Saturday, 4 February 2012

The only option - chasing joyriders

Equity derivatives is a complex place.  Microsoft shares have options (calls and puts) on them across a whole range of maturities and strikes.  Like a pair of tables.  Think of excel. Say we'll create a separate excel spreadsheet to document all of the calls and puts you can have on Microsoft.  In that sheet we'll have two tabs, one for all the calls currently in play, one for all the puts in play.  In each tab, imagine expiry dates running across column-wise at the top, and strikes running down the tab.  Now that excel sheet gets replicated, one for every equity name out there which also has options on it.  There'll be an Apple spreadsheet, one for Wal-Mart too.  Now, the set of options currently in play for Microsoft are different from the set of options which were in play last year.  Or a decade ago.  Or which will be in play a decade from now.  Likewise for all the names.  That's quite some variety.  Perhaps we can impose some organisational structure on these sheets - let's imagine storing each Microsoft sheet for a given day in a different sub-folder on our computer.  Think of all those folders.

Now for something surprising.  This can all be replaced by a single excel sheet, with a single excel tab.  This is because the theory behind the option pricing model defines an option's value in terms of a couple of parameters of the underlying share - namely its price, expected volatility and dividend payout structure over the life of the option.  Stripped down like this, all shares look identical to the option pricing model.  It doesn't care which particular share it is a derivative on.  Only that there's always a price, expected volatility and dividend payment structure available to it for any moment it is asked for a valuation.  The option also needs to know the current values of a couple of low risk interest rates, and finally it needs to know four definitional constants - start up parameters, if you like - Is it a call or a put?  Can the current holder get out of the deal during the life of the option, or only on the final day?  Just when is that final day? And what's the line-in-the-sand strike price of the option, the price point around which it operates?

This single model can actually be implemented in a couple of ways, but they'll essentially give the same result.  And I'm deliberately ignoring the existence of so-called exotic options for now.  

What about history?  Microsoft is now trading at 30.24.  Way back in 2005, Apple was also trading at 30.24.  If they had the same dividend payment structure and expected volatility, and if interest rates were the same at those points in time too, then all the calls and puts you could imagine on these two names ought to be valued at the same levels.    It doesn't matter one bit to this insight that in reality interest rates are never exactly the same, or that volatility expectations are never exactly the same.  The insight still holds - there is only one option.  

The reason for this has to do with how the recipe for the option model is defined.  Imagine a car filled with joyriders barrelling down a quiet highway.  The driver's been drinking, he's playing loud music.  his drunk friends are urging him to go faster, his cute girlfriend urging him to go slower.  That is the model for the underlying stock price.  Now the option represents a second car whose job it is to follow the first car.  It accelerates and decelerates based on what the first car is actually doing.  The interest rate could be something like the gradient of the road.

The option car doesn't need a corporate valuation model - which in this case would be a guess as to which person the driver cares to listen to at any moment, or a prediction of what his right foot will do next - the option car simply tracks the car as best it can.  The car contents can be a black box to the option car.  Once the option car learns this trick, it can track any driver on any road anywhere, for any time. 

Even more dramatically, you could pre-calculate all these numbers in an excel sheet and store them down.  From that point on, the valuation of any stock's option would be a straight table lookup in excel.  People would never do this because the lookup table would be too large; running the recipe is a better time-space trade-off proposition.