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The Skinny On Options Math

The Skinny On Expected Moves

The Skinny On Options Math

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A very common question that we are always asked is what the expected move of an underlying is and how accurate this calculation is. Many people do not understand that the expected move is not the average move, rather it is a 1 Standard Deviation move in the underlying.

Today, Jacob Perlman joins Tom Sosnoff and Tony Battista as the guys discuss the expected move of an underlying. Jacob explains how you can calculate the expected move and the probabilities associated with these price levels!

Tony: Thomas, we’re back, my friend. The Skinny on Option Math, with Jacob in the house. Jacob, what’s up my friend?
Jacob: Not that much.
Tony: Glad to have you back.
Jacob: It’s Good to be back.
Tony: It’s always good to see you.
Jacob: Yeah.
Tom: Doing good?
Jacob: Yeah. Doing pretty good.
Tom: Good.
Jacob: The cold came back and everything froze up again, but ...
Tony: It’s time to go ice-skating in my rink, if you’d like to go ice-skating in the back yard, we could .
Jacob: That’s lovely. Though there’s a rink on the Midway.
Tony: That’s lovely? That’s lovely?
Jacob: That’s lovely.
Tom: That’s lovely that ...
Tony: That’s the Skinny’s way of saying GFY to me.
Tom: Yes. That was a GFY. He says I’m not riding my bike to Glencoe to ice-skate in your backyard.
Jacob: There also is a free ice rink on the midway in Hyde Park.
Tony: Mine’s free too.
Tom: Where is this?
Jacob: On the midway.
Tom: Yeah.
Jacob: In Hyde Park.
Tom: Where in Hyde Park?
Jacob: It’s 59th street.
Tom: Oh.
Tony: That’s the piece of property that you couldn’t buy. That was the one that they held out.
Jacob: Where the Bears used to practice long ago.
Tony: It’s a park.
Tom: Right. Is that what that is?
Jacob: That’s why the monster’s the big one. Long, long ago that’s where they practiced.
Tom: Did you know University of Chicago, I think, was the National Champion one year?
Jacob: Long ago, the University of Chicago was a big football school. Then they knocked over the football stadium to put the library up.
Tony: Long ago, Tom was two-twenty five. What difference does it make, long ago?
Tom: Long, long ago, Mr. [Bat-in 00:01:17].
Jacob: There are these great murals on campus of everyone attending the football game, all the men in hats and all the women in dresses ... Of this full stadium. Now, not so much.
Tom: We live for that. Hang on. While you are talking we're getting a little ...
Jacob: Action?
Tony: Is it a frothy?
Tom: We're getting a little ... I can't even wait, actually. There's something wrong with me. But, yeah, we're getting a little frothy. Short-term frothy means that I'm selling. I justify selling something by saying it's short-term frothy. It doesn't necessarily mean anything.
Jacob: Mean anything?
Tom: Yes. As we know. Not only was the University of Chicago but the whole Monsters of the Midway ... that is where they used to practice before they ... Going back ...
Jacob: Before Soldier Field.
Tom: I have never seen ... I'm trying to think where that is. That's not the main green?
Jacob: It's ... Yeah. It's ... So off the Museum of Science and Industry.
Tom: That's where that is?
Jacob: Yeah.
Tom: And in the winter they put a skating rink up there?
Jacob: There's a rink there all the time. It's only functional during the winter.
Tom: Oh my God. I've never even noticed that. Well, supposedly they built an amazing skating rink downtown, near the Maggie Daley Center, and I have not seen it. Have you been there?
Tony: I haven't.
Jacob: I haven't been there. I have a friend who plays hockey there though.
Tom: The Maggie ... No, no. It's a ribbon.
Jacob: Oh! No. No, no. That one I haven't seen at all.
Tom: No, there's another one on the lake front.
Jacob: Yeah. That's the one I've seen.
Tom: Yeah. That's where they play hockey. Anyway, Tony, we just sold some S & P's at six and one half.
Tony: Seventy-six fifty.
Tom: Seventy-six fifty. Right in there. What are we covering today?
Jacob: Today I want to talk about expected move.
Tom: Expected move. We love a good expected move.
Jacob: I know, right?
Tony: But it's expected.
Jacob: But how expected is it? Also, what exactly is the number that gets spat out. And how does it relate to what you might think as the first definition of an expected move?
Tom: The thing that I think ... What would be important here ... What I'd like to do, I'm hoping get there, is the concept behind probability of touch and probability of reach ... Probability of being in the money, and some form of expected move. Is the coolest part about that just that it's all in real time and we're able to analyze it now in real time?
Jacob: To me, the real time is not that most important of a thing. To me the more interesting thing is how well observed data seems to fit the theoretical models.
Tom: Okay.
Jacob: Probably, as far as actually applying it, the real time nature is more important to people who aren't me but as far as I'm concerned, I'm more interested by how nicely the theory lines up with reality.
Tom: When we were trading professionally, we had ... The closest thing we had to an expected move or understanding an expected move was Delta.
Jacob: Right.
Tom: Essentially, once it got beyond Delta, it didn't mean anything to us. Even with Delta ... Delta was only important more for hedging reasons, for quick immediate adjustments so we could move on to the next trade, so you just could always just risk, define it, risk define it, risk define it. By defining it, it didn't mean it wasn't open-ended.
Jacob: You were just going to Delta flat?
Tom: It just meant Delta flat, Delta neutral get to the next trade.
Jacob: Yeah.
Tom: What you're saying now is, that an individual investor ... We're going to give you some of the math and some of the back end calculations for explaining why expected move is actually pretty accurate.
Jacob: Yeah. And for explaining what it is that ... When someone tells you an expected move is, what number they are actually telling you.
Tom: Okay. Let's see this. What is expected move? I think we haven't covered this, which would be great. If someone asked me for an expected move in an underlying, from now until sometime in the future, you might think that they were asking for the expected absolute value of the difference between the price now and the price then.
Jacob: Right. That's the first definition someone would come up with for what an expected move would be.
Tom: Really? That wouldn't be my first definition. That would be a hard one for me. Looking at the difference between the expected value and the absolute value.
Jacob: How much is it going to differ in the future from now? Then what's the expected value for that? That's all the formula on the board is. That's not actually the formula that gets used or anything. Because in practice, people generally don't compute ... That would be called the expected deviation in most areas of probability or statistics. That's not generally the statistic that people use. Instead people use the standard deviation, which is related to it but different from it. Absolute value is the square root of the square.
Tom: Okay.
Jacob: If you square a number then take it's square root, you get the absolute value of that number.
Tom: Okay. That's right.
Jacob: But actually when people compute the expected move, what they generally do is, rather than taking the square root of the square inside the expected value, they square inside the expected value and that gets the variance. They take the square root outside of it and that gets the standard deviation. You can go to the next slide if you want to see it.
Tom: Because I'm already confused. However, generally when someone refers to the expected move of any underlying, they will use its volatility to compute it as one standard deviation up or down. That's exactly what we do.
Jacob: This is the more standard thing to do. Then there's this factor of the square root of time over year because that's how you ... Because volatility always stays in the annualized form so if you're wondering how ... About the expanse of time you have to scale by the square root of time.
Tom: Our argument has always been using volatility, using current implied volatility, is a genuine ... I'm sorry let me say this. Using current implied volatility in a liquid underlying is a genuine and real way... There's no better expected move ... There's no better way to get to expected move.
Jacob: There's no better way to get to the standard deviation.
Tom: Okay.
Jacob: This formula though is a little bit off, as far as what the expected move ... If we take the current implied volatility as the correct standard deviation, that formula is still kind of off.
Tom: But it's tiny right?
Jacob: For low volatility or small times, the amount it's off by is tiny.
Tom: Right.
Jacob: If you're planning an earnings play or a one week play or maybe even for a month, in a low vol environment, it's okay. But if you were thinking about doing something on a little longer time frame, like for a year, then you're going to start noticing ... Because that's only actually correct up to first order approximation.
Tom: Okay. If that's the case, if we do it the same way every time, then doesn't, over time, it all just average out?
Jacob: What it does is it replaces what's actually a one standard deviation move, with what you have come to intuit as. If you've gotten use to this number means this amount of movement, then okay that's fine. You sort of get used to ... It's often the same way every time. But it's no longer one standard deviation. It's not that sixty-eight percent.
Tom: Right. It's a great point because ... We come from a background where we're used to trading a very limited number of products so the expectation for a certain move, should become very comfortable. Therefore you're able to assess your own risk almost every time you do something. The difference is now, as individual, self-directed traders, we want to do lots of different things. And, therefore, we trade lots of different products we don't have that ...
Jacob: That experience.
Tom: That's right.
Jacob: So you want to have something where the mathematical tools translate across them.
Tom: That's exactly right.
Jacob: If you want to do that then I think you probably want to use a slightly different computation for the expected move.
Tom: Okay. Because that's where we're at right now. We're not building a hundred positions in Apple. We're building maybe five positions in Apple, and a hundred positions altogether.
Jacob: Right. Because you want to have these low correlations, right?
Tom: That's right.
Jacob: You want this wide variety of things.
Tom: Right. And when we want low correlations, we actually want negative correlations, don't we?
Jacob: Negative correlations can help compensate for positive correlations.
Tom: Right.
Jacob: In an ideal, like, central limit theorem-type, super predictable behavior in a sense, you would want independents, which is zero correlations.
Tom: Position correlation, we'd want the inverse. But actual, if you're just trying to create as many occurrences, you want closest to zero.
Jacob: Opposite correlations aren't going to get you more occurrences, what they are going to do is hedge. That's a straight hedge.
Tom: What are hedges? It's just another opportunity.
Jacob: Right. But zero correlations, independents gets you this contracted variance.
Tom: That's what we've been talking about and arguing the last couple days here. That's very interesting. What's wrong with this formula?
Jacob: The thing that's wrong with the formula is that volatility doesn't take place in price space. It takes place in return space, which is the exponential of price or which is the log of price.
Tom: I'm not sure I understand that.
Jacob: This has to do with the Black-Scholes Model. The price doesn't move like a Brownian motion, the percent returns move like a Brownian motion.
Tom: Got it.
Jacob: The volatility is the percent return's standard deviation. This means the price falls, with log normal, should be ... We've done a whole bunch of times, right?
Tom: Right.
Jacob: Near it's peak looks similar to the normal distribution because log and x are both derivative one there, but in actuality, if you start getting to larger shifts, you'll notice a difference. The correct formula ... The other one has the advantage of it is multiplication straight across where as this one has an exponential, but at the same time, you can probably stick it into a calculator anyway, to do your multiplication and a calculator can do an exponential just as quick as it can do a multiplication.
Tony: I feel like I'm in a twilight zone.
Jacob: Microscopic world.
Tony: Somewhere I'm not supposed to be right now. I'm in a bad place. I'm in a very bad place.
Tom: I'm sorry. Just keep looking at Sports Illustrated, you'll be fine. Got to get you that.
Tony: Thank you. Tony from Mexico appreciates the extracurricular activity.
Tom: I'm starting to see, it's subtle?
Jacob: Right, it's subtle. For low volatility and small times, because it's a first order approximation, an exponential is one plus x. It's right. But as you get to a higher order ... If you get to larger values for x, the exponential function does deviate from one plus x.
Tom: Which would make sense why I trade really small in CMG. Or Priceline. Priceline's coming up this week, which makes sense on eleven hundred dollar stock with high implied volatility. Got it. This is the formula. I can't really explain the formula but this is for the small values, short period ...
Jacob: This is the formula that you're used to using which is just current price times volatility, times the time factor. That one's the one that works with ...
Tom: This is all still new for retail ...
Jacob: The correct exponential factor should be E to the volatility time factor minus one.
Tom: E to the volatility time factor minus one. Okay.
Jacob: You multiply the current price by E to the volatility time factor and if you want to know the, what was the move, you subtract one. You have to subtract what it was.
Tom: Explain the volatility time factor.
Jacob: I keep saying volatility time factor because volatility never comes without its square root of t. Because volatility has units ...
Tom: The square root of time?
Jacob: Or one over square root of time and so it always needs a square root of time on there to cancel it out.
Tom: Okay. Ultimately what that does is just give you much more accurate numbers shorter term.
Jacob: Yes.
Tom: Got it.
Jacob: It gives you much more accurate numbers longer term and slightly more accurate numbers shorter term. In short term.
Tom: As opposed to the traditional way.
Jacob: Yes. The traditional way screws up in the long term and is pretty close in the short term.
Tony: It's not you.
Jacob: This sort of ...
Tom: Thank you for confusing us all today.
Jacob: There's a quick knowledge about which way it's going to misestimate. The standard deviation ... The approximation is going to underestimate going up and overestimate going down. The log-normal distribution has bigger moves up and smaller ... Has more frequent small moves down and less frequent but larger moves up than a normal distribution.
Tom: You're saying that it's going to ... It overestimates going ...
Jacob: It overestimates going down.
Tom: Is this why we have the way skew is what it is?
Jacob: No because usually when skew is stated, it's stated in return space. There's this additional ... There's an additional skew. But you can pick up a compounding of that skew if you only ever work in price space.
Tom: I have it. I mean I have the basics.
Tony: Do you?
Tom: Very basic.
Jacob: As long as you know which way it misestimates. The other factor is just like this Jensen's inequality thing, which has to do with the average of squares is always bigger than the square of the average. Which means that the standard deviation ...
Tony: I have a t-shirt that says that.
Jacob: It's just because there is a square and if you look at the average of squares, that line crossing it, and it's above the thing you're averaging.
Tony: No two squares are created average.
Tom: Yes. Very good.
Jacob: So that's just another factor for why the standard deviation will generally overstate ... Will exceed the actual expected ... Deviation from the mean.
Tom: Would that be part of the consideration why implied volatility [Inaudible 00:16:00] actual?
Jacob: No this actually has nothing ... Everybody who works in anything already almost exclusively works in standard deviation rather than expected moves. If you were coming at this from a curious but uninformed position and someone told you an expected move, it's where they would differ from what you're talking about.
Tom: Why does very well respected finance professors and people in academia, given I assume they understand this math, right?
Jacob: I think, yeah.
Tom: Why would somebody say with stock market trading, let's just pick S & P's trading at two thousand, why would they say something that has a one percent chance of happening?
Jacob: When they say expected move, the most important thing about expected move is, this is a real fact about expected values generally, they're not expected in the sense they are going to happen. The best example is usually the roll of a die. The expected value of the roll of a die is three and one half. But you will never roll a three and one half.
Tom: Okay.
Jacob: Similarly, when someone tells you the expected move, they're not really telling you what number they expect it to move, they're telling you some sort of cut off between ... Some percentage is going to move less than that and some percentage is going to move more than that. And if it's one standard deviation, it's like sixty-eight percent to thirty-two.
Tom: I get that, but still it has to put some idea of ...
Jacob: The odds of hitting that amount of movement is, essentially, zero.
Tom: I got that. Knowing it has to average out, whatever that expected move is over time, you can't keep saying that it's going to be at one extreme.
Jacob: No.
Tom: That's what I don't understand. That's what the whole industry kind of does. It doesn't make any sense to me. How expected is expected move? If we either adjust for or choose to ignore the return over price discrepancy it is still relatively inaccurate to expect an underlying to move by its expected move. That's what you were just talking about. The properties of a normal distribution would say that the underlying will move by less than its expected move sixty-eight percent of the time. But the fat tails in the market say that this is likely an underestimation. Realized moves will be less than expected even more often than that to make up for the fact that when movements exceed expectations they can do so dramatically.
Jacob: Yes.
Tom: This goes back to that the whole reason we do what we do, this is the mathematical evidence to support you should have a lot more winners than losers.
Jacob: You definitely should have. The expected move is actually going to capture the bulk of things that occur. You have to be careful because the ones that happen outside of there are where models break down. You have to be careful about that.
Tom: Of course. The only way you can be careful about that?
Jacob: More occurrences.
Tom: More occurrences.
Jacob: More occurrences. And you can hedge.
Tom: And you can stay small.
Jacob: Right.
Tom: I mean the easiest way ... Because you can hold for longer periods of time. Talk about some heavy stuff. While expected move is something of a misnomer, it is not a useless concept. Just make sure that the next time someone talks about it, you know what it means, even if they don't.
Jacob: Yes.
Tom: Very cute. What that means is, when somebody talks about expected move, you're going to say "Hey this is why the math works but it's just an average."
Jacob: Right. It's one standard deviation.
Tom: It's one standard deviation.
Jacob: The price return discrepancy is a problem that is somewhat ubiquitous and it would be nice if it went away, but really it would be very nice if people just started saying one standard deviation move rather than expected move. Because I think the expected move confuses people. Or makes them think the wrong thing.
Tom: Yeah. It does.
Jacob: One standard deviation move is at least correct.
Tony: Glad you guys had a nice talk.
Tom: You know what the fun thing about this is?
Tony: I'm sorry. Don't. He was very sincere too. I like that.
Tom: One of the most interesting things about finance is there's no boundaries to the way ... You could almost create a complete argument/discussion on everything that's one hundred percent made up.
Jacob: Yeah.
Tom: You could start with the Sharpe ration and you could confuse an entire room of ten thousand people ...
Jacob: An entire generation.
Tom: Without saying one thing that's even remotely close to the truth.
Jacob: You can argue both up and down of off every symbol, right? This is the reason to buy and this is the reason to sell, for anything that anyone says.
Tom: Thank you for enlightening us and making us ...
Jacob: Thank you for being you.
Tom: What? Yes. For making us all think that we are ... We knew we were inadequate coming in, but now we're really -
Jacob: You're fine.
Tom: Thank you. Good job, Jacob.
Tony: We'll take a quick break, we'll come back and get a market measure next on commodity and volatility. This is tastytrade live.

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