Derivatives 101: 6

That's enough detours, at least for now. Let's grab some bulls by the horns and drag them into a china shop.

We've already dealt, at least superficially, with forwards. These puppies are nice and simple, and here's our original example:

In other words, you and I agree to exchange some known things at some known point in the future. How much is such an agreement worth? Which is to say, if either of us can sell our part of the deal to someone else, how much will we get?

Forwards are over-the-counter contracts, which means they are defined by the parties involved and the assets are not standardised. As a result, there will always be degenerate cases that are impossible to value -- a forward on the Mona Lisa, for example, or on the assassination of the President of the United States -- but these things are not derivatives in any meaningful way; they are singularities. Since neither the contracts themselves nor the things they refer to are actively traded, they are in the most literal sense priceless. Well, not the President, perhaps.

In practice, real forwards are almost always on underlyings that are reasonably fungible and reasonably liquid, and those that aren't probably aren't worth thinking about too hard anyway.

Typically, at least one side of the bargain will be some amount of cash, so it's convenient to use that as our metric of value. More complex cases can be handled as combinations of simple ones, as we'll discuss in a bit.

The thing to notice about a forward is that it is a fixed contract, fully specified from the get go, cut and dried, chiselled in stone. Its terms do not vary with time, only the values of the underlying assets do. From this point of view, there is no uncertainty: you know exactly what you're getting, for how much, when. That makes it a lot easier to plan things, and may well be considered beneficial even if in the end you wind up taking a notional loss.

This kind of arrangement is so commonplace it's a dead cert you've been a party to it yourself in one form or another: think of magazine subscriptions or railway season tickets or all you can eat for £10 buffets. In all cases, the seller offers a potential discount in return for a guaranteed income; while the buyer accepts the risk that she may not wind up using every penny of value in return for the comfort of a known outlay and reasonable certainty of getting the goods. In the retail context you always pay up front rather than at delivery, but other than that it's the same thing.

This makes forwards pretty much unique among derivatives, in that they aren't evil. A forward is an arrangement for mutual benefit rather than an exercise in attempted schadenfreude. Not at all coincidentally, even though forwards are entered into all the time they are traded much less frequently; on the whole, the participants in a forward want to make it to delivery.

None of which gets anywhere near answering the question posed a few paragraphs back. Despite the fact that forwards themselves are rarely traded, their valuation is the basis for pricing all sorts of less ethical instruments that change hands all the fucking time.

Here's how it works.

Recall from previous episodes the following:

1. Time reduces the value of things, especially cash, because of its accompanying uncertainty
2. You can always expect to be able to lend or borrow cash risk free at some minimum interest rate r
3. Arbitrage is impossible (or, at least, possible only in short-lived aberrant situations not worth considering)

(These axioms aren't exactly independent -- the second is in some senses a variation on the first -- but they're worth considering discretely.)

Although we didn't really do the maths before, there's a simple formula for calculating the present value of future cash:

P₀ = P_te^-rt

That is, the present value is the future value discounted at the risk free rate; if we invest P₀ now at interest rate r, at time t we'll have P_t.

Once again, don't worry too much about the specifics. What matters is that owning some amount of cash at some point in the future is equivalent to owning some smaller amount now, and either amount is calculable given the other. In the case of the cash side of a forward, we know exactly how much we're committed to pay in the future, and we can work out how much that equals right now.

For the asset side of the forward, we know only what the contract specifies, ie the amount of the asset to be delivered at t. Assuming the asset is reasonably liquid, we can most likely determine what that amount would cost now -- call it A₀ -- but we can only speculate about its value at delivery; which is precisely a forward's locus of uncertainty. Still, uncertainty only goes so far.

We can imagine that ownership of any asset carries certain benefits and costs. For example, owning a pig involves feeding and watering the little swine, while its growth from piglet to sow may bring some increase in worth as, if nothing else, sheer poundage of pork.

If we combine these costs and benefits into a single, necessarily approximate, net yield, then we can treat it as an interest rate -- call it y -- for (in this case) pigs; or, in the general case, for any asset. From this, we can calculate present value in exactly the same way as for cash:

A₀ = A_te^-yt

Or, conversely:

A_t = A₀e^yt

Admittedly, all we've done here is to move the uncertainty from one place -- the future value of the pigs -- to another -- their net yield. In the case of pigs, this can only be a guess, although chances are that if you rear a lot of pigs you probably have at least some idea of what goes into them. Quite a lot of forwards are on somewhat more predictable assets, and for these the yield need not be so approximate; eg, for currency forwards it is the difference between the risk-free rates of the two currencies.

Pace some serious handwaving, we now have the value of everything in the contract in terms of something vaguely knowable, and we're almost done.

The terms of a forward are defined once and once only, when it is initiated; they remain fixed and unchanging thereafter. We can assume that, at the moment of its creation, both parties are acting in their own best interests and therefore negotiate the best possible deal. Consequently, by the no arbitrage argument, the expected future value of each side should be the same:

P_t = A_t = A₀e^yt

In other words, when agreeing the price to be paid at delivery, both sides will converge on the best available estimate of what the asset will actually be worth at that time. (If they don't do this, there will be arbitrage opportunities, the specifics of which we may or may not describe next ep.) This estimate is, of course, almost certainly wrong, in the sense that by the time of delivery the spot price of the asset probably won't be exactly what they agreed all those months before, but that's not the point. It's whatever they can reach consensus on given (we assume, on average, give or take and with the usual disclaimers) the same information.

The terms of a forward are defined once and once only; the value of those terms will most likely change with the passage of time. In particular, the values of A₀ (or rather A_x, 0 < x < t) and y may vary, and with them any estimate of the net value of the forward. Whoever the imbalance favours may be able to realise a profit at that point by selling their interest; and good luck to them.

As previously noted, however, participants in a forward don't usually sign up to exploit market uncertainties; most often they just want their pigs. People who want to gamble go for futures instead.

Futures are famous. They've been in movies. Everyone and his dog Spot thinks he knows what a future is, and most of them are wrong.

A future is a forward's evil twin; a forward that's been institutionalised. Despite being essentially the same in many ways, futures have three distinguishing features that make them unexpectedly different:

1. Futures come in standard sizes with standard terms
2. Futures can be readily closed out
3. Futures are marked to market

What the fuck do those things mean?

They mean, basically, that futures are the advance scouts of an immense darkness. Unlike forwards, but like most other derivatives, they are, whether by accident or design, a sleazy, exploitative game. A numbers racket. A crap shoot. And, despite that, so much less interesting than Nathan Detroit's oldest established permanent floating etc.

Oh boy. The next installment is going to be so much fun...
Posted by matt at March 17, 2006 11:24 PM