Sunday, August 8, 2010

More on Stock-based Compensation

It was only a few flicks of the calender away, it was as recently as 2004, that one could say that generally accepted accounting principles (GAAP) in the United States didn't require employers to recognize in the books they showed the investing public that by issuing stock options to their employees they were incurring an expense.

The move of the FASB that year toward standards explicitly requiring this was caught up in the election-year debates. The FASB held its ground, although if I remember correctly it had backed down under political pressure on the same score a decade before. But the second time around it held its ground and made the expensing of stock options mandatory for all annual and interim reports, effective beginning June 15, 2005.

Over the last few years the nature of the debate has shifted. Now that we know companies are supposed to expense their stock options, the question is: how?

There are three dominant methods thus far. There's the Black-Scholes-Merton (MSN)formula, the Monte Carlo os 'simulation' methods, and the lattice model.

The lattice model is the most intriguing of the three in my eyes so I'll expound upon that a bit.

The modelers start by dividing the time between the issuance of an option and its expiration into a definite number of discrete time periods. These periods might be, for example, months or quarters.

Given input variables and assumptions that the company and its modelers must makle explicit, they can then assign a probability to an "up" and to a "down" move in each period. This yields two (or more, depending on how the particular lattice is constructed) end points, which are then starting points (nodes) for the next jump.

The model continues to generate nodes in an ever-widening pattern, looking a bit like a branching tree turned on its side (because the horizontal axis by convention represents the passage of time) until time runs out; that is, until option expiration.

Modelers then employ the whole lattice to decide what would be the value of an option in the final time segment. It helps that the ending is in a sense binary. Either the exercise price of the option is less than the current stock price or it is not, and that simplicity makes calculation straightforward. From there, one can work backward, step by step, until one comes eventually to a value for the option at time 0.

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