Derivatives 101: 8

Although we've approached the discussion of derivatives from a variety of directions, we find ourselves, eight episodes in, having really only looked at a couple of solid examples; and, for all the verbiage expended, atypically simple examples at that.

Forwards and futures (along with instruments such as swaps, to which we've fleetingly alluded, which can be represented as combinations of the above) are sometimes known as linear derivatives, where the linearity is in the relationship between the underlying price and the payoff. If the underlying changes value, so does the derivative. The profit or loss made is directly dependent on the variations in the underlying price. Graphically, it's a straight line.

More or less.

The relationship does not have to be that straightforward, however, and frequently isn't. The most common -- and most significant -- way in which derivatives become non-linear is in the introduction of choice. Derivatives involving choice are, broadly speaking, called options.

Options are the platonic ideal of derivatives; the derivatives by which all other derivatives are judged. Although they come in a daunting range of shapes and sizes, each with its own obsequious entourage of funky little demons, the basic character of an option is simplicity itself: whereas with a future or a forward you agree to buy or sell something for a particular price at some future date, with an option you agree only to consider doing so.

This choice aspect is necessarily asymmetric, or else the option would mean nothing at all. If neither side is bound by the deal, there is no deal. An option that both parties have a choice in would only be exercised when its terms were exactly the current market terms, and who would bother to enter into a contract like that? It's a void. You'd just go to the market. Buy the things you want when you want them. Maybe stop awhile to watch a hanging, or a witch being tested on the ducking stool. Fun for all the family.

The substance of an option is that one party commits to something -- for a price -- that the other party may choose whether or not to take advantage of at a later date. The committing party is the seller (often known as the writer) of the option. The buyer is (will he or nil he?) a bit of a tease.

Of course this whole arrangement forms, upfront and brazenly, a gamble; and the gambler has to ante up. A future or forward might be a gentlemen's agreement -- we're all friends here -- but an option really isn't. It's every man for himself. The law of the jungle. A dog eat dog world. Hey! Let's see the colour of your money!

Unlike futures and forwards, options have an intrinsic value of their own because they confer power on their owner: the power to compel a transaction without being compelled oneself. The power to choose. An option will only be exercised if it is advantageous to do so -- if it enables the owner to purchase the underlying asset for less than the going rate, or sell it for more. Let's face it, when you can buy the same thing down the street for half the price you've opted, you ain't gonna opt.

Where a forward contract binds the parties together in common purpose -- for good or for ill -- an option makes them opponents, glowering at each other across the green baize, each fingering his revolver under the table. Eventually they'll have to show their cards and count their losses, but there's so much macho posturing to get through before that: the narrowed eyes, sweat-beaded brows, endlessly rearranging stacks of chips and, uh-oh, that nervous tap-tap-tap with the cigarette lighter...

A ten-spot says the guy on the right cracks first.

Meanwhile, the underlying asset -- you remember those squealing pigs? -- is going on its merry way, getting fatter, getting stronger, unexpectedly succumbing to an epidemic of Foot & Mouth. Demand goes up, supply goes down, whatever. Maybe there's a mad rush of people converting to Judaism or Islam and everyone forswears bacon. Or a new post-Atkins diet fad makes grilled pork chops de rigueur. Who knows? Who cares?

The underlying asset behaves as underlying assets do: unpredictably, vexatiously, stochastically. Ladies and gentlemen, place your bets.

Now, If you were really interested in the pigs -- for their meat, or their bristles, or their fine dinner table conversation -- you wouldn't be fucking about with options, you'd be mucking out the sty. By definition, if you're fucking about with options, you're pretty much only interested in the underlying as a random variable.

Which is not quite to say that options are only about rolling the dice, however much it sometimes seems that the people involved would be better off indulging their predilections at the nearest William Hill.

An option derives (oh-ho!) its randomness from its underlying pigs -- which is what makes it gambleable -- but there is at least one context in which its behaviour is not random at all, and that is with respect to those pigs. Assets and the options on them are intimate partners, bound to the same source of uncertainty in a morbid formal dance: promenade and do-si-do, take your partners, quick-quick-slow. Consequently, if you happen to be exposed to porcine uncertainties already, then gambling on them can make a lot of sense.

Let's say I'm a swineherd expecting to have a lot of surplus stock to shift in three months time. I might well want to buy the ("put") option now to sell my charges at that time, even if the price I'm buying to sell at, taking into account the cost of the option, is less than I estimate the market will be paying when the gavel falls. At least I'm guaranteed the sale; and if it turns out the gammon market is sizzling come September, well, I can just choose not to exercise my rights. This is, in every important way, a kind of insurance.

If, however, I'm a speculator who thinks the price of pigs is going to rise, I might be willing to fork out for a ("call") option to buy in September at a slightly-higher-than-now price, on the basis that I'll be able to sell them on the spot for even more. In other words, I can gamble on the price of pigs without either owning any or having any intention to do so for more than a femtosecond.

In either case, my exposure -- which is to say, what I'm putting on the line -- is only going to be the price of the option, which is typically much less than the price of the actual pigs.

Reader, leverage. Leverage, reader. Oh, you've met before?

Unlike futures, which, despite the comparative ease of closing out, are at least nominally binding on both parties, options are a straight bet against the house: you pay your money, wait, and either count your winnings or nurse your loss. As such, there has to be a house. Someone has to write those options. Which is to say: someone has to make book on the underlying.

Bookmaking, a profession as ancient and venerable as language itself, has a lot in common with writing options -- but no-one has yet won the Nobel Prize by providing Ladbrokes with a simple mathematical formula (or even a stochastic partial differential equation) for pricing a 7/1 bet on Horatio Nelson in the 4.20 at Epsom.
Posted by matt at June 29, 2006 10:31 PM