Tony: Thomas, we're back, my friend. The Skinny on Options Math. Jacob in the house. Jacob, how you doing?

Jacob: I'm pretty good.

Tony: Good. Glad to have you back.

Jacob: It is my favorite kind of day, cool and gray.

Tom: Cool and gray? What would be your favorite day?

Tony: I like a nice warm sunny day.

Tom: You like…

Tony: 75 degrees and sunny. You know, the classic. Jacob: Yeah.

Tom: Classic?

Tony: Mm-Hmm. (affirmative)

Tom: I don't mind 66.

Jacob: Yeah?

Tom: I prefer sunny.

Jacob: I like, you know, 65 to 70, at least mostly cloudy.

Tony: That would be…

Tom: I like sun.

Jacob: Yeah.

Tom: Seattle.

Tony: Yeah that would be Seattle.

Jacob: Yeah. This is why I lived in Portland for 4 years.

Tony: Okay.

Jacob: I love that weather.

Tony: Now, Jacob comes on, because I like when you come on.

Tony: The market usually sells off.

Jacob: What?

Tony: Jacob comes on, usually the market sells off.

Tom: Yeah, S & P's down 7 now. They just sold off.

Jacob: Now, we can do a piece about correlation and causation here.

Tony: Mm-hmm. (affirmative)

Tom: That would be very interesting. I'm not sure how much statistical validation we can create, but…

Jacob: I think we have some small sample size problems.

Tom: Today, we are covering, What are Monte Carlo methods?

Jacob: Right. Monte Carlos are sort of this under the hood thing that occurs throughout most of the models that get used in finance. Because a lot of the models that get used, they sort of have this relatively well-founded theoretical idea about, "Okay, the underlying's going to move like this, so things will be valued like that." That's great, except in the case of Black-Scholes. Pretty much only Black-Scholes reduces down to a closed form integral solution, where you can just sort of compute it and find out the answer.

Tom: Right.

Jacob: In pretty much every other situation, when you want to get a number out of a model, you have to do something clever. Monte Carlos are this trick in order to extract usable information from our models.

Tom: Are we talking more about… What's the focus? An individual investor is not going to go to a Monte Carlo model.

Jacob: Method.

Tom: Method, right.

Jacob: The Monte Carlo is a thing you do to model.

Tom: Right. I'm sorry, method, right.

Jacob: When you go to model, let's say you're using something like the model for your underlying prices and you want to know what's my probability of expiring in the money. Right?

Tom: Right.

Jacob: The actual trying to do this computation, there are integrals to do, but there's not really super trackable… What you wind up doing, is someone wants to estimate it. This is more about not what an investor's going to do, but where that number comes from. When you look at the platform, and it shows you that number.

Tom: Right.

Jacob: How did it get that number?

Tom: Would this be also something you could say that, "The reason you're watching this today is because the takeaway is the Monte Carlo method is probably what's used in figuring out pricing for lots of things."

Jacob: For lots of thing, yeah, figuring out pricing. It's also used throughout. Away from finance, it's used to do all sorts of physics.

Tom: Right

Jacob: It's used any time there's a good mathematical model, but it's computationally hard to do. The Monte Carlo method is this lovely trick you can do to replace whatever computation time you had with, pretty much, a quadratic computation time, which is not that bad in most situations.

Tom: Monte Carlo methods are a way to estimate probabilities or expected values in models where the exact computation is not feasible for one reason or another. The way they work is to run many independent random situations of the system being modeled, and then guess that the proportion of these trials where something happens is close to the actual probability of that event happening.

Jacob: Right. It's a pretty simple idea, right? It's called Monte Carlo because it's sort of based off the guy who broke the bank at Monte Carlo…

Tom: Okay.

Jacob: Who's name I'm not going to remember now. Because what he did is he went and he sat by the roulette table, and he went, I don't know the distribution of the numbers that come out of the roulette table. Maybe it's uniform. Supposedly, it's uniform, but I'm going to guess it might not be. I want to know what the probabilities of all these numbers are. He just sat there and he wrote it down. Then, at the end he got, "Oh, no. It's not quite uniform. It's a little skewed." He came back the next day, and just kept betting, knowing the skew, and broke the bank.

Tom: You know, I did not know that.

Tony: I didn't know the story either.

Jacob: Yeah.

Tom: Is that a real story?

Jacob: Yeah. No, the man who broke the bank. I'm not going to remember his name, but it's a famous story.

Tony: Uh-huh. (affirmative)

Tom: They're used in a wide variety of situations where there is a clear stochastic model in use, such as Black-Scholes model, but explicitly the formula is not trackable, tractable.

Jacob: Tractable.

Tom: Tractable, such as pretty much anything other than the Black-Scholes model. That's what you just said. For example, getting prices out of the binomial model pricing model is almost always done via the Monte Carlo methods to avoid exponential growth in the size of the binomial tree.

Jacob: Right, if you remember the binomial model when Vol was doing this.

Tom: Sure.

Jacob: Constructing this tree …

Tom: Yeah.

Jacob: The tree you have more time steps, so it doubles every time.

Tom: Right. Right.

Jacob: That, computationally, becomes essentially impossible.

Tom: Right, when you get to a certain level.

Jacob: Right. Once you want good accuracy, it becomes almost impossible to do it. Instead you can just run a whole bunch of random Monte Carlo simulations.

Tom: Which is why they didn't build a binomial model at the roulette table.

Jacob: Right. Which is why he ran the Monte Carlo methods.

Tony: Which by the way, Charles Wells.

Jacob: Charles Wells. That sounds right.

Tony: Mm-hmm. (affirmative)

Tom: My son the other day was at the roulette table.

Tony: He was not breaking any banks, was he?

Tom: He was not breaking any banks. He was at the roulette table and he told me that they had a method. Now I didn't know…

Jacob: System.

Tom: A system.

Jacob: See, everyone has a system.

Tom: Everyone has a system, but in the end it turned out his system was finding somebody that won a couple of hands and just doing whatever they did.

Jacob: I mean, there are worse systems. Charles Wells and this method is why they now on roulette tables, they change the wheel all the time. They change the tosser all the time. They want to keep it cycling, because they don't want anyone to be able to get enough data. Because it's never going to be quite uniform. There's always going to be some skew. And they never want anyone to have enough data.

Tom: Do you mean skew because the tables are going to be a little bit slanted, or?

Jacob: The table's not perfect. The wheel isn't perfect. The tosser isn't perfect.

Tom: Right.

Jacob: It's not perfect. If it's not perfect, then there's some skew. If there's some skew, then, right there … The house edge in roulette, it's not super huge, you know, 1 in 37.

Tom: 2 in 37.

Jacob: 2 in 37, actually. There's a 0 and the double 0. Right.

Tom: Yeah, 0 and double 0, there's 2 greens.

Tony: So you're saying, is that there's a little bit of a variable. There might be an edge there and that's why.

Jacob: Yeah. Right. If anything's off, in a perfect little uniform thing, the house just has this 2…

Tom: Yeah, but here's what I don't understand. What I don't understand, okay. Now this, you call me crazy or whatever.

Jacob: Mm-hmm. (affirmative)

Tom: With all the technology we have nowadays, why can't …

Tony: It all be automated?

Tom: Why can't somebody have, like, effectively like a GoPro, I'm just saying, whatever it is.

Jacob: Mm-hmm. (affirmative) Yeah.

Tom: Then, within a few hundred hands, it just does it, while you're just sitting there.

Jacob: All right. I mean, so many people try this. This is why casinos try to change the roulette. They change the wheels. They change the tosser.

Tom: They don't change them every day.

Jacob: They do, don't they? They change it, like, every couple hours.

Tom: Do they?

Jacob: Yeah. I think so.

Tom: Wow. I did not know that.

Tony: I didn't know that either.

Tom: You mean they change the ball.

Jacob: They change the ball, they'll rotate the wheels, and they'll change the person who's doing the tossing.

Tom: Oh.

Tony: They'll change the person. That I know. I didn't know they changed the ball…

Jacob: Well, changing the person is probably enough to, like, reset… It will change what the skew is.

Tony: Mm-hmm. (affirmative)

Tom: Yeah.

Tony: He's got the scheme figured out.

Tom: I know.

Jacob: Well, they didn't, and then Charles Wells took all their money.

Tony: Mm-hmm. (affirmative)

Tom: I love the idea that people will spend so much time trying to beat a stupid game for a 1 or 2%

Jacob: A game of chance.

Tom: Yeah, a game of chance for a 1 or 2% edge, but they basically won't even spend 30 seconds. "okay, this may be cool because I can do this right now, right here."

Jacob: Right.

Tom: Right now.

Jacob: Right.

Tony: Right.

Jacob: You can play the game with a better edge

Tony: Unbelievable.

Jacob: In the market.

Tom: Perfect. Let's go on. How many trials do you need to run?

Jacob: Right, so the idea is… This is the question. Why can't someone with a GoPro sit there and just figure it out in a couple hands?

Tom: Yeah.

Jacob: The answer is that your accuracy … The accuracy comes from, instead of whatever the core property of the model is, just the central limit theorem, that these independent trials are going to converge like a square root. That means if you get the quadratic behavior. If you want to, you know, double your accuracy, you need to run 4 times as many trials. If you want 10 times as much accuracy, you need a hundred times as many trials.

Tom: Got it.

Jacob: As far as a person actually doing it in a casino, this is not actually super good. Right? You do run out of time if they are willing to change it relatively often. However, on a computer, quadratic growth is not one of the fastest… It is faster than you might like. It's not linear. It's not log linear, but it's not exponential bad. It's not a high order polynomial, right. The other thing this gets used for in finance a lot is trying to evaluate how much a portfolio is worth. Because doing that requires that you have to integrate over all your different holdings.

Tom: Got it.

Jacob: That's a big multi-dimensional integral. That, in general, is a runtime of N to the dimension. Going to N squared is much better.

Tony: I got it.

Tom: Actually, I'm very clear of the concept right now, which is amazing. The convergence of Monte Carlo methods rely on the central limit theorem, and so only converges at the square root rate. This means, this is what you just said.

Jacob: Right.

Tom: If you want to be 10 times as confident in your Monte Carlo results, you need to do a hundred times as many trials. This can be improved on slightly by using the quasi …

Jacob: Quasirandom numbers

Tom: Quasirandom numbers of random, psuedorandom numbers.

Jacob: Right. Most of the time these things get run … In principle and in theory, you'd make these simulations random. You'd use random numbers, but in practice, people generally don't have random numbers. People have pseudorandom numbers. Psuedorandom numbers are pretty good.

Tom: I'm using that word from now on, pseudorandom numbers, because I love it so much.

Jacob: Yeah, psudorandom numbers are [crosstalk 00:09:54]

Tom: I've never seen it before.

Jacob: There's a somewhat modern trick called quasirandom numbers, which aren't random at all. Pseudorandom numbers aren't random at all, but they're even less random than pseudorandom numbers. They get a wide… They become well-distributed faster than randomness would have it be.

Tom: I understand.

Jacob: They make the Monte Carlo methods converge a little bit faster.

Tom: The reason I would use Monte Carlo, as a trader, would just be to check on my other stuff.

Jacob: Right. You go, "okay, do I agree? Does this work?"

Tom: Right. Monte Carlo methods are still only appropriate for situations where the complexity level is significantly above quadratic, such as computing the value of a portfolio, which generally requires doing it at a high dimension integral.

Jacob: Mm-hmm. (affirmative)

Tom: Because…

Tony: You having a stroke?

Tom: No, no, no. I'm trying to figure out how to…

Tony: Make sure.

Tom: How to…

Jacob: For example, in Black-Scholes, you always get these integral formulas.

Tom: Yeah.

Jacob: Even, worst case scenario, you get these integral formulas that come out.

Tom: Yeah.

Jacob: Computing a single integral to a high degree of accuracy is, like, a like a linear time approximation, which just means that doing it by Monte Carlo isn't going to speed it up. It's going to slow it down. You're going to be getting a random, close guess, and it's going to take you longer to get your close guess.

Tom: Slightly above quadratic, what does that…

Jacob: Quadratic, you have to go, how long this takes to run, scale with how much accuracy I want, and how many things I need to compute.

Tom: Got it. Okay. I got it.

Jacob: Quadratic means X squared.

Tom: How well do they work?

Jacob: Generally, very well. As far as we can tell, they're highly accurate. Largely because this quadratic convergence is pretty fast. There are 2 things that can go wrong. 1, if your model is bad, then you're out of luck, right? Monte Carlo method is only going to tell you what the model tells you. It's never going to tell you what reality tells you. When you're paying attention, in real life, you can notice a discrepancy between reality and your model. In particular, Monte Carlo methods, because it's doing random simulations, while a large number is to us is a small number to the theory, it can miss [tail 00:12:00] events. Monte Carlo methods are not really good at distinguishing between these heavy tails. So, this is one of the worries about using them in finance, because we are aware that these tail events are already the part that we don't understand well, that our models aren't predicting well, the Monte Carlo methods aren't going to pick those up particularly well either. There's a little bit of danger to that. At the same time, because we only really trust our models in the bulk, it's nice that the Monte Carlo methods, we also only trust in the bulk.

Tom: Yeah, but how does Black-Scholes pick those up?

Jacob: What?

Tom: They don't pick up the tails either.

Jacob: Black-Sholes has its tails. It has perfect forms.

Tom: It has the tails.

Jacob: Right. It has tails. Its tails are too narrow, right?

Tom: Right.

Jacob: This is the usual complaint about Black-Sholes.

Tom: Okay.

Jacob: Monte Carlo won't notice this. It won't notice.

Tom: I got it. I got it.

Jacob: It won't notice that your tails are narrower than they should be or fatter than they should be.

Tom: I got it. It just takes them for what they are.

Jacob: Right. It takes them for what they are, and then also, just doesn't see tails very well. Because it does off of this random simulations…

Tom: You can't see stuff, because you can't see stuff that's not there.

Jacob: To see the thing that only happens 1 in a thousand times…

Tom: That's right.

Jacob: Or 1 in a million times, you need to run a million simulations, right?

Tom: Right. Right.

Jacob: The Monte Carlo method, the point of it is that you don't run a million simulations.

Tom: Right.

Jacob: You run like, maybe 30 thousand or something.

Tom: How do you ever see those tails?

Jacob: What?

Tom: The only reason I'm even saying this…

Jacob: I mean, 1987.

Tom: Yeah.

Jacob: That's how you see those tails.

Tom: Yesterday. Right, right. You go back in time…

Jacob: Right. They happen sometimes.

Tom: We had that move yesterday, and yeah.

Tony: We had that move yesterday?

Tom: Yeah.

Tony: Correct.

Jacob: Right.

Tony: On a lot of products, like bonds.

Tom: Assuming that we are, we had a crazy move yesterday in the market…

Jacob: Yeah.

Tom: …and the bond market did something it's never done before.

Jacob: Wow.

Tom: We had kind of that 3, 4 standard deviation move.

Jacob: Yeah. There was this tail event where our models sort of break down.

Tom: That's right. All the models broke down yesterday in bonds. It's interesting because it just happened yesterday, and that's the part you can't see in any of these methods.

Jacob: Right.

Tom: Assuming that you're running sufficiently many trials, there are 2 potential problems with Monte Carlo methods. 1, the randomness, and second, the model. Since the simulations are generally run on computers, true random numbers are hard to come by. Why would that be? Why can't you generate true random numbers on computer?

Jacob: It's possible there aren't true random numbers period. Well, except for like, quantum events, maybe. Computers are binary. They're digital.

Tom: Yeah.

Jacob: Everything that happens inside a computer is deterministic. It is.

Jacob: Everything that happens in a computer is determined. This is where pseudorandomness is used. Right? You get these long strings.

Tom: Oh, that's the definition of pseudorandomness.

Jacob: It's these long things that look random.

Tony: Now he can't use it anymore.

Jacob: I'm sorry. Yeah, pseudorandom are these deterministic things that look random, that will pass randomness tests. They'll be sort of uniformly distributed, and if you reorder them, they'll look kind of like they did before. There are these various tests for pseudorandomness that…

Tom: I can't use those numbers, because they just

Tony: Pseudorandomness

Tom: They're pseudorandom. They're not really random.

Jacob: Yeah.

Tom: Okay.

Jacob: This is a complaint that a pure theorist will have to a computer scientist. There are people out there who sell …

Tom: That's it, that's another [crosstalk 00:14:58]

Jacob: True random numbers, websites where you can pay them per number.

Tom: Yeah.

Jacob: They will give you a supposedly true random number that they got out of a quantum experiment or something.

Tom: I can only imagine having to listen to that argument from 2 people. At this point there are many good approaches to pseudorandomness and quasirandomness.

Jacob: Yeah.

Tom: I love both of those words.

Jacob: It's pretty much not that big a deal to not have real randomness, most of the time

Tom: Less tractably, running a Monte Carlo trial relies completely on our model for the future. Errors in the model, especially tail errors, can be very hard to notice during the simulation process, and only beome apparent once the Monte Carlo predictions face reality.

Jacob: Yeah.

Tom: I think that one of the things that might, because I kind of remember vaguely having this discussion with T.P. years and years ago when we were building the models for Think or Swim.

Jacob: Yeah.

Tom: This was his argument, that there's too many errors in the model. But I guess I didn't, I couldn't figure it all out, you know. It's too hard.

Jacob: Right. You have to be careful, because we use all these models, so all of our probabilities, and all of the evaluations come from these models. We have to be a little bit careful about how much we really trust them because they're pretty good. As far as we can tell, they seem to hold up, but they have these high probability events where we won't know that they're wrong then, until they're wrong.

Tom: What's your biggest fear? I know you're not a big investor…

Jacob: Yeah, yeah.

Tom: Or that kind of stuff, but what's your biggest fear about the markets. Is it that our models, that our risk models are kind of messed up?

Jacob: Oh.

Tom: Is that something that makes you nervous? If you had money at a brokerage firm, or an exchange, whatever it is, would it make you nervous that those models are messed up? What makes you most nervous.

Jacob: I would be nervous because I had money there, because really, if everyone loses their money there, probably everyone loses their money.

Tom: Okay.

Jacob: Whether or not your money is in the banks when the banks all crash, if the banks all crash, money sort of stops being meaningful.

Tony: Fair. So what's the risk?

Tom: So what's your biggest fear?

Jacob: I think the biggest fear would be some catastrophic tail event, either direction. Either everything super spikes all of a sudden, or everything collapses all of a sudden.

Tom: Mm-hmm. (affirmative)

Jacob: Because all the investors make the same decision at the same time. If that happens, sort of everything starts looking meaningless.

Tony: Got it.

Tom: All right. This is Cool. S&P's down 14.

Tony: But they were just down 18. The only thing that has not been, what is it, pseudorandom,

Jacob: Pseudorandom, yeah. Quasirandom?

Tony: Yeah. Is your appearance on tastylive having a negative effect on the market, a 1 to 1 correlation, and I love that. We got Bootstrapping next. Good job, out of you. You're listening to tastylive live.