Read The New Market Wizards: Conversations with America's Top Traders Online
Authors: Jack D. Schwager
The point is that option trading decisions should be based on conditional probability. I may have thought that an option was worth X, but now that someone else wants to bid X + Y, I may have to revise my estimate of the option’s value. The lesson we try to teach our traders is that anything that seems very obvious should be double-checked.
A great example to illustrate this concept is a puzzle posed years ago by Fisher Black of the Black-Scholes option pricing model fame. Imagine that you’re on “Let’s Make a Deal,” and you have to pick one of the three doors. You pick door No. 1. Monty Hall says, “OK, Carol, open door No. 2.” The big prize is not behind door No. 2. Monty Hall, of course, knows which door the prize is behind. The way he played the game, he would never open the door with the real prize. Now he turns to you and asks, “Do you want to switch to door No. 3?” Do you stay with door No. l or switch’? [Reader: You might wish to think of your own answer before reading on.]
The obvious answer seems to be that it doesn’t make a difference, but obviously that must be the wrong answer.
The correct answer is that you should always switch to door No. 3. The probability that the prize is behind one of the two doors you did
not
pick was originally two-thirds. The fact that Monty opens one of those two doors and there is nothing behind it doesn’t change this original probability, because he will always open the wrong door. Therefore, if the probability of the prize being behind one of those two doors was two-thirds originally, the probability of it being behind the unopened of those two doors must still be two-thirds.
I don’t understand. This show was watched by millions of people for years, and yet no one realized that the odds were so heavily skewed in favor of switching!
You have to remember that you’re talking about a show where people had to wear funny rabbit ears to get picked.
The thing that confuses people is that the process is not random. If Monty randomly chose one of the two doors, and the prize was not behind the selected door, then the probabilities between the two remaining doors would indeed be 50/50. Of course, if he randomly selected one of the two doors, then sometimes the prize would be behind the opened door, which never happened. The key is that he didn’t randomly select one of the doors; he always picked the wrong door, and that changes the probabilities. It’s a classic example of conditional probability. If the probability of the prize being behind door No. 2 or door No. 3 is two-thirds, given that it’s not door No. 2, what is the probability that it’s door No. 3? The answer, of course, is two-thirds.
Ironically, four weeks after my interview with Jeff Yass, the
New York Times
ran an article on the exact same puzzle. The
Times
article reported that when Marilyn Vos Savant answered this puzzle correctly in her
Parade
column in response to a reader’s inquiry, she received nearly a thousand critical (and misguided) letters from Ph.D.s, mostly mathematicians and scientists. The
Times
article engendered its own slew of letters to the editor. Some of these provided particularly lucid and convincing explanations of the correct answer and are reprinted below:
To the Editor:
Re “Behind Monty Hall’s Doors: Puzzle Debate and Answer?” (front page, July 21): One reason people have trouble understanding the correct solution to the puzzle involving three doors, two with goats behind them and one with a car, is that the problem uses only three doors. This makes the assumed, but incorrect, probability of picking the car (1 in 2) appear too close to the actual probability (1 in 3) and the solution difficult to arrive at intuitively.
To illustrate better the right answer—that a player should switch the door picked first after one of the other two has been opened by Monty Hall, the game-show host—suppose the game were played with 100 doors, goats behind 99 and a car behind 1.
When first offered a door, a player would realize that the chances of picking the car are low (1 in 100). If Monty Hall then opened 98 doors with goats behind them, it would be clear that the chance the car is behind the remaining unselected door is high (99 in 100). Although only two doors would be left (the one the player picked and the unopened door), it would no longer appear that the car is equally likely to be behind either. To change the pick would be intuitive to most people.
Cory Franklin
Chicago, July 23, 1991
To the Editor:
As I recall from my school days, when you are dealing with tricky, confusing probabilities, it is useful to consider the chances of losing, rather than the chances of winning, thus:
Behind two of the three doors there is a goat. Therefore, in the long run, twice in three tries you will choose the goat. One goat-bearing door is eliminated. Now two times out of three when you have a goat, the other door has a car. That’s why it pays to switch.
Karl V. Amatneek
San Diego. July 22, 1991
And finally there was this item:
To the Editor:
Your front-page article July 21 on the Monty Hall puzzle controversy neglects to mention one of the behind-the-door options: to prefer the goat to the auto. The goat is a delightful animal, although parking might be a problem.
Lore Segal
New York, July 22, 1991
The point is that your senses deceive you. Your simplistic impulse is to say that the probabilities are 50/50 for both door No. 1 and door No. 3. On careful analysis, however, you realize that there is a huge advantage to switching, even though it was not at all obvious at first. The moral is that in trading it’s important to examine the situation from as many angles as possible, because your initial impulses are probably going to be wrong. There is never any money to be made in the obvious conclusions.
Can you give me a trading example of a situation where the obvious decision is wrong?
Let’s say a stock is trading for $50 and an institution comes in with an offer to sell five hundred of the 45 calls at $4 1/2. The instinctive response in that type of situation is: “Great! I’ll buy the calls at $4 1/2, sell the stock at $50, and lock in $1/2 profit.” In reality, however, nine times out of ten, the reason the institution is offering the call at $4 1/2 is because it’s fairly certain that the stock is going lower.
Does this type of situation ever happen—that is, an institution offering to sell options at a price below intrinsic value [the minimum theoretical value, which is equal to the difference between the stock price and strike price—$5 in Yass’s example]?
It happens all the time.
I don’t understand. What would be the motive to sell the option below its intrinsic value?
In the example I gave you, the institution may be very certain that the stock is going to trade below $49 1/2, and therefore a price of $4 1/2 for the 45 call is not unreasonable.
Even if they have good reason to believe that the stock will trade lower, how can they be
that
sure of the timing?
The straightforward answer is that they know they have a million shares to sell, and that they may have to be willing to offer the stock at $49 to move that type of quantity. It all comes down to conditional probabilities. Given that this institution is offering the option at below its intrinsic value, which is more likely—they’re so naive that they’re virtually writing you a risk-free check for $25,000, or they know something that you don’t? My answer is, given that they want to do this trade, the odds are you’re going to lose.
When I first started out, I would always be a buyer of options that were offered at prices below intrinsic value, thinking that I had a locked-in profit. I couldn’t understand why the other smart traders on the floor weren’t rushing in to do the same trades. I eventually realized that the reason the smart traders weren’t buying these calls was that, on average, they were a losing proposition.
If it’s not illegal, why wouldn’t the institutions regularly sell calls prior to liquidating their positions? It seems that it would be an easy way to cushion the slippage on exiting large positions.
In fact, that is a common strategy, but the market makers have wised up.
How has the option market changed in the ten years that you have been in the business?
When I first started trading options in 1981 all you needed to make money was the standard Black-Scholes model and common sense. In the early I 980s, the basic strategy was to try to buy an option trading at a relatively low implied volatility and sell a related option at a higher volatility. For example, if a large buy order for a particular strike call pushed its implied volatility to 28 percent, while another call in the same stock was trading at 25 percent, you would sell the higher-volatility call and offset the position by buying the lower-volatility call.
I assume these types of discrepancies existed because the market was fairly inefficient at the time.
That’s correct. At that time, a lot of option traders still didn’t adequately understand volatility and basic option theory. For example, if a call was trading at a 25 percent volatility, which was relatively low for the options in that stock, many traders didn’t understand that you didn’t have to be bullish on the stock to buy the call. If you were bearish on the stock, you could still buy the underpriced call by simultaneously selling the stock, yielding a combined position equivalent to a long put. The more mathematical market makers understood these types of relationships and were able to exploit pricing aberrations. Now everybody understands these relationships, and you no longer see situations in which different options in a same stock are trading at significantly different volatilities—unless there’s a good fundamental reason for that difference in pricing. Now that everybody understands volatility, the major battle is in the skewness in option pricing.
Can you explain what you mean by “skewness”?
To explain it by example, the OEX today was at 355. If you check the option quotes, you will see that the market is pricing the 345 puts much higher than the 365 calls. [The standard option pricing models would actually price the 365 calls slightly higher than the 345 puts.]
Are options prices always skewed in the same direction? In other words, are out-of-the-money puts always priced higher than equivalently out-of-the-money calls?
Most of the time, puts will be high and calls will be low.
Is there a logical reason for that directional bias?
There are actually two logical reasons. One I can tell you; the other I can’t. One basic factor is that there is a much greater probability of financial panic on the downside than on the upside. For example, once in a great while, you may get a day with the Dow down 500 points, but it’s far less likely that the Dow will go up 500 points. Given the nature of markets, the chance of a crash is always greater than the chance of an overnight runaway euphoria.
Did the markets always price puts significantly higher than calls for that reason?
No. The market didn’t price options that way until after the October 1987 crash. However, I had always felt that the chance of a huge downmove was much greater than the chance of an upmove of equivalent size.
Did you reach the conclusion about the bias in favor of larger downmoves based on a study of historical markets?
No, nothing that elaborate. Just by watching markets, I noticed that prices tend to come down much harder and faster than they go up.
Does this directional bias apply only to stock index options? Or does it also apply to individual stock options?
The options on most major stocks are priced that way [i.e., puts are more expensive than calls], because downside surprises tend to be much greater than the upside surprises. However, if a stock is the subject of a takeover rumor, the out-of-the-money calls will be priced higher than the out-of-the-money puts.
Do your traders use your option pricing models to make basic trading decisions?
Anyone’s option pricing model, including my own, would be too simplistic to adequately describe the real world. There’s no way you can construct a model that can come close to being as informed as the market. We train our market makers to understand the basic assumptions underlying our model and why those assumptions are too simplistic. We then teach them more sophisticated assumptions and their price implications. It’s always going to be a judgment call as to what the appropriate assumptions should be. We believe we can train any intelligent, quick-thinking person to be a trader. We feel traders are made, not born.
Essentially then, you start off with the model projections and then do a seat-of-the-pants adjustment based on how you believe the various model assumptions are at variance with current realities.