Thursday, August 06, 2009

Derivatives: an Example

Nobody asked me, but maybe I can use it in class someday. I take a shot at explaining why derivatives can cause so much trouble.

Here's the framework: we know that when prices go up, demand goes down, and when prices go down, demand goes up. It's basic economics.

Except that it isn't true. Charles Morris gives the example of Andy Warhol. As the price of a Warhol goes higher, we may find ever more buyers getting ever more frantic to get a piece of the action. So also with housing in, say, Las Vegas in, say, 2006--as the price goes up, demand goes up as well. And it works the other way. As the prices collapse, we find owner offering more property, not less--and provoking a panic, driving still more buyers to drive prices down still further. So also in, say, the stock market on a bad day.

Why do derivatives matter? Derivatives, as they say, don't kill people; people kill people, and derivatives are just the weapon.

True, but misleading. The fact is that some of the most popular and intelligible derivative techniques are designed to aggravate just such a problem as I set forth above.

Here we go. Consider BigCo. You can buy a share of BigCo ("S") today (t=0) for $100. We predict that next year (t=1), that the price of BigCo will move up by 25 percent, or down by 20 percent--so, to $125 or $80. You can also buy a call option ("C") that would allow you to buy a share of BigCo at t=1 for $105. Oh, and you can also borrow/lend money risk-free ("b") at (say) 10 percent.

We know the price of the stock. What should be the price of the call? To answer this question, we proceed from the known to the unknown. We construct an "artificial call"--a security using the same payoffs as a call--using the "known" stock and the "known" risk-free borrow/lend. How do we construct it? We set up a pair of equations:

125s+1.1b= $20
80s+1.1b = 0.

In words, here is a package of "S" and "b" that mimics the payoff on the option. So, how do we assemble this package? Find out by solving the equation. Subtract the second equation from the first and solve for S. S turns out to be -.4444= 44.44 percent. Translated: we buy 44.44 percent of a S (=$12.12). We borrow enough to cover the downside (=$32.32). We reach into our own pocket to cover the spread ($44.44-$32.32=$12.12). That pocket money: that is the only skin we've got in the game. That is what we are paying to buy a package that imitates a call. So that must be the price of the call.

So far, yawn. What does this have to do with the panic selling? Bear with me. That "percentage of a share"--for convenience, give it a name: call it "delta," Δ. What we have just said is that the price of a call should equal "delta stock" minus risk-free borrowing. Formally:

C= ΔS-b

But if this is true, then


This is huge. This is the pivot point. This means that you can can mimic a risk-free investment by buying stock, and writing (selling) calls.

This is alchemy. This is magic. This means you can transmute unsafe into safe by mixing stocks and calls.

But we are not done yet. We still haven't seen why this leads to panic selling. So: suppose you want an absolutely safe portfolio. You create it as above by mixing delta-stock and call. But now, suppose the stock price fall. Inorder to keep safe, you're going to have to remix. And this is the devilish part. When the stock falls and you want to restore safety, you sell stock.

This is just the opposite of what you learned in Econ 1A. In 1A you learned when stuff is cheap, youy buy. Here, when stuff is cheap you sell. You dump a bunch of stock on the market which makes it even more cheap--and which, also, scares the bejabbers out of all the people who do not play your silly game.

If you are truly keeping score at home, try repricing the example above on the assumption that the stock falls to 90, with the same percentages--i.e., next year it may move up to $112.50, or down to $72. You will find that the new delta is 19 percent. That is, you want only 19 percent stock in your portfolio instead of 44 percent.

Of course, this is only the beginning of the game. You can play it so many other ways. For example forget about calls, C. Suppose you want to construct a "put," P, a floor under your portfolio, a stop-loss, a kind of insurance. You can create an imitation put the same way you did the call--same equation. Only this time, you find that instead of simply selling the stock on a drop, you short the stock--you bet that it will fall even further. Anyway, the direction turns out to be just the same--as the stock falls, you are motivated to dump more.

And we are still in preschool. Recall that all over the world, their are 85-pound kids with square glasses, starting at computer screens all night long while the rest of us are asleep, trying to figure out newer and more exotic trades. The real wonder is that the world isn't in a bigger mess than it is.

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