Tom Sosnoff: Perfect. We are back. I hear her, I'm not used to ... Tom Preston: I know. I'm still getting used to this whole ear phone thing, but thank God Jacob's here. Jacob P: I'm here. Tom Preston: Saving the day. Tom Sosnoff: At least you didn't say you're getting used to a whole ear full. What up Jacob? Jacob P: Not much. Tom Sosnoff: How did you get here today? Jacob P: I took the train. Tom Sosnoff: You took the train. I told Tony, "Tell him that he's not riding his bike here from the South side, okay?" Jacob P: It's clear, it's sunny, the roads are fine, but it's really cold out. Tom Preston: We need to protect the genius. Jacob P: I see. Tom Sosnoff: Remember what we used to tell Roman? Tom Preston: That's right, protect the head. Tom Sosnoff: Protect the head. Tom Preston: Protect the head. We'd get these stories, "Hey, we're going on a helicopter tour." Tom Sosnoff: No you're not! Tom Preston: A helicopter tour? Things like that, protect the head. Tom Sosnoff: When he was our lead developer, we'd always say, when he would go skiing, we'd go, "Protect the head." Tom Preston: Protect the head. Tom Sosnoff: Protect the head. Tom Preston: You can lose anything else, but protect the head. Tom Sosnoff: These guys would do stupid stuff, like jump out of airplanes, you know ... Tom Preston: Well they were ... These Russians, they have a different sense of the value of life than we do and so they would take risks that maybe we would not. Jacob P: I assume they enjoy them. Tom Preston: Well, I think they did. Jacob P: Yeah. Tom Sosnoff: I remember on of our lead guys broke both his legs because they came down too fast on a parachute jump. He was ... Tom Preston: But he protected the head. Tom Sosnoff: But he protected the head. Jacob P: And two broken legs ... You can still sit at a computer and program. Tom Sosnoff: That's right. Tom Preston: The fingers still work. Tom Sosnoff: Exactly our point. Did they cancel school today? Jacob P: No. Tom Sosnoff: They did with Chicago public schools. Jacob P: Oh, yeah. We've only had cancellation for cold once. It was last year, the first day of winter quarter, and I think it was honestly because a lot of people got snowed in wherever they had been over winter. Tom Sosnoff: I don't remember a bad snow last year. I don't remember that. Jacob P: It was just cold last year. Tom Preston: It's just been a little colder. It's been a pretty mild winter here, hasn't it? Jacob P: This winter's been OK. Tom Preston: Yeah. Tom Sosnoff: It's going to be about 70 degrees in Austin today. Just to let you know. Jacob P: 70? Tom Sosnoff: 70 Jacob P: About 70. Tom Preston: It was 80 in L.A. the other day, somebody was telling me. Jules was out there. Tom Sosnoff: I almost kind of prefer this because I don't perspire as much here. Jacob P: Sure, of course. Tom Sosnoff: It's that gross stickiness you get down there. Jacob P: I really do often miss the northwest, where it would be 40 ... The whole winter is 40 degrees, a little bit gray, and it's all fine. Tom Sosnoff: I hate that. Jacob P: I really like it. Tom Sosnoff: Really? Jacob P: Yeah. Tom Sosnoff: Horrible. Anyway. You ready? Today we are going to do ... What are we doing today? Jacob P: I wanted to talk about where probabilities come from. We have these models and we have this software and the software tells us about ... This is 50% to do this or 23% to do that. Where is it getting these numbers from? Tom Preston: The stork. Jacob P: Yeah. Tom Preston: There's got to be something. Is it evolution, creation? Jacob P: The Tooth Fairy comes and leaves the numbers under your pillow. Tom Sosnoff: That's right. All the good little traders and that sort of thing. Tom Preston: If you're not a good kid, if you don't believe ... Jacob P: You're not going to get your probability models. Tom Preston: There's no probabilities for you. Jacob P: Right. Tom Preston: Try that with your kids. No probability models for you. Tom Sosnoff: Maybe I will. Tom Preston: Yeah. Tom Sosnoff: Just to stick it to them. Jacob P: Just [inaudible 00:03:15]. Tom Preston: I lost my ear bud again. Jacob P: It wasn't in your ear, ever. Tom Preston: It's a low prob ... My ear bud's a low probability event. Tom Sosnoff: Anyway. All right. Let's hit it. Tom Preston: Where do probabilities come from? See, TV, in the wrong hands, like Tony's hands, this would be a horrible question. With Jacob and you, I think we can do this professionally. Tom Sosnoff: It's safe. Tom Preston: Where do probabilities come from? Jacob P: All right. The main thing is that we have these models. We have some sort of description about how underlyings or the market or various things move. Usually, they're probabilistic. They're random models. We have some sort of description about how these things are going to behave. After that, we need to think about where we're going to actually ... How to compete with this. How to come up with, all right, now what is 25% to happen? How likely is it to wind up here? Tom Preston: Right. Jacob P: There's sort of 2 general ... There's a number of different techniques. The nicest one is if there's what's called a density function. Some random variables are sufficiently nice that we can get a nice function, who, when you graph it, then describes the random number in the sense that the probability that the random number is between any two points is just the area under the graph. The total area under the graph would be 1 and the area under the graph between 1 and 2 is the probability that the random number is between 1 and 2. Tom Preston: Got it. Tom Sosnoff: No matter what that distribution? Jacob P: Right. That is what makes a distribution. The variable determines the distribution, the distribution determines the variable, and some variables have distributions. Nice ones do. Tom Preston: All probabilities in finance ultimately come from models which are specific mathematical descriptions of how various objects in the market, most often the prices of the underlyings, tend to change over time. When these models use randomness, the state of the future needs to be described by probabilities of certain things occurring. It was really interesting. A guy sends me ... One of our viewers sends me an email yesterday and it was of a Bloomberg news report that discussed the probabilities of a certain underlying hitting ... Basically ... What the guy who was reporting it was talking about, he was ... It was a discussion over an analyst that was making a decision with respect to some stock variance. A potential high and low for this one stock. The analyst was saying one thing and then this reporter, who I guess was a reporter or an analyst for Bloomberg, he actually said, "Well, that sounds good, but here are the real numbers." I thought to myself- Tom Sosnoff: Was he a viewer? Tom Preston: This person thought he was a viewer because they sent it to me and said, "Oh my God. You're finally making a difference because this columnist just called out this analyst in a very nice way, suggesting that his 4.68 range, whatever it was, was ridiculous because here are the real numbers." I thought it was really interesting because I just don't understand how any of this is that hard. How hard it is to apply it to the common sense piece. Tom Sosnoff: That's a different question because this has been out there for ... Jacob P: Since the beginning of time. Tom Preston: Of course. Tom Sosnoff: I mean, when we build the image, you can get it for free off Think or Swim. Tom Preston: Of course. Tom Sosnoff: Why hasn't ... That says more about the industry and the way people are paid and ... Tom Preston: It says more about- Jacob P: There's a flow of, you have the model, then the probability, then the probability's going to be communicated to the people, the people need to use those [crosstalk 00:06:59]- Tom Sosnoff: If it doesn't benefit you ... In other words, there's no ... In other words, if- Tom Preston: It shows you how powerful the fear message is. Tom Sosnoff: That's what it is. I think that's what it is. Tom Preston: It's not a question ... People aren't going to sit here and go ... No one's going to write us an email and say, "I complete object to everything Jacob just said. This is ridiculous." Tom Sosnoff: Yeah. If Jacob's wrong ... No, you can say, "Well, gee, Jacob. Maybe your assumption about the ..." You can maybe have some mathematical disagreement, but the logic is correct. Tom Preston: What happens is everything gets overridden with the whole fear argument. It trumps everything. It just ... Oh, you're right, I can't do this myself. That's all. Jacob P: It seems confusing. Tom Preston: It does. If I could, everybody would do this, and this wouldn't even be a question. All right. Let's keep going. For very nice models, the future values will have density functions. That's what you just talked about. The non-negative function ... I don't know. F's the function? Jacob P: Yeah. Tom Preston: Where the total area under the graph is 1 and the area between the numbers is the chance that the future number will lie between these numbers. Jacob P: Right. The probability ... If capital X is some random number, then the probability that X is between A and B should be the integral of F from A to B. Integrals are the thing from calculus where you look for the area under the curve. If you graph the function to the left, then the area under the graph is it's integral. Tom Sosnoff: Jacob dreams in integrals. Tom Preston: Yes. I think he does. Tom Sosnoff: Which isn't bad. Tom Preston: What do you dream in? Tom Sosnoff: Fear of goats, at this point? Jacob P: Goats? Tom Sosnoff: I'm terrified of them. Tom Preston: I dream in- Tom Sosnoff: I have goats now, Jacob. 3 goats. Jacob P: They're harassing you? Tom Sosnoff: In my dreams, yes. Jacob P: OK. Tom Preston: As you get older, you dream of how long you can sleep before the next time you have to go to the bathroom is. Tom Sosnoff: Jacob doesn't need to hear this. Jacob P: I don't want to scare the children. Tom Preston: The guy's half our age. Let him live his life in joyous- Jacob P: In ignorance of my future bowel problems. Tom Preston: He doesn't need to know this. Tom Sosnoff: I never worried about it either, but now I'm just starting to lose it all. Jacob P: You just have to share with everyone. Tom Preston: Yes. For example, you talked about him dreaming with density functions. Jacob P: That was all you needed. Tom Preston: For example, if the price is going [crosstalk 00:09:08]- Tom Sosnoff: I dream in density functions. Would that make more sense? Tom Preston: Let's get through this. Before Tony left yesterday, he talked about himself dreaming in density functions, Brownian motion, whatever. Tom Sosnoff: I would sell that until my hands bleed. Tom Preston: For example, if the prices of underlyings moved as geometric Brownian motions, as they do in the Black-Scholes model, then, at a fixed time, the price will be logged normally distributed and have a density function. What do you mean by this? Jacob P: Right. The general premise for the Black-Scholes model is that the returns functions as a Brownian motion. Tom Preston: OK. Jacob P: Which means that the actual amount of money or the value of the underlying, because of the geometric Brownian motion ... Which is literally just defined as the exponential of a Brownian motion. Tom Preston: OK. Jacob P: Brownian motion is pretty much the nicest random object that we have. Anytime you look at it's future, it's normally distributed. Tom Preston: All right. Call it tractable. Jacob P: Tractable. It's the most convenient. It's nice. It solves a lot of problems. It computationally works very well. The geometric Brownian motion is a little bit messier, but still, because exponential is a nice function and the Brownian motion's so nice, it still is nice enough that it cuts a density function. The normal density function is one that I think a lot of people have seen a lot of times. It's that normal bell curve. Tom Preston: Yes, absolutely. Jacob P: The long normal distribution has a different density function and the formula for it is up there, but there are pictures of it on the next one so you can- Tom Sosnoff: Let's go to the next slide. Jacob P: Yeah. There are some pictures of just different levels for the mean and the variance of- Tom Preston: Which one is the one that you're talking about? Jacob P: All 3 of those are different log normal distribution curves. Density curves. They just depend on different means and variances. Tom Preston: In order to calculate the area under the curve between the specific values, the most common thing that happens is that a computer chops the base into very narrow rectangles and uses the sum of the sizes of these rectangles as a good estimate. Jacob P: Those have to do with how do you actually use these density functions. Tom Preston: OK. Jacob P: Right. I said that you want to find the area under the curve, but asking for the area under the curve is sort of a weird question to ask [inaudible 00:11:11] because these curves have weird shapes. They're sort of these weird lumpy things. The theory of calculus is based originally around this idea and it works pretty well. There were many years of humans getting very clever and tricky about it and then computers got good at it and now computers do it way back in the oldest original definition way. They just grind it out. You want to know what the area under that curve is? It goes, "I don't know. But I do know the area of rectangles, so I'm just going to see ... I'm just going to make a really fine thing out of rectangles that looks a lot like that curve and see what the area of that is and call that good enough." Tom Preston: They don't actually program integrals into computers? Is that what you're saying? Jacob P: There are some things that will do what's called symbolic integration where it will get you the axis and do it how a human would have done it. For a lot of things, there is no closed form analytic solution. About as good as you can do is a numerical approximation. The numerical approximations can be made arbitrarily close, very quickly. Tom Preston: Computing power, at this point, is so cheap. Jacob P: Right. Tom Preston: Just use that. Jacob P: Also, there have been good algorithms. It's a pretty easy problem to give it a function, get a very, very good estimate for it's area under the graph. Tom Preston: Do all models have associated densities? Jacob P: Right. This is the problem. The Black-Scholes model is the really nice one because of Brownian motion, geometric Brownian motion, all of these things are normally distributed, [inaudible 00:12:35] distributed. We have this nice density functions and we can always do these integrals. In practice, a lot of models who try to take into account more issues, wind up ... Maybe they have some sort of theoretical density, but they don't have a closed form, nice, easy solution density where you can actually plug into a computer and have it do integrals for you in the same way. When you want to run into models that don't have density functions, you have to come up with something else you're going to do because you can't just ask a computer to estimate an area for you anymore. Tom Preston: Finding explicit density function for real world random objects is essentially incredibly fortunate- Jacob P: Right. Most things that happen in real life won't have these closed form density functions. Tom Preston: Yeah. I understand that. Many models, including various generalizations of Black-Scholes, describe how the price of an underlying may evolve over time, but they don't have a nice closed form expression of future probabilities. However, if we trust our random algorithms, then many simulations can be run on the potential ways that prices might evolve. My question for this is, let's say you don't have a derivative. How do you do it then? Jacob P: All of these models that are generalizing it- Tom Preston: They're all based off derivatives? Jacob P: No. They're based off of ... They sort of are. They're based off of what's called a stochastic derivative, which is allowed to be much rougher. A lot of functions don't have derivatives as a rate of change, but they do have ... Being stochastically differentiable is an easier condition. It's- Tom Sosnoff: I think what you're talking about is derivatives as an option. Tom Preston: I am because- Tom Sosnoff: Not derivative as- Jacob P: As a rate of change. Tom Preston: What happens is as long as you can give Jacob a mean and a variance- Jacob P: Yes. Tom Preston: You can apply that to a density function and get a probability. The trick is saying, "Show me a product, an underlyer, that has no options." As option traders, we say, "What's the volatility?" We look at the options the derivative implies. Jacob P: Right. [crosstalk 00:14:38] backwards about how we get these probabilities. Tom Preston: Right. How are you supposed to figure ... That's kind of where I'm getting to. It all makes ... Tom Sosnoff: Bitcoin. Tom Preston: Right. [crosstalk 00:14:47] Tom Sosnoff: What's the density function for Bitcoin? Jacob P: Zero. Tom Sosnoff: Whatever, right? Jacob P: Delta's back at zero. Tom Sosnoff: Right. That's just it. There's no options. You and I can guess about what the volatility is or we can look at historical price changes, but what's the number? Jacob P: Even when you have options to get an implied volatility, you still don't know. You just have to come up with a good guess. Tom Preston: That's exactly right. You never- Jacob P: At least it's tradable. Tom Sosnoff: You only have a confidence interval in the future. You are this confident about the future. Tom Preston: At least knowing that it's tradable- Tom Sosnoff: Of course. Tom Preston: ... Gives you some confidence to make that- Tom Sosnoff: This is the difference between let's say, doing the math part and figuring out the volatility and the [inaudible 00:15:31] and making money. Tom Preston: People always ask ... People always have this big question about finance. This is the, again, leading back to that whole skeptics thing. What value do you add? What value does the options ... What value- Tom Sosnoff: I get that question a lot. Tom Preston: OK, but the answer is, in the real world of ... If we're going to be proactive as individual investors, understanding the contexts around the strategy we're about to use and the reasonable parameters of expectations or reasonable expectations or creating reasonable expectations for ourselves, then what the options do is they provide that basis for making reasonable bets. Or no. Am I wrong on this? It sounds like ... Don't you need those markets to make reasonable bets? Jacob P: If you want to ... In order to try and place fair bets ... If the objective of the market is to provide everyone with these fair bets, these efficient market hypothesis type things, then you do need this high liquidity. Tom Preston: Of course. Jacob P: For that, the options are just much easier to trade because they don't require as much capital or investment. Tom Preston: Otherwise, every single person and every single ... I've always wondered why you can't apply this kind of specific theoretical pricing to things like sports betting. Tom Sosnoff: Of course you can. Tom Preston: Of course you can, but there's not a 2 sided market to know ... Tom Sosnoff: Right. Tom Preston: It's not nearly as exact. It should be. Jacob P: You don't have bookies betting against each other, right? Tom Preston: That's right. Jacob P: If you had bookies betting against each other, you'd get much better odds. Tom Preston: They're not taking both sides of the ... You're a straight facilitator. Tom Sosnoff: Yes. Tom Preston: That's all. It's the same thing. Tom Sosnoff: Your Goldman Sachs. Tom Preston: The comparison is, in the world of professional trading, professional traders don't trade with each other. Just like professional poker players don't play poker with each other unless it's a tournament or something. They're waiting for the person that doesn't know. Tom Sosnoff: That's where the edge is. That's a defining edge. Tom Preston: Of course. My point is, the reason that these markets are so good and so special is because you have this secondary option market. Tom Sosnoff: Yeah. Tom Preston: All right. Whatever. I'm just- Tom Sosnoff: I think this is a separate- Jacob P: It's a lot. Tom Preston: My doing many many of these simulations, then counting the proportion that do do something, such as make more than 50% max profit, we can estimate what the probability of that happening is by just looking the proportion. Such methods are generally known as the Monte Carlo. Jacob P: Right. This goes back to when you have a random model that doesn't admit a density formula, it usually does so because it describes how a thing moves. It says at each time, this has a little bit of a push this way and a little bit of a push this way and there's some randomness to it. If you can model that randomness well enough, like you have a good coin you flip or you have some good algorithm that produces decent random numbers, then rather than needed to figure out the density formula and doing an integral to figure out it's probabilities, you simulate it. You simulate it thousands of times. If, at the end of thousands and thousands of simulations, 75% did this thing and 25% of them didn't do this thing, then you have a pretty good idea that the odds of that happening are probably, if your model is right, are 75% or close to it. Tom Preston: That makes sense. Jacob P: That is Monte Carlo. Tom Preston: I'm not sure that's that easy to do, but ... Tom Sosnoff: It's easy to do with computers. Jacob P: Right. Tom Sosnoff: That's what it is. You could do it on pieces of paper if you had the time. Tom Preston: How trustworthy are the probabilities? Jacob P: This comes into the question about ... There are 2 problems here. The first one is relatively minor and it's how good are your Monte Carlos and how good are your numerical integrators. The answer there is very good. Tom Preston: They're good enough to trade on. Jacob P: Right. Tom Sosnoff: They're good enough depending on what your inputs are. That's what [crosstalk 00:19:17]- Tom Preston: They're good enough to trade on. Jacob P: How much do you trust your probabilities is how much do you trust your models and how much do you trust your parameters? Tom Preston: Hold on. If something's good enough to trade on, that's a real ... Tom Sosnoff: Yeah, OK. Yeah. Tom Preston: It's very trustworthy. If you're willing to put up X amount of money, it's- Tom Sosnoff: In fairness, most Monte Carlos are done by, let's say, banks. Traders don't use a whole lot of Monte Carlo- Tom Preston: I'm not suggesting Monte Carlo. I'm just saying how trustworthy are the probabilities. Tom Sosnoff: Banks put 120 parameters in because currency- Tom Preston: OK. Whatever. Tom Sosnoff: It doesn't matter. Anyhow. I'm sorry. Let's keep moving on this. Jacob P: The question on probabilities comes down to how much do you believe in your models and then once you've decided you believe in your model, how much do you trust the parameters. There's sort of a conflict here because Black-Scholes isn't a perfect model. It doesn't have any room ... It has a constant risk-free rate of return, it has a constant volatility, it assumes that everything is going to be Brownian motion, which may be it isn't quite. Maybe it's only kind of like that. It only needs 2 parameters. It only needs a volatility and a risk-free rate. That's all it needs. If you can feed it those and you have good estimates for those and maybe you decide that IV gives you a good idea of what volatility is, then anything the Black-Scholes model tells you is really what the Black-Scholes model tells you. Its problems notwithstanding, you at least trust it. When you start trying to relax these things. The Heston model, which includes a stochastic volatility term. There comes a problem where now the model does a much more realistic job of describing the things are actually moving, but now it needs a whole bunch more parameters. Now, not only do I need to describe the volatility of the underlying's returns, I also need to describe how that volatility changes over time. That's going to need its own mean and variance and more parameters. Every time I have to have these more parameters, these aren't observable things that you can see in the world. They're not the temperature where I can put a thermometer out or a barometer where I can measure the pressure. They're abstract concepts. For volatility, there's only one at a time. We can look at the price and decide to reverse engineer it and try to come up with a good guess. Once there's more than that, once there's a whole space of parameters, you're sort of at a loss to come up with a good way of describing, of knowing which ones are good. Except for back testing, but then we've done segments before about how this is sort of extra recursive where of course you can find numbers that make it look like it fit the past and that doesn't' necessarily give it any predictive power. Tom Preston: If we trust the models, then both of these methods are actually very good. Especially because of how fast computers are, which is what you're saying. The numerical approximations to integrals is one of the oldest problems in computation. At this point, we are very good at estimating the area under a given density function. Similarly, it is possible to run a very large number of Monte Carlo trials and wind up with a close approximation to the predicted probabilities. This is everything that you just said but here's the rub. Probabilities are necessarily dependent on the models that we use and we can never be totally confident in a particular model. Maybe they make too many simplifying assumptions like Black-Scholes, or maybe they have too many adjustable parameters like the Heston model. For this reason, it's really important to have a clear idea of how far you trust your models when deciding how far to trust your probabilities. This is, again, another reason why I love the whole trading space because isn't the answer to this that if there's 25 ... Nobody runs the same model. They all run their own variations of a model that's also fit around your positions. Even though you assume most people have the same positions on, they idea of how much do you trust your models is how big is your market? Tom Sosnoff: That's exactly right. Tom Preston: Right. In other words, if you have no position on, are you 10 up or 100 up or 1,000 up or 1,000,000 up? Tom Sosnoff: We're not the only people trading the S & P circuit. Jacob P: This is perhaps ... The biggest different between trying to use math to describe the market and trying to use math to describe physics is that when the baseball gets hit and I write down an equation trying to describe how the baseball flew, that doesn't change how the baseball flew. Tom Sosnoff: That's right. Jacob P: When I write down a formula trying to describe how the market's moving and then people start trading off of it, this does change how the market moves. Tom Sosnoff: Here's one of the things. This goes back to the whole idea of curve fitting because I think there's some curve fitting applied to the choice of these things. Jacob P: Right. Tom Sosnoff: When people see perceived failures in Black-Scholes or the normal distribution or the large normal distribution of prices, they start scouting around at some of these other distributions because they see short-term behaviors and, "Oh. The market's changed now. I'm going to find a new distribution that predicts the values better, that does a better job at modeling." You know what? They don't work. Either because they require too many parameters that you're still guessing at. What's volatility going to be in the future? Good luck. Tom Preston: This is also why, if you think about it, sports gambling should be gigantic but it's not relative to the amount of notional money that's bet in the stock market. Tom Sosnoff: Right. Tom Preston: The stock market has this incredible depth of liquidity and all these players. Jacob P: It also has some claim at providing some sort of service to society that sports betting never even tries to say it has. Tom Sosnoff: Right. Tom Preston: Let's not push it past what it really is. Me going to make a bet at Caesar's Palace, $500 on a college football game, is- Jacob P: Is the same. Tom Preston: Yeah. It's not that much different. The difference is that people ... There's so many models and all the models are trusted and so you end up with this- Tom Sosnoff: They all get to basically the same option price. Tom Preston: That's exactly right. Tom Sosnoff: That spider market and that option is a penny wide. Tom Preston: That's why the world trades here. I don't think that we've figured that out. I still talk to exchanges all around the world and they have no clue why they can't get any business. None. This is a whole other discussion for another day, but they can't figure out ... Our stock exchange is all got the greatest blah, blah, blah. Jacob P: You want to trade on the busiest one. Tom Preston: Exactly. Jacob P: Right. It's the Facebook problem. It doesn't matter is the best social network. The best social network is the one that everyone else is already on. Tom Preston: Thank you. Jacob, thanks. We're way over our time. Jacob P: Yeah. Tom Preston: That was awesome. Again, just another crazy interesting discussion. Thanks so much. Jacob P: Great. Tom Sosnoff: Stay warm out there. Jacob P: I'll try.