Black-Scholes and Monte Carlo

Aside from the lattice method discussed in yesterday's entry, the two most important methods of valuing the expense of stock options as compensation are: the Black-Scholes-Merton (BSM) formula, and a Monte Carlo simulation.

BSM is here.

This amounts to valuing an option to buy a share of stock by making certain simplifying assumptions. The devisers of the formula were explicit about these assumptions. One of the more important of them is that extreme price changes are very rare, because the movements in the value of the underlying stock follow lognormal distribution. Many scholars have quarrelled with this, but BSM does still offer a useful approximation of the value of such options.

Still: the "Monte Carlo" method has by far the cooler name, paying homage as it does to the city whose image I've uploaded with this entry.

Through the miracle known as a digital computer, modelers can run thousands of simulation paths covering possible price movements in the underlying asset -- the stock - and can assign a value to the option for each of these paths. The present value of the average of all those values is the Monte Carlo value of the option.

Brokers and those routine votes

Last week I mentioned that the SEC rules on "routine" broker votes are once again in flux.

Individual investors usually hold their stock in brokerage accounts, registered in its "street name," i.e. the name of the broker. The proxy then, or its agent ADP Proxies, has often voted on routine matters. So: what counts as routine?

Until 2002, most increases the availability of stock options were considered routine, and thus subject to such broker-level rubber stamping. That stopped because of widespread complaints that stock options had become throughout the 1990s an instrument for the distribution of wealth away from shareholders, toward management and employees. Part of the fall-out of the post-dotcom and post-Enron re-appraissal of corporate governance was a change on this point: no more routine broker votes on stock options.

Uncontested directorial elections, though, have continued to be treated as routine. That is what is now up for change.

The first-line regulator on such matters is not the Securities and Exchange Commission but the New York Stock Exchange itself, though the NYSE must submit its rule changes to the SEC for approval. Accordingly, the NYSE has submitted to the SEC a rule change that would prohibit discretionary voting by brokers in uncontested director elections.

Experts at Wachtell Lipton have argued that "the proposed rule change could significantly increase the power of institutional shareholders generally and activist shareholders specifically in influencing director elections and corporate affairs."

The public comment period on this rule change expired Tuesday, March 28.

Here's a link to one of those public comments, what the Investment Company Institute (the trade group for mutual funds and such) has had to say.

The ICI isn't happy. It has concluded "that the proposal would have a disproportionate impact on investment companies and would create significant difficulties for investment companies in achieving quorums and electing ... directors."

For now, I'll leave the issue there, although of course I'll be very happy to hear reactions from readers of this humble blog.

Google's options

Google announced this week that it has re-priced 7.64 million stock options belonging to 15,642 non-executive employees.

Why? The stock's price (NASDAQ: GOOG) has taken a beating in the last year, not unlike everybody else's. It peaked at $594.90 in early May 2008, then tumbled to $259.56 on November 20. That represents a loss of about 56% percent of value, peak to trough.

There was some recovery (post-election optimism? Your call) so that on February 9, GOOG closed at $378.77. The optimism has worn thin, and the stock was back down to $308.57 when they made the announcement changing the options terms Tuesday.

A stock option has an "exercise price." The idea of giving options to employees of course is that it gives them the incentive to work to get the actual price well above the exercise price, so that they can garner the difference when they cash in.

But Google's employees have of late found themselves with options on $300 stock with an exercise price that was reasonable when they were at the peak -- $500 a share or higher. That far underwater, incentive effects are hard to imagine.

Look for such deals to become a contentious issue in the months to come, though, as stockholders worry about the dilution of the value of their stocks for the benefit of employees. Likely ticked-off shareholder argument: "Do we really need to worry about incentive effects in today's labor market? Will the top talent in Google's fields leave? If so, where will they go? who is hiring?"

BS v. BS II

So: what do I think about the "Black Swans versus Black-Scholes" dispute I tried to chronicle in yesterday's entry?

By way of answering, let me re-introduce you to Emanuel Derman. I mentioned him yesterday -- he was a co-author, with Taleb, of "The illusions of dynamic replication," the 2005 article that seems to have started the whole anti-Black-Scholes campaign.

Derman dropped out of the story then, because Taleb started shifting his line of attack and acquired new co-authors for the new approach.

But Derman, a one-time subatomic physicist, and the 2006 recipient of the Wilmott Award for Contributions to Quantitative Finance, has re-appeared. Now he shows up more as a defender of Black-Scholes than as a critic. See his New Years' Day blog comment on the Wilmott website.

So far as I understand these things: I think Derman is right. the Black-Scholes-Merton model was a real advance in the understanding of financial economics, one that has the benefit of being very clear about its simplifying assumptions. To oppose simplifying assumptions, after all, is to oppose a good deal more than this particular model for the pricing of stock options. It is to oppose a crucial step in the achievement of every major scientific advance on record.

And yes, I think with the anonymous recent poster at FT, that BSM has been on the whole a force for good in this crazy mixed-up world.

One intruguing sidebar to this controversy is the whole "teaching birds to fly" meme. Taleb and others on his side of the debate sometimes speak as if the idea of teaching birds to fly is inherently ridiculous, and if Black, Merton, and Scholes were trying to teach options traders to trade, describing methods the options traders already (instinctively?) possessed.

The linguist Noam Chomsky has invoked the same meme, as it happens, in his evaluation of primate-language research. Chomsky believes that langauge is a distinctively human attribute, that humans are hard-wired for it, and that the efforts to teach human language to other primates have failed in a predictable way, making this point. In an interview he gave in 1983 link, he said: "That's exactly what we should expect, I think. Why should we expect it? Because, if it turned out, contrary to what has so far been shown, if it turned out that apes really did have something like a capacity for human language, we would be faced with a kind of biological paradox. We would be faced with something analogous to, say, the discovery on a previously unexplored island that there is a species of bird with all the mechanisms for flight that has never thought of flying, until somebody comes along and trains it and says, look, you can fly. That's not impossible, but it's so unlikely that nobody would take the possibility very seriously."

Chomsky and Taleb, then, share the teaching-birds-to-fly meme.

But it seems to me, from either source, a bit presumptuous. There is nothing inherently absurd in teaching birds to fly. In fact, when I googled that phrase just now, I came up with this.

Birds are, it appears, taught how to fly, just as Elsa was taught how to be a free-ranging lion.

Cue the Born Free theme music please.

BS v. BS: Black Swans against Black-Scholes

On December 30, the Economist's website posted an anonymous column about the Black-Scholes-Merton options pricing formula. Access it here if you please.

The columnist observes that BSM has come under attack of late, and it has even been used as a prime example of why the Nobel Prize in Economics ought to be discontinued. Scholes and Merton received that august prize in 1997 for their work on this model in articles published in 1973 [Fischer Black had died in 1995 and the award is never given posthumously.]

The critics of the BSM model to whom the Economist alludes include Nassim Nicholas Taleb, the author of Fooled by Randomness (2001) and The Black Swan (2007), who attributes much of the recent financial dislocation to what he sees as the fallacious view of financial risk and risk management of which the BSM model is an important part.

In 2005, Taleb and co-author Emanuel Derman wrote "The illusions of dynamic replication," in the journal Quantitative Finance. This was a brief nerdy paper in quant jargon. In essence, "dynamic replication" refers to the replication of the value of a derivative by the continuous ("dynamic") trading of its underlyings.

Presumably, dynamic replication would render the derivative itself redundant, and this fact allows for the determination of its value by means of the no-arbitrage principle.

The Derman-Taleb paper made the case that dynamic replication is much too complicated—there are simpler ways to get the results. The most important of these simpler ways is static replication. Their idea was to drop the principle of continuous trading and construct a portfolio consisting of a long position in a call and a short position in a put. The traditionally discounted expected value of their payoffs must replicate a forward contract.

The Black-Scholes option pricing formula could have been discovered much more rapidly than it was, Taleb and Derman maintained, had this simpler route to that goal been adopted.

In 2006, the same journal published a responsive comment from two students of Robert Merton, Doriana Ruffino and Jonathan Treussard, who took issue with the Derman-Taleb reasoning. Ms. Ruffino and Mr. Treussard wrote in their paper that although it's conceivable the same result could have been reached by mathematicians making use of put-call parity without dynamic replication, it would have been a fluke, a "matter of pure chance."

In 2007, Taleb changed partners and somewhat changed his line of attack. He wrote a paper with Espen Haug entitled "Why We Have Never Used the Black-Scholes-Merton Options Pricing Formula." They proposed that the name "Black-Scholes" itself be relegated to the dustbins. They would call the options pricing formula Bachelier-Thorp, after the early stochastic-process theorist Louis Bachelier and blackjack-sharp hedge-fund pioneer Ed Thorp.

The Taleb-Haug paper attracted more attention than did its Derman-Taleb precursor. I wrote about the controversy myself at that point.

On December 7, 2008, the Financial Times ran an opinion column by Taleb and another co-author, Pablo Triana, using the intervening market chaos to reinforce their argument against contemporary financial risk management in general and the BSM model in particular.

"Ask for the Nobel prize in economics to be withdrawn from the authors of these theories," they urged their readers. "Boycott professional associations that give certificates in financial analysis that promoted these methods. The fraud can be displaced only by shaming people, by boycotting the orthodox financial economics establishment and the institutions that allowed this to happen."

So it was that the same Pablo Triana wrote an open letter to the Swedish central bank, which gives out the Nobel Prize in this field (it isn't one of the real Nobel Prizes, instituted by the old peace-loving dynamiter of that name.) Triana said: "Stop doing this! You only encourage theorists who come up with these terrible ideas that lead us all off a cliff!" -- that my paraphrase of his letter, but I submit it's a fair one.

And so we come back to the column in The Economist, which takes issue with Triana and with the whole campaign, contending that the BSM model has been "a force for good," i.e. that the world is much better off in terms of how risks are managed than it would have been had the three authors written nothing at all.

So much for the history of this dispute. I'll wade into it (unworthily, for I'm a humble scribe who has just named several people all of whom are much smarter than am I) tomorrow.

Payment for order flow

There is an intermittent controversy among those whom manage stock exchanges, brokerage houses and related institutions -- and among those who regulate them -- about a practice known as "payment for order flow."

Back in January 2003, for example, the then-chairman of the Securities and Exchange Commission, Harvey Pitt, wrote to the heads of each of the five US exchanges where stock OPTIONS are listed, just to give them what one might call a heads up.

"Hey guys, we're looking at this issue down here in DC. I'm not saying nothing, I'm just sayin'." [Not his exact words].

The idea was that an exchange would pay a brokerage firm for routing an order to them rather than elsewhere -- the payment might be a penny per share.

The controversy arises because your broker is suppsoed to be working for you, the investor, trying to get you the best deal. If he can get you a better deal for certain options on exchange A than on exchange B, shouldn't he rout your order through exchange B? If a payment from exchange B persuades him to do otherwise, aren't they cheating you?

The same might well be asked also if you're trying to buy the underlying stocks, though Pitt's January 2003 letter involved options for reasons I won't get into today.

Likewise, the same questions might be asked when it is a market maker, rather than an exchange, that is paying to keep orders on some form of security or other flowing. From whomever the money is coming, the broker who receives that money may be putting itself into a conflicted situation vis-a-vis its client.

I'm thinking about such matters today because an investment manager named Bernard Madoff is all over the news this weekend, even putting the continuing controversy over the auto bail-out in the shade for the moment.

Prosecutors claim that Madoff told senior employees at his firm, a market maker, that his operations were "all just one big lie," and "a giant Ponzi scheme."

If there is anything to the charges, the ongoing scandal may further discredit the whole idea of anyone -- exchange or market maker -- paying for any kind of market flow. Because Madoff had been closely associated with the practice, and was in fact a public voice in its defense.

He once told a reporter from CNN who interviewed him on the subject (May 2000): "If your girlfriend goes to buy stockings at a supermarket, the racks that display those stockings are usually paid for by the company that manufactured the stockings. Order flow is an issue that attracted a lot of attention but is grossly overrated."

The analogy is borderline absurd. The stocking manufacturer isn't in a relationship of contractual privity with the shopper, so such issues don't normally arise.

Here's some further reading for the curious.

Anyway, when this is all sorted out we may think of the whole idea of payment for order flow as an important warning sign. For the mark of a pyramid schemer is an increasingly desperate desire to keep increasing order flow.