Black-Scholes Option Pricing

Factual Questions
1

I have a couple of points on Cecil’s most amusing (“let a professional help you out,” indeed) column on a couple (fission and Chicago School economics) of the glorious achievements of UofC affiliates (http://chicago.straightdope.com/sdc20090115.php).

First, what is with Mr. James T. Struck and his honorifics? “James T. Struck, BA, BS, AA, CNA, MLIS?” Bachelor of Arts, Bullshit, Alcholics Anonymous, Certifed Nurse’s Assistant and Master of Living in Sin, right?

Second, the Black-Scholes option pricing formula is not what caused problems for Long Term Capital Management or the current markets. Not in itself. Not exactly. Yes, over-reliance on too complex mathematical formulas to price derivatives while ignoring correlative risks and, you know, reality, is a bad thing. But, the Black-Scholes formula itself, appplied to stock options, serves a useful function and is endorsed by the SEC.

The SEC requires the top executives of public companies to disclose their compensation in proxy statements. How do you disclose the value of options? An example: a stock option issued as compensation is typically a right to buy shares at $100 any time during the next 10 years, where the price of the stock on the date of issuance is $80. The executive is suppposed in be incented by this to improve the stock price so that he can exercise the option for a profit. However, the stock price may never reach $100 (in which case it will never be exercised) or the executive may never have the $100 to exercise the option (simplified–there are often cashless exercise rights).

So, for a long time, executives took the position that options couldn’t be valued and option value had no place in compensation disclosure–even if the stock is at $99 and the option still has 9 years to run.

Failing to disclose potential option value was misleading to shareholders. Various suggestions were made–assume an annual 5%/10% increase in stock price, etc. The Black-Scholes formula solves this problem and results in better disclosure to shareholders of the bloated compensation packages of corporate America’s fattest cats.

So, the formula itself: NOT bad, for exemplary and theoretical purposes.

Using related formulas to actually buy and sell derivatives on a market-wide basis: Bad.

Even more flavors of Bad: buying and selling financial instruments created from mathematical models too complicated for any one but your pusher from Citigroup/Bear Stearns/etc. to understand. Never let anyone who implies that you are too stupid to grasp the details of a bond or annuity sell you anything.

2

I have no idea what’s up with Mr. Struck. He doesn’t seem like the shy type and I wouldn’t be surprised if we hear from him again - rather hope we do, in fact. Perhaps at some point he’ll favor us with an explanation of his titles.

I hope I didn’t give the impression the B-S formula was inherently bad. On the contrary, I happily join in the consensus that it’s a brilliant piece of work by some very smart people. However, more trust was placed in it (and in mathematical modeling generally) than was warranted, and what’s more, some of said unwarranted trust was placed in it by the people who invented it, as witness the swift rise and collapse of LTCM, in which two of the three B-S folks played important roles. I don’t suggest the formula itself was the proximate cause of LTCM’s problems, but everything I read on the subject suggests Scholes et al. thought they were too smart to screw up. In hindsight, LTCM’s story ought to have forcibly suggested to the financial community the limitations of an approach that reduced the interplay of market forces to something akin to the ideal gas law. I imagine they’re paying more attention now. I think we may confidently predict that the growth industry in economics is going to be the study of bubbles and panics.

Tru dat. Lots of money managers (or former money managers) assumed everybody else in the room understood the complicated financial products being touted to them, and now have learned to their sorrow that nobody did, sometimes including the people doing the touting. Here’s a lesson you don’t need to have scored 800 on your math boards to grasp: If you don’t know what you’re doing, don’t do it.

Glad you enjoyed the column.

3

Read all about it. Mr. Struck is quite possibly the only certified nursing assistant in human history to write a book debunking the Big Bang theory.

4

Former librarian now working at Dominick’s, I see. Interesting career path.

5

Black-Scholes was the first option pricing model, and the guys who won the Nobel[sup]1[/sup] deserve it for getting the conversation started, but it’s not the last word on option pricing, and the practice of using it to price executive compensation options is completely unjustified.

The first problem you run into is that Black-Scholes was derived for pricing European options, which can only be exercised on their expiration date. But nobody trades European options; rather, we all trade American options, which can be exercised any time up till their expiration date. Since you have more choices about when to exercise an American option than a European one, if all else is equal, the American option will be the more valuable of the two[sup]2[/sup]. That’s not the only problem, though. The assumptions underlying the model are as follows[sup]3[/sup]:
[ol]
[li]Money can be borrowed at the risk free rate.[/li][li]The risk-free rate is fixed (i.e., known and constant).[/li][li]The dividend structure of the underlying stock is fixed.[/li][li]Market volatility is fixed.[/li][li]The returns on the underlying stock are normally distributed.[/li][li]There are no taxes or transaction costs.[/li][li]There are no costs associated with short-selling.[/li][/ol]Out of the list, the first assumption is pretty reasonable, and the last two are false but necessary for doing any kind of reasonable calculations. The middle four are the interesting ones. On a short-term basis (say, less than three months), it’s pretty unlikely that they’ll be violated, and so they are reasonable approximations to the world. But if you go out beyond that, market volatility is clearly not fixed, and stock returns are clearly not normally distributed. If you apply a model to a situation where it’s not appropriate, can you trust the results?

If you try to price options more realistically, you run into the problem that no exact solutions are known, and you have to use numerical or Monte Carlo methods. Since there’s some uncertainty associated with the answer from those, people who would rather be precisely wrong than approximately right tend not to favor them.

My take on it is that the current mess is going to play the same role in financial economics that the Great Depression plays (or played?) in macroeconomics: everybody’s going to be trying to figure out what the hell happened, and how we can make sure that it never happens again.

[sup]1[/sup]: Emanuel Derman suggests in “My Life as a Quant” that the Nobel committee would not have awarded the prize to Scholes and Merton had Black still been alive, as he was employed in industry at the time, and the committee has no great love for industrial researchers.
[sup]2[/sup]: The issue of vesting is a red herring here. As long as the option vests before its expiration date, an American option is still more valuable than a European one. No one is likely to be paid in options that expire before they vest.
[sup]3[/sup]: Taken from Robert McDonald’s “Derivatives Markets”, second edition.

6

So you are saying that Monte Carlo or “numerical” formulas should be used instead? Is there agreement on the best one to use? If not, I can see why the SEC might prefer a known quantity with limitations to “more approximately right” if it leaves open room for argument by executives who have an interest in making it come out a certain way. Not that I am vested [ha, I killl myself] in Black-Scholes–if they can build a better mathematical mousetrap, I’ll buy it.

Compensatory options generally cannot be traded as I am sure you know, and they may expire before they vest–continued employment is a common vesting term. I would also estimate that the majority of compensatory options are exercised at expiration–in addition to the inertia of employees who are not otherwise active market participants there is an incentive not to exercise to keep the free-ride (some exceptions of course).

I didn’t realize; I apologize for poking fun.

Economic bubbles, gas bubbles, close enough. They think randomness is like gas bubble movement? I say we put the formulas in a box with lump of uranium and a gieger counter.

7

Black Scholes and other option pricing models assume an awful lot. Rather than debate Black Scholes, Monte Carlo and Binomial Lattice, I would prefer an argument about adopting a standard. Maybe the standard could apply across industries, the goal being to serve the compensation analysts favorite pasttime: reporting on what the other guy is doing.

For years Black Scholes seemed to be the standard, but now it is all mucked up. Expensing options has seemingly added another layer in how we declare an options value. Is it messier than our all using Black Scholes (which we all recognized was BS), or better for this layer of complexity and variation?

8

The most famous equation in finance, the Black-Scholes/Merton equation, came from physics. It launched an industry worth trillions of dollars

Veritasium (31:00)
https://www.youtube.com/watch?v=A5w-dEgIU1M

9

You’ve nailed BS, but the degree program progress follows with MS - “More of the Same” and PhD- “Piled high and Deep”.

10

Myron Scholes was also employed in Industry at the time of the award (1997). It’s true he still had a Stanford appointment but he’d completely given up teaching before the award was granted and had been working in industry most of the time since the early '90s.

11

The Black Scholes model assumes stock returns are lognormally distributed not normally distributed.

12

A study in 1996 by Huddart and Lang found that many options were exercised well before expiration. In addition, if the options have a reload feature (it’s complicated), then it is actually optimal to exercise as soon as they vest and are in the money because new options are granted.