Jacob explains the meaning behind fugit, a statistic for American style options that describes the expected time to optimally exercise a position. With complex math, he shows through fugit, Brownian motion, and theta decay when it is the best time to manage trades.

Tony: Thomas, we're back my friend. The skinny on option math.
Tom: Ah yes. We sold some more calls and tests at 1090. The same ones we sold at 990 earlier, that's always fun right Pat?
Tony: That's correct.
Tom: You know Jacob, when you sell things at 990 in the morning and then sell them at 1090-
Jacob: The same morning.
Tom: The same morning, that usually means things aren't going exactly the way you planned.
Tony: You left out the point about buying them at $9.
Tom: That was good too.
Jacob: That's good then.
Tony: Okay, now you're reloading.
Tom: Now we're reloading, but we have other problem children of course. Managed fugit.
Jacob: Fugit.
Tom: Managed…
Tony: Fugit.
Tom: Fugit.
Tony: I'm going to have a lot of fun [crosstalk 00:01:04] time with this one.
Jacob: I could have called it something else but I figured…
Tom: I was thinking of [Pugit 00:01:09] sound, I was thinking of yes we're going boating on the pugit sound. Fugit, flying.
Jacob: Flying.
Tom: In Latin.
Jacob: Yeah, as in time flies.
Tom: As in time flies.
Tony: Tell me you came… TP helped you with that. There's no way you came up with that on your own.
Tom: I took Latin when I was in 11th grade.
Jacob: He has a computer right in front of him.
Tony: You are giving him way too much credit if you think he looked it up because-
Jacob: He has like four computers in front of him.
Tony: He'll tell you what it means long before he looks it up.
Tom: I took Latin in 11th grade mind you, I didn't learn one thing. Have you ever taken a class for an entire semester and you didn't learn one thing? That was Latin in 11th grade for me.
Jacob: I took Latin for four years and learned only grammar.
Tom: I processed absolutely nothing in Latin class, and then the first time I met Tom Preston I said "What's your degree?" And he goes "Well I've got a masters degree from University of Chicago" He's got a masters degree in something I can't even remember, it was business school, in finance. He also goes "I have a masters degree in Latin". I go "Why would anybody have a masters degree in Latin?" He goes "Good question".
Jacob: Spare time.
Tom: I literally could not process one thing. Jacob hang on one second, because one of the things I like to do is make money when Tony doesn't.
Tony: [inaudible 00:02:30]
Tom: Thank-you, thank-you very much. To get a little five points out of those.
Tony: [875 00:02:35] yes.
Tom: Yes, very nice.
Jacob: But so-
Tom: We know we love this more than anything, come on. Anyways so-
Jacob: It's the only part of the show my mom watches.
Tom: Which one?
Tony: When you're on?
Jacob: Until we start actually going through the slides.
Tom: Then she turns it off when you start going through the slides, you're mom?
Tony: That's a good mom, that's what she's supposed to do. It's called supporting you I like it.
Tom: What's your moms name?
Tony: [Felice 00:02:58]
Tom: Felice, hello Felice.
Tony: Felice you've done good by the way. This guy's good.
Tom: Yeah but I'm not sure if he's monetizable.
Tony: We're trying.
Tom: You made a smart baby, the question is can we monetize him? God knows we're working on it.
Tony: She says that's your problem, she just IM'd me.
Tom: Managed fugit… There you go.
Jacob: Yeah.
Tom: I got a new word, Tony's got a new word. He's going to go home today and he's going to go…
Jacob: Forget it.
Tom: Yeah, his wife's going to go "Time flies Pat." And he's going to go "You mean-
Tony: Fugit.
Tom: "You mean fugit?"
Jacob: Tempest fugit.
Tom: "Tempest fugit?" Or "Fugit?"
Tony: Just put mine back at 79.
Tom: Nice.
Tony: Just an FYI.
Tom: Nice, no that's good I'm happy for you.
Tony: No, I don't think you're really happy.
Tom: I am happy. All right managed fugit, your ready?
Jacob: Yeah.
Tom: I in all my years of trading, i have never heard this time.
Jacob: Right, it's-
Tony: Neither have I.
Jacob: It's generally not… a fugits not a commonly considered thing, but it is a statistic that associates to American options. Because in American options you can exercise them earlier, most people don't, but in order to price them you use the binomial tree model and you…
Tom: Give us an example of an American option.
Jacob: I think all of the options you trade are American style.
Tom: Any equity options are American style.
Jacob: Yeah they're all…
Tom: Give me an example of a European option, would be an option that you can't exercise early.
Jacob: Right, but most of the options we deal with are American, which is a little bit unfortunate given the fact that Black-Scholes is built for a European options. Thankfully they come out very close given the fact that you can always take trades off, you know you can put on the opposite trade.
Tom: To be fair, they have long considered that one of the… Maybe a flaw in Black-Scholes, the reality of it is that when interest rates are low, there's very little difference, it's marginal, I mean it's a tiny difference when interest rates are low.
Jacob: Right and we'll sort of see the term that makes that the case.
Tom: Okay.
Jacob: The fugit is… the pre-existing definition of fugit which is the expected time until you optimal exercise for American option. Right, so when we're…maybe go to the next one, we'll see the binomial-
Tom: So fugit is an occasionally used statistic for American style options which describes the expected time of optimal exercise. General it is approximately computed by a binomial model approach in parallel to the pricing of the American option. Now if you have any questions about today's segment it's bat, B-A-T at tastylive dot com.
Jacob: If we put it to the next one we'll have a binomial model and we'll be able to point some stuff.
Tom: Of course. [crosstalk 00:05:32]
Jacob: All right so there's the binomial pricing model right…
Tony: [crosstalk 00:05:34] be able to fugit and forget about it.
Jacob: We're trying to figure out what the probabilities are for these various end prices, and we get those by just following the tree, and then when we go back what we want to know is should we exercise the option then. Well it's worth either whatever it would be worth if you exercise it in the earlier part of the tree, or the weighted average of the later leaves or branches.
Anytime that you come up with, oh it's worth more now than its average is later, then you should exercise it then. So if you just consider those as a time average you get what's the fugit, how long you expect to have the American option before you exercise it.
The statistic exists and has since the 80's, but it isn't super useful because exercising the American options doesn't really get you anything back. Right it just is when you do this thing. But there's sort of a parallel concept for managed trades, which is how long are you going to leave them on? That's actually a useful concept because it tells you how long your capitals going to be tied up for.
Tom: Yeah, we traded a product throughout our careers, it was the [SMP 00:06:34] 100, which was the one of the only… there was a couple but that was the only actively traded one, which was an American style index product.
Jacob: Yeah.
Tom: And…
Tony: You could get in an earlier exercise for cash.
Tom: Which is very confusing to most customers, because it's a cash settled American index option, where there wasn't a perfect hedge but there was other futures.
Jacob: Right.
Tom: It was a huge advantage to the trader, not to the customer. The customer never knew what to do.
Jacob: Right.
Tom: It was kind of a poorly structured… product's not really in business anymore it's very small, OEX. But at the time that…you know so we're very familiar with our own kind of version of what we're looking for here, but know exactly what you're talking about.
Jacob: Right, so that's for American options. But in practice usually when we're trading we're not really worried about exercising the American options, we're more worried about when we're at a certain percentage of max profit and when we're going to take them off.
Tom: Yeah that has to do with the whole managing winners piece right.
Jacob: Right.
Tom: Exactly.
Jacob: So now I want to talk about how long are we going to keep a managed trade on? Because that's a more meaningful question, because you want to get an idea right? This will get you your profit and losses per day, right.
Tom: Yes, would you like me to bring you… we have a cheat sheet we put together. Didn't know that did you?
Jacob: I've seen the cheat sheet.
Tom: Oh you have seen the cheat sheet.
Jacob: It was out last week.
Tom: Oh okay, because the cheat sheet has been really… we've been building a lot of research around it which is quite fascinating. We're going to add to that cheat sheet today are we?
Jacob: A little bit. We're going to get a concept, the cheat sheet has to do with how long, if it's been how long what percent should you be taking it off at.
Tom: That's right.
Jacob: This is sort of the inverse. If you're waiting to take it off at a certain point, how long are you going to have to wait.
Tom: Okay, let's do it. So for managed trades, we really don't care if our options are American or European that is correct, but still might want to know how long to expect to have a trade on. In order to do this, the simplest thing to do is to decide on a price where you'd like to take the trade off, if the underlying reaches that level and call that level Q.
Jacob: Right, so this is not quite the same as taking it off at a-[crosstalk 00:08:41]
Tony: Not the Q we know, but another Q.
Jacob: That's not quite the same as taking it off at a percentage of max profit, because doing that sort of has this moving Q that moves as time goes, and also depends on volatility. There are two difficulties there, one is doesn't depend that much on time, it doesn't move that much with time so it's sort of a fair assumption, and changing volatility we always earn Black-Scholes if it calculates probability, just sort of assume that our volitilities are constant.
Tom: Okay, so what does this mean?
Jacob: So-
Tom: It's what you just said?
Jacob: No, no this is not what I just said. Now I want to do [crosstalk 00:09:14]
Tony: Exactly. Welcome to my world, come on sit down.
Tom: Jacob's going to make me learn, Ruby on Rails coming up next is just a piece of cake.
Jacob: We decided we're going to take our option off if the underlying reaches Q. Right that's the statement we've made.
Tom: Okay got it, yeah.
Jacob: But all of the math is much easier to do if you go to return space right? Black-Scholes says the returns are Brownian motion, which means that the underlying is geometric Brownian motion but, Brownian motion is at least where I'm more comfortable and I think where people have a better intuition.
We'll go over return space by switching Q to this alpha, which is just log Q over the initial spot. Then we have our Brownian motion with drift that determines the returns and it's down there.
I think one of the good take aways for today is to keep that R minus sigma squared over 2 term, that's always the drift in your returns, and that's one of the more important things to keep in mind and it's the most relevant number for figuring out it's expected time.
Tom: Tony reminds me every day. No questions.
Jacob: I don't think it ever occurs but, if you want just the quickest, back of the envelope approximation for how long to keep a trade on, it's alpha over drift term, R minus sigma squared over 2.
Tony: You have to have some testosterone for the alpha, which you do not have [Salsoff 00:10:28] so forget about this equation, you just keep scalping away monkey.
Tom: Alpha over sigma squared by 2 times whatever, piece of cake. Everybody can do that, please tell me that your mom does not understand this.
Jacob: I don't think so, no.
Tom: All right because I would feel really bad, it would take all the manhood I have left which is not very much, and knock it down to you know. Because if the world gets this, I'm in big trouble.
Jacob: No she's a medical doctor.
Tom: Okay, perfect.
Tony: She understands its.
Jacob: She understands different things, psychiatrist.
Tom: Since the time we take this trade off is capped at expiration, we get what, what is this I don't-
Jacob: Here's the actual formula for the expected, here's the full formula if someone wanted to like enter it into Excel or Mathmatica and just get arbitrarily approximations. You would know your alpha, you would know your sigma, so you would just plug in this formula and it would give you the expected time, about how long you're expected to keep an option on.
Tom: So the take away from all this is, don't lose your cheat sheet.
Jacob: Yeah. There's a little bit of a cheat in this formula, which is that drift term, R minus sigma squared over 2, could be positive or negative. If it's positive and you're waiting for the option to go up, this formula is right, and if it's negative and you're waiting for the underlying to go down, this formula is correct.
If it's positive-
Tony: This formula is always correct.
Jacob: And you're waiting for the underlying to go down, or vice versa, there's a little bit of a extra correction you need to make, because there's some chance that… there's additional chance you're going to hold it till expiration. Because there's a larger chance of it never making it to the point you were looking for.
Tom: Okay. I'm pretty sure-
Tony: Did you just say okay?
Tom: I'm pretty sure I've never been more lost.
Jacob: It's okay, the next one's a picture I think.
Tom: Okay, good.
Tony: Good, that's where I excel.
Tom: Hold on, I'm going to put an offer in above because I have these-
Tony: NASDAQ futures?
Jacob: Make you feel better.
Tom: Yeah.
Tony: Yeah, the sale seems easy and these buys, I mean it's like we're painting the bottom. Tread lightly.
Tom: I am treading lightly.
Jacob: Painting the bottom.
Tony: Yeah.
Jacob: There's a picture of this distribution to make it a little bit… Before that I had these formulas. But you can sort of see it has a pretty concentrated spike, this is the expected time to hit a thing.
Tom: So…
Jacob: That spike is at alpha over the drift term.
Tom: Okay, so I'm still not getting this take away.
Jacob: So the amount of time you need to wait for a Brownian motion to reach a point-
Tom: Right.
Jacob: Is if I'm… described by what's called an inverse [dowsing 00:12:55] distribution.
Tom: Right.
Jacob: What's up here is the picture of the inverse dowsing distribution, so that you can try to get an idea for it. You can sort of see it's pretty concentrated, it has a pretty good spike around its mean.
Tom: Does anybody use this?
Jacob: For trading?
Tom: Yeah.
Jacob: Not that I know of, but the idea of managing winners is relatively novel.
Tom: That's true.
Jacob: So the idea of, now when you want to know on a managed winner, you want tp know how long is that going to tie up your capital for.
Tom: If we were building into Dough, I've got to understand this better.
Jacob: Right.
Tom: If I was building something into Dough, which I am right now.
Jacob: Right.
Tom: Which suggests when we're going to build… How we're going to present the optimal times for managing winners, and we're building it right into our technology. What would we use? What we already have? Would we use this? Would we use a combination of both? I'm just confused.
Jacob: The inverse dowsing is probably the correct thing to use.
Tom: The inverse dowsing.
Jacob: Right that's the distribution, and it will just give you the number of… in addition to telling you, so it won't tell you the optimal time to take it off, but it will tell you if you're take it it off, how long you expect to be waiting for.
Tom: Okay.
Jacob: Right, so when you're putting on the trade you know this trade I'm going to manage at 50%, and I'm going to expect to have it on for you know, seven days.
Tom: If there's volatility… is it tied to volatility?
Jacob: Yeah volatility is an important factor, it's got that R minus sigma squared over 2.
Tom: Okay got it.
Jacob: The sigma [crosstalk 00:14:17]
Tom: That's the volatility piece, okay.
Jacob: Right.
Tom: Because as volatility contracts it's got to accelerate.
Jacob: Right you'll expect to have it sit on longer, because it's going to move around less. So you're not going to get your cross points as quickly.
Tom: Right. Got it, okay that part I get, okay.
All right, good still hard, not saying I'm getting very much of it. What do we have here?
Jacob: But I said before that you needed the point you're waiting for and the drift-
Tom: R minus sigma squared over 2, I've got that part.
Jacob: Right, that drift, you needed those to be going in the same direction, in order for the formula to add up, in order for the inverse dowsing to correctly describe these things.
If it's not-
Tom: [crosstalk 00:14:54]
Jacob: There's a little bit of a correction you need to make, and there's sort of two corrections you can make. One of them is complicated and the other ones simple, and so I'm going to put up the simple one, the simple one is more of an approximation, which is just, if R minus sigma squared over 2 is zero then the inverse dowsing becomes degenerate. Which is not great but it has a nice limiting distribution called the [Levy 00:15:16] distribution, the Levy distribution has a similar looking but slightly different shape than the one that was up there.
It's a little bit wider and flatter, it has more concentration out towards the high times, but you can still get a good formula for it and still get the same things.
Tom: I just want to remind people, if you want to trade options and you think you never can because you don't understand this.
Jacob: You don't need to.
Tony: You're okay.
Tom: You're okay. We like to challenge ourselves every week and Jacob does an amazing job of doing that, but you don't have to run right out now and turn in your option handbook okay? You're still okay.
If the drift term is against you, what does that mean?
Jacob: The drift term is that R minus sigma squared over 2-
Tom: Okay, got it.
Jacob: So if that's negative and you're waiting for the underlying to go up-
Tom: Okay.
Jacob: Or if that's positive and you're waiting for the underlying to go down.
Tom: Okay got it. If the drift term is against you, then there will be an additional term in the probability that you hold it until expiration, given by the probability that the Brownian motion with drift never reaches your target point.
That would be for like the worst case.
Jacob: This has to do with-
Tom: This would be like if you have a spread that goes to zero.
Jacob: [crosstalk 00:16:29]
Tom: Or it's max.
Jacob: Right, it just get's away from you.
Tom: Yeah.
Jacob: It never comes back.
Tom: So it's max loss or max profit.
Jacob: Right.
Tom: I'm sorry if it's-
Jacob: If it's max loss.
Tom: [crosstalk 00:16:37]
Jacob: If it's max loss and it just goes way over there, which is a much more likely event when the drift term is against you
Tom: That part, that's easy, because we would take that exact same approach. That's the only condition when there would be no management whatsoever.
Jacob: Right.
Tom: That's the condition for holding.
Jacob: That's when you'd let it run till expiration.
Tom: That's right you have no risk.
Jacob: Right.
Tom: Right, so that part we get.
Jacob: Right, but that's-
Tony: We're good at managing the no risk part. We're fine with that.
Tom: We're good at managing your $2 credits when its trading for $2-
Tony: Yes.
Tom: And you're $2 debit's probably trading for zero. We get both of those.
Jacob: Those are good, and so when those happen there's an extra term you would need in the formula.
Tom: Yeah. [crosstalk 00:17:19]
Jacob: Which I don't have in there, because those are sort of rare events, or unfortunate events, but there's an easier formula which on the next page, I mean easier for-
Tony: Very easy formula, you got a profit in that guess.
Jacob: It's a very similar formula of the same sort of integrals are going on, the same sort of expectation your taking, but the density functions a little bit different it looks like that, remember that alpha is just a log of the point we're taking it off of from the initial point. Right, so it's the returns you need before you take it off.
This gets you, this is the Levy distribution which describes how long it takes a Brownian motion with no drift to get somewhere.
Tom: Yeah, you had me at R minus sigma squared divided by 2, because the…
Tony: The research team says you owe them an apology because they vetted all this for you, to make sure that it was correct.
Jacob: Okay.
Tony: That's a joke.
Jacob: I don't think they've seen this before, I think I walked in and they asked what we were doing today.
Tom: So I think you've blown everybody away, I'm not sure that… well the only person that will try to figure this out actually, is Woody.
Jacob: Yeah.
Tom: Because now he's going to go back and say that the formula that we gave him for figuring out optimal management of winners, needs to be you know the best it can be, so…
Jacob: These two things are side by side, one is optimal management-
Tom: No I get it.
Jacob: And the other is, okay so let's say your managing it like that, how long is it going to be on there for. Right and I think this is a good like bit of extra information that can hand you value so-
Tom: No i'm telling you, he's going to take this and he's going to try to figure out through every single thing.
Jacob: I think that this is the sort of thing which while a person might not be able to understand, well software will easily tell you, these formulas are not that hard for a computer to do, and you can have the software immediately spit out.
Oh you're putting this trade on? And you intend to manage it at this point? This is how long it will stay on for.
Tom: Okay, got it. Jacob that was awesome, I mean it was really heavy today, but it was great that's what we love.
Tony: Thanks Jacob, appreciate you coming in, we're going to take a quick break and when we come back we're going to have [Bootstrapper 00:19:17] next. You're listening to tastylive live.