Stock is more efficient than options are more efficient than stock

November 18, 2009 in Finance,Risk

There is a very interesting debate taking place on the profitability of options as opposed to the underlying stock. It originates in this post from Ultimi Barbarorum on options volume following the Palm/3Com announcement, and continues in the comments on Felix Salmon's coverage of that post.

The crux of the argument is the spike in options volatility immediately preceding the merger announcement, which many took as a clear sign of insider trading.  Baruch argues that options are notoriously volatile, so one spike is hardly a smoking gun, and I agree (see also: superstitions regarding trading on option expiration days). However, I echo Felix in noting that it's very hard, therefore, to draw any conclusion about insider trading whatsoever. Baruch's second point is that if it were insider trading, it was misguided - the insiders could have made more money and attracted far less attention by trading the underlying stock rather than the options. This is where the debate lies - and I confess up front that my immediate impulse was to say "that can't be right." In fact, it could be right, depending on your point of view.

(p.s. hats off to Baruch for introducing his post with "Before the Zero Hedge folks get the pitchforks out, let’s stop and think a bit.")

Baruch writes:

Had someone concrete knowledge of the 3Com deal, it would be far more efficient to buy the stock. The most important of the “Greeks”, as options dudes call the panoply of statistics surrounding options, is “delta”, the rate of change in the value of the option relative to the value of the shares (it’s a function of volatility, time to expiry, a whole lot of stuff, don’t trouble your head), and this is always less than one. 3Com options buyers made far less money on the takeover by buying options than they would if they had bought the stock.

It will be instructive here to discuss delta - I know some of TGR's readers are already familiar with concept, and I hope you will excuse this detour.

"Delta" is a mathematical (as opposed to financial!) derivative of the option formula, as described by Baruch - but it is easier to understand as a "hedge ratio." It tells you how many shares of stock you need to hedge your exposure to an option (I'm going to assume from here on that we are discussing calls). To see why, run back to the math for one second and consider that delta is the amount that the price of the option will rise if the stock price goes up by $1 - ok, now ignore the math. If the option is way in the money and trades at its intrinsic value, then it will gain $1 for every $1 the stock rises - a delta of 1. It the option is at the money, then its as likely  about whether it will ultimately pay off at all, and so it only gains $0.50 for every $1 the stock rises - a delta of .5. Thus, if you want to hedge your option exposure, you would short [delta] shares for every option you hold. Delta is always less than 1; no option will gain more than $1 for every $1 the stock price rises.

The key to this delta business is that as long as delta is less than 1, you need to short fewer shares than the amount you control via options in order to hedge your option exposure. Put another way, it takes more options than shares to create the same exposure (on a per-share basis) to the underlying stock. If the stock price moves up, the dollar gain from holding that stock will be greater than the dollar gain from the options, hence the argument that stocks are "far more efficient" than options.

The closing price for COMS on November 10th, the day before the option purchasing frenzy, was 5.41. The $5 November calls cost slightly more than their intrinsic value at $0.55, trading with a delta of 0.72.  On November 12th, the stock closed at $7.46, representing a gain of $2.05, whereas the options finished at $2.50, gaining just $1.95. Share for share, the stock outperformed.

However, shares controlled is an poor metric for comparing investments. This is particularly true for options, where you may not know until the day they expire if you actually control those shares or not! Instead, for risk management purposes we think of the number of shares the position is likely to control, given the current state of the world: the probability-weighted number of shares. Unsurprisingly, it's the same as the number of shares it takes to delta-hedge the position. From this observation, a nice property of delta is revealed: it may be roughly interpreted as the probability of an option finishing in the money.

The important philosophical point here is not to make the mistake of thinking that the number of options you buy is equal to the number of shares you own - that's only true the day they mature in the money. To set up the same exposure in options as we have with shares, at the time of purchase, we need to buy a few extra options. Specifically, for the November calls with a delta of .72, we need 1/.72 or 1.39 options for every share. Run the numbers and you'll see that this results in a final profit of $2.71 on the option side, vs $2.05 for the shares. If an insider bought options on a delta-adjusted share basis, he'd find the options more profitable than the stock.

(If you constantly adjust the number of options to correspond to the prevailing delta, you'll wind up making $2.05 on your options - this process is called dynamic delta-hedging [that's a real aside for this post, because the discontinuity in COMS stock price would make the rebalancing futile].)

So, on the basis of shares-at-maturity, stock yielded a better dollar profit. On the basis of shares-at-trade, options would have been preferable. There's an argument to be made that, as an insider, you know the options will finish in the money, so shares-at-maturity is the right way to consider it. But there's a third exposure metric: capital at risk.

You can look at capital at risk as either 1) the maximum loss you could experience OR (if you're an insider who knows the trade will be profitable) as the opportunity cost of capital. This is very straightforward to explain: those options only cost $0.55; the stock cost $5.41. The percentage gain on the options is 355%; for the stock it's just 38%. If you consider your exposure in terms of dollars invested, rather than shares controlled, you'd find the options a far better bet: they cost almost 90% less than the stock but return nearly as much per contract! So for every dollar you could put into the stock, you could instead put into options and return 10x as much. Options, from this perspective, are far more effective.

So this all depends on how you look at your risk and exposure. Baruch assumes that his insiders want to control a certain number of shares, and from that perspective they should absolutely have transacted stock instead of options (assuming that, with their perfect knowledge, they skip over the delta-adjust share argument). Personally, I would look at it from a capital at risk perspective - if I'm willing to spend $5.41/share to make $2.05, why not put that to work in options and make $19.18?

It all depends on your perspective - both answers could be correct, given some set of portfolio constraints and different definitions of risk/exposure.

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