below.

Assuming you won't get a rebate on the stock you borrowed to short the stock to hedge (which is reasonable and the way to look at the ''value" of the option even if you don't short the stock to hedge), BS would give you about $2.46 in value for a this warrant.

But this is lower than intrinsic value. By giving you 1.0356 shares per warrant instead of 1 share, this would bring the warrants value to: 3.65 * 1.0356 - 1.2 = $2.58. The way you describe the structure above is a little unclear; I treat the $1.20 US as the strike. As you can see, there is basically no time value in this warrant (deep in the money) at this volatility (you need to use a much higher implied volatility, giving the option a higher probability of going below the strike) in order to get any time premium. Also, if for some reason you can get a rebate on short stock there may be time premium and therefore they may not be an exercise.

The bottom line is since BS gives you a value below intrinsic they should be an exercise and have no "option" value.

Prof. Succo

Prof. Succo,

I am quite surprised that BS can come in below intrinsic value on a very long-term warrant. I am very grateful for your time and explanation. To me, it appears that Black Scholes is just one more arena where inefficiencies "can" be sought and found.

Have a great day!
Minyan Jeff


The BS is a very inadequate model for explaining option values in general. For example, it assumes a normal distribution for stocks, all stocks, when we know that is not true. It explains nothing about the discrepancies in "tail" risk": the propensity for certain stocks to move in great degrees more than others. Various adjustments can be made to help, thus some claiming "proprietary" models, but none of these models can incorporate the subjectivity inherent in stock price movements.

The idea of the BS actually came from an engineer's attempt to describe the flow of water, the same who came up with the natural log.

The major variable in the BS giving only intrinsic value to your warrant is the lack of rebate on the short stock. If you are not short against these warrants, you may decide not to exercise. Subjectivity does come into play.

Prof. Succo

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