25 Feb 2011

## Weekend Reading Assignment on Option Pricing

Back in high school I was a Star Trek geek. I remember one episode where they capture one of the Borg and come up with a plan to feed him a brain teaser that’s so complex that it will cause the Borg collective mind to explode. (Picard eventually quashes the idea because he apparently believes moral principles apply even when the fate of humanity is at stake.) I think of this episode whenever I go to Steve Landsburg’s blog.

The latest example is his post on option pricing. I am going to be a paternalist and “nudge” you to read it by posting the meat of it here:

When I was young, the pricing of stock options and other derivatives seemed like an obscure black art. Then one day Don Brown showed me a simple example that made everything crystal clear. Today I’ll share an even simpler version of Don’s example.

Imagine a stock that sells for \$10 today. A year from now it will be worth either \$20 or \$5. (Yes, I know that real-world stocks have a wider range of possible future prices. That’s why I called this a simple example.) What would you pay for an option that allows you to buy the stock next year at today’s \$10 price?

You might think you’d need a whole lot more information to answer that question. You might expect, for example, that the answer depends on the probability that the stock price will go up to \$20 rather than down to \$5. You might expect the answer to depend on how much traders are willing to pay for a given dollop of risk-avoidance.

But the amazing fact is that none of that matters. The only extra bit of information you need is the interest rate.

Let’s assume, for example, that the interest rate happens to be 25%. (Yes, I know that’s unrealistic.)

Now let’s price the option. The key is to focus on my imaginary cousin Jeter, who never buys stock options. Jeter happened to wake up with \$12 in his pocket today. Then he went out, borrowed \$8, and used his \$20 to buy 2 shares of stock.

A year from now, one of two things will happen. Either Jeter will get lucky, sell his 2 shares for \$40, use \$10 to repay his debt (\$8 plus \$2 interest), and pocket \$30. Or he’ll get unlucky, sell his 2 shares for \$10, use that \$10 to repay his debt, and pocket \$0.

I, on the other hand, bought 3 stock options today. A year from now, one of two things will happen. Either I will get lucky and use my 3 options to buy 3 shares of \$20 stock at a price of \$10 each, pocketing a \$30 profit. Or I will get unlucky and the stock price will plumment, in which case I will throw my option away and pocket \$0.

In other words, Jeter and I are guaranteed exactly the same outcome next year. Either the stock price goes up, and we each pocket \$30, or it goes down and we each pocket \$0. In that strong sense, Jeter’s strategy and mine are perfectly interchangeable.

Jeter’s strategy costs him \$12 out of pocket. Therefore my strategy must also cost \$12 out of pocket — otherwise, nobody would ever pursue the pricier strategy. Since buying 3 stock options costs \$12, the price of a single option must be \$4. Problem solved.

I submit that there’s something really fishy with the above argument. In particular, I claim that Landsburg is wrong when he says that you can correctly price options without caring about the probability of a stock’s future prices or the risk tolerance of investors.

But the fallacy/hidden assumption is tucked very deeply into Landsburg’s demonstration, such that halfway through writing my Mises.org critique of it–which will run Monday–in a moment of panic I actually thought Landsburg was right.

But then I came back to my senses and finished the article.

So, in the meantime, ponder the above and see if you come to the same conclusion by Monday.

#### 11 Responses to “Weekend Reading Assignment on Option Pricing”

1. Jon O. says:

Based on his simple example he is correct. With binary outcomes the probability is irrelevant because the long stock and long call position are essentially identical; both require \$12 of capital that can turn into \$12 of loss or \$18 in net gains.

With that said this excercise is a waste of time because it has little to do with the real world. IThe problem with this kind of thinking can be seen when it is extrapolated into the Black Scholes model where the important variables are historical volatility of the underlying and the interest rate vs. the dividend rate of the underlying. (I believe it also assumes a European style option that can only be excercised at maturity; it’s much more complex to figure out American style options which are mostly what are traded)

When the BS model came out it allowed option market makers to calcualte a “correct” price for the option based on some of the variables I mentioned but as time went on people began to realize that it may work most of the time in normal conditions but it doesn’t completely explain option pricing. The obvious problem is trying to use an instruments past volatility to understand future volatility. The idea of a random-walk is faulty; markets don’t follow a standard distribution of outcomes and have fat-tails mostly because of human behavior/psychology.(The LTCM guys learned that the hard way). There are other problems with the assumptions about rates and liiquidity that are necessary for the model to hold.

Plenty of option writers have gone broke because markets are made of humans who can get wacky at times. The point is that although his example is correct, ithe underlying insight is irrelevant, and in fact dangerous if dogmatically applied to actual option pricing. There is no correct option price, just the market price that buyers and sellers determine through an auction process.

• bobmurphy says:

Actually Jon I don’t think he’s correct. Even with all that information, I don’t think it’s necessarily true that “you would pay \$4” for that stock option.

• Jon O. says:

bob, if you have two identical assets (in a simplistic example) they should be priced identically. If the option was priced \$3 traders would arb the two (shorting the stock and buying the call) for risk-free profits until the two were back inline. (Lets say someone else wants to buy all the stock they can at \$10 – so the short-stock side of the arb would not push the price lower – it would move the option back to \$4. If that big buyer wasn’t there the stock would drop a little, the option would rise a little until they were priced in accordance with landsburg’s logic. )

If the stock went to \$5 the short-stock position would make more than the long call. If the stock went to \$20 the long call position would make more than the short stock. This shouldn’t happen if the option was priced “correctly”. Of course its only priced correctly in this fantasy land example.

• Captain_Freedom says:

Jon O.,

I think the disagreement here stems from the assumption concerning the expectation of the future stock price.

If both Steve and Jeter’s expectation of the future stock price is either \$20 or \$5, then they should price the option at \$4.00, for as you showed, arbitrage profit opportunities will arise if the option price (or stock price) is too far away from \$4.00 (or \$10.00).

However, and I think this is Murphy’s point, there is no reason why Steve and Jeter *should* have the same expectations of the future stock price. If Steve expects a stock price of either \$20 or \$5, and Jeter expects a future stock price of, say, either \$30 or \$5, then the option price does not *have* to be \$4.00. Since Jeter expects a greater volatility in the future stock price compared to Steve, he would more than likely be willing to pay a higher option price, and if Jeter represents a large enough market, then the option price will be likely be higher than \$4.00.

So assuming that Steve and Jeter have the exact same expectations of the future stock price, then it makes sense that they would price the option the same today. Steve relays this by saying “the option price MUST be \$4.00.” I think alarm bells went off in Murphy’s mind because as an Austrian, he knows that value is subjective to the individual, and so when he hears people say “the price of this thing SHOULD be such and such”, he philosophically rejects it, because the statement invokes some kind of objective value mechanism, a “fair price”, etc, that it independent from the price that actually exists, which must be subjectively determined.

So in the real world, it is the case that there is no option price that “must exist”. In the real world, the concept of an option price that “must exist” is a fully CONTINGENT concept, it is contingent on the subjective values and expectations of investors (Steve and Jeter), who may not agree. But IF they agree that the stock will be either \$20 or \$5, then given their separate, subjectively determined, but nevertheless equal, expectations, then the option price will be \$4.00. Option price is a derivative phenomenon, i.e. it is a contingent necessity, given other subjectively determined factors are true.

2. Silas Barta says:

As I said in my brilliant, ignored comment on Landsburg’s post:

It would be more accurate to say that “the probability distribution on its future price is irrelevant except to the extent it influences the current price”. Or, in the language of causal networks (see Judea Pearl), “Current prices screen off future prices”, or “Once you know the current current price, you learn nothing else upon being told the probability distribution on the future price”.

You can certainly value the option *yourself* more if you believe certain things about its future price. All this means is that such a belief implies that you (to avoid inconsistency/dutch-booking of your beliefs) should _also_ value it more than its *present* price. In other words, that the two are unrelated. Beliefs about the stock’s future price imply beliefs about its present price.

Landsburg has merely worded this basic phenomenon so that it sounds strange.

• Silas Barta says:

Argh! That should ‘read “In other words, that the two are not unrelated.” Or, without the double-negative, “In other words, that the two are related.”

• bobmurphy says:

Yeah that was a good post. I am saying a lot of the same stuff that you and other commenters said at Landsburg’s site, but I think I added a finishing move that will either seal the deal or show I’m an idiot.

3. RG says:

Stocks are not money.

4. Art says:

His trick (mistake? lie?) is in the assumptions of the problem. He states that Jeter’s out of pocket cost is \$12, but that is not true. What if he borrowed the entire 20? Would that mean that the option in that case would cost zero? Or, what if he had borrowed nothing? Would the option then be worth 6.66? The fact is that the pricing of the option has nothing to do with how much “out of pocket” someone can buy the same underlying security today.

An option’s price is a combination of its “intrinsic value” and the time value. Intrinsic value is the difference between today’s price for the security and the strike price. If today’s price is lower than the strike price, the option is “in the money”. For instance, if the stock is \$10 today, and the strike price is \$12, the option is in the money, and the intrinsic value of the option is \$2. If today’s price is higher than the strike price, the option is “out of the money” and has no intrinsic value.

In Landsburg’s example, the option is “at the money”, that is, today’s price is the same as the strike price, so the premium is only the time value.

5. Art says:

another trick he’s using is he’s assuming a certain outcome as far as the future price that ensures that the result of buying 2 shares or 3 options is exactly the same. but if you move the assumptions, then the outcome is not identical. the reality is that his exercise makes sense if and only if, one of the 2 exact outcomes he models, results, which is obviously impossible.

6. Brian Bergfeld says:

Ah, it seems so obvious now! You don’t need to know anything about the probability distribution of future stock prices because that information is already embedded in the current stock price.

So when he gives you the current stock price, he is giving you a lot more information than it seems.

Thanks, Bob.