Tony B: Thomas, we are back, my friend, the Skinny on Options Math. Jacob back in the house. Jacob, say hello to everybody. We missed you.
Jacob P: Hi everybody, it's been awhile.
Tom S: Where've you been?
Jacob P: You all were off for July 4th, or July 3rd.
Tony B: Yes.
Jacob P: You all took July 3rd off, then I was in Austria for a little over a week, which managed to span two Thursdays.
Tom S: Austria! What were you doing in Austria?
Jacob P: My father lives in Austria. I have a step-mother and a little brother, who's very little.
Tom S: How old?
Jacob P: He's 6.
Tony B: So it was a family trip?
Jacob P: Yeah.
Tony B: Where in Austria?
Jacob P: In Graz, which is in the Steiermark, it's the southern part of Austria.
Tony B: I have been to Austria one time when I was in college. I went skiing in Kitzbuhel and, what's the town next to it?
Tom S: You turn to me, like I'm going to know the town-in Austria. I watched a movie; all I know is that little mountain. Were you near that mountain?
Jacob P: Yeah, I don't know.
Tony B: Kirsch? Maybe it was Kirchberg? Kitzbuhel and Kirchberg. Were you near the mountains, or the other side?
Jacob P: They're on the south of the mountains.
Tony B: South of the mountains. It's a spectacularly beautiful country. It really is, but I was in college and I didn't really care how beautiful the country was, I just cared if there was snow on the mountains.
Jacob P: The beer is also very delicious.
Tom S: That's it. That's what I want to hear. You're diverse. You go to Burning Man. You go to Australia.
Tony B: Austria.
Tom S: Austria, same thing with an A.
Jacob P: They're very similar.
Tom S: Just seeing if you were listening.
Tony B: We have a few issues, Jacob, with Mexico/New Mexico and Austria/Australia.
Tom S: Don't even get me started on Maine.
Tony B: There's a great scene in … If you could pull up that scene, AC, I want to show it to … In Dumb and Dumber. Have you seen the scene with Austria and Australia?
Jacob P: I probably have, but I don't recall.
Tony B: AC, pull that up, please, and I'll show it, or we'll use it eventually. Let's talk, what are we covering today?
Jacob P: I want to talk about the specific effect that number of occurrences has on how reliable our …
Tony B: Did you wipe out or something?
Jacob P: Yes.
Tony B: I was going to say, you're all scratched up.
Jacob P: I'm a little bit scratched up.
Tom S: What happened?
Jacob P: My front wheel hit some mud, I was taking a turn. I slid.
Tony B: Tony, you send the car for him from now on.
Tom S: I'll do my best.
Tony B: We're not taking anymore risks with this.
Jacob P: I wear a helmet, it's fine.
Tony B: Protect the head. We'll send the car for you.
Jacob P: It's fine.
Tony B: Let's start over.
Jacob P: All right. I want to talk about how a large number of occurrences gets you this predictable behavior, but what I want to do today, is a more specific, "How much so?" At least one viewer had the question of "How many occurrences do I need to get predictable results?"
Tony B: That's pretty fair, because we could sit here and say "You need a large number of small occurrences" until we're blue in the face, but how does somebody who doesn't know … Like if the industry average, let's just give an example. Let’s say throughout the whole financial service space, most people don't even look at their accounts every year, so there are zero occurrences. When was the last time you looked at a bank account you have? Outside of checking, you've never looked, right? You never look.
Tom S: No.
Tony B: I haven't looked for years. Most people treat their brokerage accounts like that. But if you think about active, online traders. The definition of active is somewhere between fourteen and twenty four trades a year. Once a month is considered active. Three times a quarter, and you're off the charts.
Jacob P: I think the punchline at the end of this is we want more on the order of a hundred trades, than even thirty isn't quite enough.
Tony B: Are we going to talk about it as the super-active or are we going to be able to make this kind of broad?
Jacob P: The estimates all apply generally, and you'll be able to see which number you need to make be high.
Tony B: So how can we get more predictable results from our trades?
Jacob P: The answer is, you need more occurrences. The simplest example is, if you bet a dollar on a coin flip, then you're going to make a dollar or lose a dollar. If you bet a penny each on a hundred different coin flips, then you're very likely going to come out very close to even. You're going to get this extracted value result. We can do this more specifically, it comes from the central limit theorem, and how we add these things up. But as you take averages, if you average n independent things, the standard deviation is going to shrink like the square root of n. If you're concerned about your expected moves, do it more. The thing to remember is if you want to cut the amount of variance you have in half, or the amount of standard deviation in half, then you need to quadruple your number of occurrences.
Tony B: In all this, by the way, there's another side to this which is the liquidity aspect of what you're doing plays a big role too, because if you want to, like you just said, cut the variance in your …
Jacob P: I did say standard deviations, variance is technically the square of the standard deviation, so …
Tom S: Technically.
Tony B: Tony usually corrects me on those, he missed that one.
But if you want to tighten it up even more, it requires way more occurrences if you're using less liquid markets. The more liquid the marketplace, the better … Theoretically, it's less challenging.
Increase the number of occurrences. If you bet a dollar on a fair coin, you're betting 50/50 to make or lose that dollar. But if you bet a penny on each of a hundred coin flips, then you are very likely to come close to even.
Just so you know this interesting stat, because I've been reading a lot on randomness, certain coins are actually a little bit more weighted front an back because they weren't built for the purposes of [crosstalk 06:14]
Tom S: Right, I would imagine the head side is more …
Tony B: The tail side.
Tom S: I would've thought it was the head side.
Tony B: It's the tail side on most coins, especially the quarter and the dollar coin, whatever it is; it's the tail side which has a minuscule little edge, which is why ….
Tom S: Less cut out?
Tony B: Yes. The other thing, which I didn't know, is which coin do you think is most likely to land on its side?
Jacob P: Nickel?
Tom S: It's got to be that or …
Tony B: Yes, you're exactly right. The nickel will land on its side like one out of every nine thousand times.
Tom S: Like you. I'll walk into his office, and I'll go, "Is Tom down?" No, he just landed on his side, he's fine.
Tony B: That's a damn good point.
For specific math, there's a fancy version of the, what is that?
Jacob P: Pythagorean.
Tony B: Pythagorean theorem, and I knew that.
Tom S: Oh, come on, even I knew that one.
Tony B: Really? Because I can't see that well.
Pythagorean identity, I need at least a little help here. The Pythagorean identity that says "When we add up independent random variables with standard deviation, the sum has the standard deviation and the average has standard deviation. I don't really understand that.
Jacob P: I'm sorry, this line is a little bit jumbled. If you add up n independent random variables, the sum has standard deviation sigma, the sum has standard deviation sigma root n. The average has standard deviation sigma divided by root n.
Tom S: Thank you for clearing that up.
Tony B: What does that mean?
Jacob P: The standard deviation goes up. If I take a bunch of random variables, and they each have standard deviation one, like betting a dollar and a penny, that's a standard deviation one. If I'm going to [inaudible 07:58] If I'm going to bet on a hundred coin flips, and I'm going to bet a dollar on each of them, then I go in with ten for standard deviation because they go up by the square root of a hundred.
Tony B: I got it.
Jacob P: If I take the average, then I'll divide by the square root of ten, so I'll end up with a tenth for my standard deviation.
Tony B: So thus, in order to divide the standard deviation by two, you just need to make four small trades instead of one large trade.
Jacob P: Yeah.
Tony B: I think that's the takeaway for most individuals, because they hear us say all the time about independent occurrences. The key to achieving your goal is not what the industry has traditionally suggested about some sort of portfolio diversification or portfolio allocation.
Jacob P: Right.
Tony B: The real answer is to achieve a predetermined number of wins, not solely P & L. This is not about P & L. It's really about the number of occurrences.
Jacob P: Right. It's about [inaudible 08:53] occurrences. Though it's important, this portfolio diversification helps you get independence. When I say you need to make four small trades instead of one large trade, those need to be four separate small trades. It doesn't count … You're not betting …
Tony B: It doesn't count if you buy IBM, then sell a put in IBM.
Jacob P: Right. Or if you bet a penny, if you bet on the same coin flip every time, that doesn't help you.
Tony B: But I think most people get that. These are what we call independent events, independent occurrences.
Jacob P: Right. So now the question is, where do we find independents?
Tony B: Where do we find independents?
Jacob P: And the answer is that it's hard to find true independents. It's one of those things that's maybe philosophically impossible to totally establish, but even in practice, it's hard to find good measures for given two data sets and you're asked where are these dependent or independent of each other. It's very hard to tell. But one thing that's easy to tell is are they correlated? If they are correlated, then they're not independent.
You can have things that are uncorrelated but aren't independent, and that's going to be hard to notice and you're going to have to hope that it's okay. As far as getting these good averaging behaviors, it sort of is because even though they're related to each other, they work in a way where one cancels the other out.
Tony B: "The independence above is important, making one hundred penny bets isn't any different than making a single dollar bet if the penny bets are all on the same coin flip. However, in practice, determining whether two random variables are independent or not is either very difficult or impossible. However, independent variables must also be uncorrelated, which is much easier to check for since correlation can be computed between any two data sets."
I would say the way we look at the markets, we can create … I know people are going to make this argument that they're not "Perfect independent events." For us, they're close enough. I don't consider … there's so many stocks, like for example today, the market's up but Netflix is down $8. You can have positions on … Nasdaq … The Russell's up or the Russell stocks are down.
Jacob P: You're never going to have perfect independence, but you can find some very low correlation pairs of things or sets of things to have work with each other. And one of the easier ways to do this for people who have easier access to looking at beta values than they do for correlations is to beta weight one against the other.
If you do beta with the other as the benchmark, then correlation and beta are very closely related. There's a factor of the volatility from one of the two in it. If you don't want to take out an Excel spreadsheet and cut and paste in your data tables and compute correlations and try to figure out low correlation, because that sounds like a lot of work, you can just put a column on for beta and put a column on for implied volatility and just change the underlying on beta to whichever two you're trying to compare.
Tony B: "While we might not be able to determine independent trades, trading on low correlation underlyings can make for a good approximation. If you are more used to looking at betas than correlations,"- and I think every single person watching today is more used to looking at betas than correlations-"Remember that beta is just correlation scaled up by the volatility of the non-benchmark underlying."
This is very close to what we just covered on the market measure two seconds ago when we replaced volatility by using beta to express a correlation. Virtually the same thing, everything is so interrelated.
Jacob P: You could, rather than put them in the S and P beta, you could do a cross beta between the two in the pairs trade and do your beta weighting off of that factor.
Tony B: Sometimes I cross foods and it gets ugly, so we have to be very careful here. Sometimes we don't like certain foods touching other foods, and sometimes we will cross foods, like …
Tom S: If you're as mature as you are.
Tony B: Like coleslaw in his fried chicken sandwich, we will mix and match.
Jacob P: That sounds delicious.
Tony B: It is delicious.
"If I put on thirty trades each with a probability of profit, POP, of 70%, I expect to get twenty one winners, but how am I sure of that?"
Jacob P: Right. Or, how sure of that are you?
Tony B: But how sure of that am I? But how sure am I of that?
Tom S: Can it core an apple?
Jacob P: I think most people can do the math and go, thirty things each 70% to succeed, twenty one is expected … Now you want to know … You don't get exactly what you expect all the time, you're going to get some bell curve off of twenty one, in other words, how tight is that bell?
Tony B: Assuming, because the underlying is the most efficient marketplace in the world. Because it's a two sided auction marketplace. You're never going to find something that's any better.
Tom S: You've got to go with that assumption, yes.
Tony B: You have to. There's not a better marketplace in the world. The difference here, which I don't want anybody to get confused on is, twenty one winners doesn't necessarily generate a positive P and L.
Jacob P: You're still going to be neutral on P and Ls assuming all the trades are fair, which they probably are.
Tony B: "If we decide to assume that each of the trades is independent, can we use Hoeffding's …"
Jacob P: I think it's Hoeffding's, but I do not guarantee my own pronunciation of that, guys.
Tom S: I just came back from Australia. I'm just kidding!
Tony B: "Hoeffding's inequality to at least get a safe estimate. If we put on a certain number of trades each with probability of success, then the number of winners we get follows the binomial distribution, we'll expect to get …"
What is np?
Jacob P: The product of the number of trades and the probability of success. You multiply those two numbers together, that's how many you expect to have win.
Tony B: Like I was saying, "We'll expect to get a certain number of winners and Hoeffding says that for any …"
I'll skip that.
Jacob P: Positive number epsilon.
Tony B: "Positive number epsilon the probability that we get more than a …"
Jacob P: Epsilon n, so epsilon is some fraction. It will be like a quarter of your total n.
Tony B: You do realize that I'm reading this off during the show, right? You're just doing this to kill me.
"Away from expectation is at most two, Tony what's this one?
Tom S: Epsilon squared.
Jacob P: Two e to the minus two epsilon squared n.
Tom S: I knew that from going to … [crosstalk 15:31]
Jacob P: Hoeffding's is not a sharp estimate, it's not an exact quantity but it's a very convenient thing you can calculate that you can see how it goes down. So n there is your number of occurrences and epsilon is the fraction of the total number of occurrences you expect to be off by. We'll do an example.
Tony B: I'll let you do this.
Jacob P: "So in the example …"
Tom S: Are you sure … Hold on a second. Jacob, I don't mean to interrupt you. Are you absolutely positive you want Jacob to take this or do you want to show your brilliance some more?
Tony B: Conceding brilliance.
Tom S: Jacob, please take it away.
Tony B: I'm off the brilliant train.
Tom S: Understood.
Tony B: I love when Jacob's here, but I hate the fact that I have to scramble and babble my way through some of these math equations.
Jacob P: We'll get back to the example of we're putting on thirty trades and they're each probability of profit 70% though, we're not going to see that for a second. We'll say, what's the odds that of these thirties I'm within … So my expected number is twenty one, but I'm within one sixth of my total trades.
That's going to be plus/minus five, so between sixteen and twenty six winners. I know that the probability that I'm outside of that is at most the two e to the minus two, one over six squared thirty.
The probability that I wind up inside there is at least one minus that. Which is about 62%, which is not great. You would like to be more confident than that.
If my epsilon is small then I need to make my n big. As epsilon gets small, n needs to grow the square of epsilon.
I've made epsilon be a sixth, thirty wasn't a big enough n, so I'm going to triple my n up to ninety. Now I'm expecting to get sixty three successes, where I'm just tripling everything. Now I'm in within a 99% likely to come out in that middle sixth, either side of the expected number.
Tony B: You're 99% to get between forty eight and seventy eight.
Jacob P: Right. Which is that middle sixth plus or minus.
Tony B: How tight can we get that number?
Jacob P: You can adjust it … If you want a smaller epsilon, that forty eight to seventy eight, you want that to cut down? Your percentage is going to get worse but if you bump up n, you can get it back up.
Tony B: The numbers just keep getting tighter and tighter and tighter?
Jacob P: As n goes up it’s going to get tight. It's going to get exponentially tight as you get n up.
Tony B: So how many trades does somebody have to make to get to where there's … Let's say I don't want thirty, I don't want forty eight to seventy eight, I want something like …
Jacob P: Like sixty to sixty three?
Tony B: Yeah.
Jacob P: Sixty to Sixty six. Right, the average of Sixty three so you got to be center to there. That's plus or minus three which is one thirtieth of ninety, so you need n to be a thousand or so.
Tony B: Right. Let’s say every year I may make five to six thousand trades, but about three thousand are option trades. That's why we can say …
Jacob P: That you come out real close to your expected number.
Tony B: … That we come real close to our expected number. That's why. There's nothing else to it, it's amazing! Why isn't this what finance is all about? Why isn't this the whole story of finance, this is crazy.
"Also, note that if Hoeffding …"
Tom S: I'd like to see you prove Hoeffding's inequality to Jacob, please.
Tony B: "… Does it need the trades to be all the same probability …"
Jacob P: You need to do Hoeffding's lemma first.
Tom S: Yeah, do that one first. Please, I have my pen, I have my paper. I'd love to learn from you.
Tony B: Maybe I did my thesis on Hoeffding.
"It still tells you at least how likely you are to vary from it."
It basically all boils down to the same thing.
Jacob P: My example was all the trades were the same 70% probability. That's not important for this.
Tony B: Right, because what if you do one in ninety one at sixty, it's just the average of those.
Jacob P: You just need to do a lot of them and then the average moves and that's it.
Tony B: The takeaway from all this is, mathematically the more trades you make the closer you're going to come to whatever your statistical expectation is. If you use tight liquid markets, create lots of number of occurrences and stay small; although there's no guarantee to your positive P and L.
You'd much rather be right 70% of the time than not.
Tom S: Sure.
Tony B: That's it!
How's that not changing the world of finance? How's that not what every single person should be doing/teaching, whatever it is?
Tom S: I agree, I don't know.
Jacob P: I have one more thing, which is that the specific question of what is a large number of occurrences? You need to decide how much variance you want to accept and then square that and that's your number of occurrences that you need.
Tony B: Oh, that's how it works. You square the variance.
Jacob P: If you want to be within a tenth, you should be doing a hundred times as many occurrences.
Tony B: I got it.
Jacob P: If you want to be within 1%, you need to be doing ten times the trades.
Tony B: Yeah, no, this is great. This is awesome. You want to know how this high frequency firm makes money every day? They have millions of occurrences. It's pretty damn simple to figure out. It's not magic.
But in Washington, and from the political side and from the media side, we beat these companies up because we suggest that it's rigged. When it's really just a number of occurrences.
Jacob P: There's maybe something rigged about the fact that no person could possibly be doing that many. They have some unfair advantage in that they can do so many more occurrences than other people can.
Tony B: That part of it doesn't bother me because that's just about capital business. Because I can't go and drill an oil well in Dubai either, so you can't pick on that.
Jacob, this was awesome man.
Tom S: Great job Jacob, you actually made the market a little bit softer too. It's been strong since you've been gone. All hail the mighty Jacob!
We'll be back in ninety seconds, we got bootstrapping next. This is tastylive live.