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The Social Function of Call and Put Options

December 13, 2006

Tags Financial MarketsFree MarketsEntrepreneurship

In two previous articles we examined the "social function" of stock speculation and futures contracts. Rounding out the trilogy, today we shall discuss the beneficial effects of a free market[1] in call and put options.

Despite their intimidating complexity, these derivative contracts are just another example of the financial market's growing ability to shield itself from unnecessary risk. As such, options allow producers and consumers to lengthen the time horizons of their plans and permit a better utilization of resources.

Call and Put Options

A call option gives the holder the right, but not the obligation, to purchase the underlying asset at a specific price (called the strike price) on a specific date, called the expiration date. So-called European options restrict the purchase privilege to the expiration date alone, whereas American options grant their holders the right to purchase the underlying asset at any date prior to and including the expiration. In contrast to calls, put options allow the holder to sell the underlying asset at the strike price, either on the expiration date (for European puts) or at any time during the life of the option (for American puts).

Other things equal, call options are more valuable when the spot price of the underlying asset rises, while puts appreciate when the spot price of the underlying asset falls. On the expiration date, if the spot price exceeds the strike price, then a call option is worth the difference between the two, while a put option is worthless. In contrast, if the spot price is below the strike, then a put option is worth the difference and a call is worthless.

For example, suppose Jones owns a call option on Microsoft stock, with an expiration date of July 1, 2007 and a strike price of $80. When July 1 rolls around, if the actual spot price of Microsoft stock is $80 or less, then Jones's call option is worthless. Jones isn't out anything (except what he initially paid for the call); after all, it's an option and so doesn't force him to take any unprofitable action. However, if Microsoft's share price has risen to $95 on July 1, then Jones's call is worth $15. This is because the call gives him the right to buy a share of Microsoft at $80, which he can then turn around and sell on the open market for $95.

Things are exactly reversed for Green, who (let us suppose) owns a put option with the same expiration date of July 1 but a strike price of (say) $20. On July 1, if the actual price of Microsoft is $20 or higher, then Green's put expires worthless. But if Microsoft stock has fallen to, say, $18, then Green's put is worth $2, because he can buy a share of Microsoft in the open market for $18, and then exercise his put to force the other party (i.e., the writer of the put) to buy it from him at $20. (For more details, as well as a very helpful graphical depiction, see this entry on options.)

Textbook Treatment

Unlike forward and futures contracts, where the initial market value of the contracts is zero, the writer of a call or put option receives an upfront payment from the buyer. This is quite intuitive, for the option buyer can only benefit (after the purchase) from its ownership; at worst it will expire without being exercised. Consequently the buyer is willing to pay a premium upfront for this one-sided arrangement.

Although the market value of an option is elementary to determine at the time of exercise, its price beforehand is far more complicated. The famous Black-Scholes formula gives the price that an option must satisfy in order to eliminate arbitrage opportunities. However, as with so many elegant results in mathematical finance (and economics), the Black-Scholes formula relies on unrealistic assumptions.

Even so, the Austrian economist who wishes to truly understand options has no choice but to delve into the conventional literature, if only to understand what everyone else is talking about. Fortunately, for those who are not intimidated by mathematics, the results can often be quite surprising and compelling.

My favorite example — both for its initially counterintuitive result but also the relatively mild assumptions — is that the theoretical price of an American and a European call option on a share of stock are identical (if the stock pays no dividends). This is shocking at first; it means that a call option that grants the ability to exercise only on July 1, 2010 has the exact same value as an otherwise identical option that gives the ability to exercise today, tomorrow, and every other day up to and including July 1, 2010. The novice could be forgiven for assuming that the latter option would command a far higher price.

The textbook approach for this result is to demonstrate that it would never be optimal to exercise an American call option before the expiration date; hence, it must have the same market value as an otherwise identical European call. Early exercise of an American call option is faulty because there exists a strategy that will either make as much, and possibly more, money.

To take a concrete example, suppose Jones owns an American call option on Microsoft stock, with a strike price of $20. Today it is Monday, and the option expires in five days on Friday. Further suppose that Microsoft is currently selling at $35, so that the call is "in the money." Now Jones could exercise immediately for a sure $15 gain. Some readers might think that Jones should indeed do so, especially if he thinks Microsoft stock will likely come down in the next few days.

Such reasoning is dead wrong. Rather than exercising his option early, Jones can short sell a share of Microsoft and hold this position (as well as his call) until Friday. Now if the price has risen by Friday, Jones still nets $15 overall. It's true that he loses a buck for every dollar that Microsoft stock rises (since he shorted one share), but this is exactly offset by the higher value of his call option (when he exercises it on Friday). For example, if the price on Friday is $38, then Jones uses his call option to buy a share at $20, then returns it to the original owner and now can claim the $35 (its price when he shorted the stock) held by the broker. Hence, Jones still nets $15, just as he would have done by exercising on Monday.

On the other hand, the price of Microsoft might fall. In this case the value of the call drops, but again this is offset by the enhanced value of the short position. However, if the price of Microsoft by Friday has fallen below the strike price (so that the option is "out of the money"), then Jones will have made more money by our strategy than if he had exercised on Monday. This is because the lowest the call option can go is zero, when Microsoft shares hit the strike price of $20. For every dollar below that, Jones's short position still goes up in value, but the call has stopped falling.

For example, if on Friday Microsoft is selling at $8, then Jones's call is worthless — why would he use his call option to buy Microsoft at $20 when he can buy it in the open market at $8? — but his short position is worth $27 ($35 - $8). This $27 is more than the $15 Jones would have made by exercising on Monday. (The $12 gain reflects the fact that Microsoft has fallen $12 below the strike price.)

When we extend the time involved, we must worry about interest rates and the time value of money. However, this extra complication only strengthens the result (since the present discounted value of the strike price falls as we push the expiration date further into the future). The conclusion remains that for a stock that pays no dividends, it is suboptimal to exercise an American call option early, because it would always be better to short the stock and wait until the expiration date to exercise the call.[2] Hence, an American and a European call must have the same market value during their lifetimes, since in practice the two will always be exercised on the last day possible.

How Do Options Promote Efficiency?

Now that we have a basic understanding of how they work, the question remains: What good are options? As we shall see, options allow people to tailor their financial exposure to various events in a much more sophisticated way than simpler derivatives such as a forward or futures contract.

For example, if a particular speculator is very bearish on the prospects for the US economy next year, he might sell futures on the S&P 500 index. (Note that this would tend to push down the current spot prices of these major stocks, which is just what we want to happen.) However, in reality it's far too crude to ask someone, "Are you bearish or bullish on the market?" In response to such questions at his weekly meetings with colleagues, Nicholas Taleb would often say something like, "I think the market will probably go up this week, but I also think there is a small chance that it will drop sharply."

Depending on the specifics, someone with this type of prediction might not disagree with the prevailing futures price, and thus might behave in exactly the same fashion as someone who is very confident that the market will not deviate more than 1% in either direction. But this simply shows the limitations of a futures market! Our two hypothetical speculators have very different views of the future condition of the stock market, and yet neither wants to "transmit" this information to the rest of the world through financial activity.

This all changes with the introduction of call and put options. Suppose that currently the S&P 500 index is trading at 1400. As before, the speculator who thinks the market is very likely to stay where it is won't have any reason to trade. However, suppose a speculator believes (if we had to assign numbers) that there is a 90% chance that the index will end up between 500 and 1500, a 9% chance that it will collapse to 500 or lower, and a 1% chance that the index will rise above 1500. He could act on these expectations by (say) selling European call options on the index with a strike price of 1500 and using part of the proceeds to buy European put options with a strike price of 500.

Of course the desirability of this strategy would depend on the specific prices of these different options, but if (by hypothesis) this speculator's expectations differ from those of most others, then the calls that he sells would be far more expensive than the puts that he buys. This would allow him to net a large gain upfront, which he could lend out at interest to partially offset his losses if the index should rise above 1500.

Now consider the speculator's position: he believes there is a 90% chance that his long puts and short calls will all expire out of the money, allowing him to retain the initial net gain (due to the higher purchase price of the calls versus the puts). He believes there is a 1% chance that his puts will expire worthless and that he will lose $1 per call for every dollar that the index rises above 1500; however, he has set aside the initial gain to help compensate should this happen. Finally, he believes that there is a 9% chance that the calls will expire worthless and he will be able to exercise his puts, earning $1 on each for every dollar that the index falls below 500; plus, in this scenario he would also retain his initial net gain.

Again, it would depend on the specific numbers and other factors, but clearly the speculator is more likely to notice a "bargain" with a host of calls and puts with various strike prices, rather than simply being able to buy or sell a futures contract on the S&P 500 index. Consequently, the options market allows the unorthodox wisdom possessed by mavericks to more easily be implemented into action and speeds the equilibration process if market prices really are out of whack.

Two Oil Examples

We can see the enhanced flexibility of options (versus futures) if we adjust the oil examples of the previous article. Suppose a speculator believes that the rest of the market is currently being far too pessimistic about future supplies of oil. Now one avenue would be to sell futures contracts on crude oil for delivery in, say,  eighteen months (if that's the time frame in which the speculator's forecast differs from the consensus).

There's a danger with this approach, unfortunately. Once the speculator enters into the (initially) costless transaction, he could lose an indefinite amount of money depending on how far crude prices rise by the delivery date of the futures contract. Therefore, depending on his level of risk aversion, he might not sell many crude oil futures even if he thought the futures price on oil were far too high.

A much safer approach for this speculator is to buy put options (at an appropriate strike price) on oil[3] with an expiration date in eighteen months. Here, the speculator's investment is laid down right at the beginning, when he buys the options. After that, no matter what happens to the price of crude, the worst that can possibly occur is that his options expire worthless. With options at his disposal, this speculator will be willing to enter the market with his minority viewpoint (and hence influence the price of crude) whereas the riskier futures market may not have enticed him into action.

We close with another oil example. It is admittedly contrived, but I hope that it conveys the real effects of a mature options market. Suppose the owner of large oil deposits is trying to decide how much to invest in his extraction equipment. To make his decision, he needs to forecast the average price of oil in the upcoming years.

If the price is between $0 and $20 per barrel, he won't make any new investments, but will simply let his current equipment run its course and then shut down once operating costs per barrel exceed the spot price. If the price is between $20 and $60, he will invest enough to maintain his current pumping capacity and will (naturally) pump at a higher rate depending on how high the price ends up, within this range. However, if the price should jump beyond $60 per barrel (and the oil man could see this coming), he would invest more heavily in order to expand his capacity.

Finally suppose that the numbers are such that, averaged across all different outcomes, the oil man would definitely be willing to invest in the more efficient, large-scale operation. However, the oil man's future will not be an "average" of the various scenarios; the price of oil will be some number (or in some small range), and he will either make a killing (if it's high) or lose his shirt (if it's low). Because the oil man — like most people — is risk averse, meaning that he would prefer a sure $1 million rather than a 50/50 chance at $2 million, he might go with the smaller investment.

It's true, the introduction of a futures market could alter his decision. If he locks in a guaranteed future price of oil, that might be enough to induce him to invest in the more efficient, large scale operation. However — as we saw with the S&P index example above — in the real world things aren't always neat and proportional. By selling oil on the futures market, the tycoon is committed to the target futures price; every dollar that oil goes above that hurts him, and every dollar below it helps him (as far as the value of the futures contract goes).

But this may not be enough to offset the "discontinuities" in his costs of production for various ranges of oil prices. For example, it's possible that his field's market value (if only he could know how much capacity to invest in beforehand!) would go up by 300% if the price of crude were $100 rather than $50. The simple futures contract wouldn't reflect this sudden jump in the payoffs, and so might not really allow the oil tycoon to plan as if he didn't care what happened to the actual spot price of crude.

If the tycoon can invest in options, though, he can far more precisely reconfigure his exposure to crude prices, offsetting the jagged cost structure he faces. For example, the tycoon could massively sell call options on oil with strike prices ranging from $60 to $100. With the money raised, he could invest in the larger capacity equipment. Then, if crude prices remain moderate, he isn't ruined; no one will exercise the calls and he has gotten his high-volume output for free, as it were.

On the other hand, if crude prices go through the roof, he is perfectly happy. This is because (we assumed) the value of his operation jumps up sharply once crude prices break a certain threshold, so long as he has beforehand invested in the superior equipment — which he will have done with the proceeds from the sale of the calls.

As this contrived example illustrates, an options market might allow for more efficient output decisions whereas a futures market (let alone a simple spot market) might not. Just as fire insurance allows homeowners to do the efficient thing by "averaging" their house's future over all possibilities, so too do options allow producers to mitigate the inefficiencies posed by uncertainty.

Conclusion

In theory, the most efficient decisions would occur if all producers and consumers could enter into very specific bilateral contracts, spelling out every possible future state of the universe, and what each party in the contract would do in each scenario. In the real world, the transactions costs involved render this Coasean point moot.

By providing a standardized yet flexible method of reconfiguring financial exposure, calls and puts (as well as more exotic options) allow people in real markets to approach this theoretical ideal more closely. The result is a more efficient use of society's resources to satisfy consumer desires.

Notes

[1] Many readers were troubled by my naïve endorsement of futures markets in the last article. They argued that the Federal Reserve's willingness to bail out companies that are "too big to fail" skews the incentives and renders the multi-trillion dollar derivatives market a giant bubble. I personally think these fears are a bit inflated (pun intended), and at worst would simply increase volatility, not the accuracy of a given forward or futures price. (After all, nothing in the critics' arguments suggested that the Fed would bail out only those holding short positions, so I was not convinced that the possibility of a bailout keeps spot commodity prices systematically lower than they ought to be.) In any event, I think we can safely say that given a fiat currency and paternalistic government, the addition of a futures market helps. To put it differently, if in addition to its other meddling the government were to outlaw derivatives trading, we would all be worse off in the long run.

[2] Two clarifications are in order. First, this argument does not apply to puts; American puts are worth more than their European counterparts. Second, the argument doesn't mean that shorting the stock and hanging on to the call is the best strategy available; all we have shown is that it weakly dominates the strategy of early exercise of the call. For example, in certain circumstances perhaps an investor wants to close out his position entirely. What we have shown is that it would be silly to exercise his "in the money" calls. If the investor really needs cash, he could sell his (unexercised) calls to another party who would then enjoy the "optionality" inherent in their remaining life.

[3] It is my understanding that calls and puts are often written on the relevant futures contract, rather than the ultimate commodity. Thus, the speculator would buy put options, not on barrels of crude, but rather on futures contracts on crude. The reason for this is that the futures contracts are standardized and very liquid.


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