When looking to place a trade, it is very important to know your Probability of Profit (POP). This ensures that we are placing a trade that has a high probability of success so that we are able to increase our overall profitability. However, there are many things that can effect our POP in one way or another.

Today, Tom Sosnoff and Tony Battista are joined by Jacob Perlman has the guys discuss Probability of Profit. Jacob explains how POP is calculated from the Log Normal distribution of stock prices. Additionally, they discuss how POP is different for define and undefined risk spreads. Finally, the guys close out the segment by discussing what happens to POP when you manage a winning trade!

Tony Battista: Thomas, we're back my friend the Skinny on Options Math.
Thomas Sosnoff: Cool, Jacob's in the house. How are you?
Jacob P: I'm pretty good. How are you?
Thomas Sosnoff: I haven't seen you since the …
Tony Battista: Yeah, you guys, this is like reunited for you two.
Jacob P: Yeah … it's been a month.
Thomas Sosnoff: Yeah, reunited … Sorry I'm just entering an order. All good?
Jacob P: All's good. How is your trip, your travels?
Thomas Sosnoff: Mine was work. I didn't get the Burning Man experience.
Jacob P: Work …
Thomas Sosnoff: I'm burning myself. I'm about to become the burning man here if this market doesn't turn around pretty soon. It's been an unbelievable … Let's put it this way, it's been an unbelievable market move. What else? Has school started?
Jacob P: No, classes are still a week and a half out.
Thomas Sosnoff: What is that, what kind of a semester …
Jacob P: The quarter system.
Thomas Sosnoff: Oh, the quarters system.
Jacob P: Literally everyone who has to be exposed to it would much rather start a wee earlier and have an extra week off between winter and spring.
Thomas Sosnoff: Sure.
Jacob P: But somehow that's not an option because you have to be off for Christmas and New Year's, so it's just how it fits.
Thomas Sosnoff: Whatever.
Jacob P: It's terrible. Semesters and trimesters are better.
Thomas Sosnoff: Yes, I would think so. Anyway, the Skinny on Options Math. What are we going to cover today?
Jacob P: So I wanted to go back and review probability of profit calculations and then talk about the effect tht you'll get on that going from an uncovered call to a defined risk trade or from managing your winners.
Thomas Sosnoff: You know, thank God you shaved off your mustache. Because I couldn't have handled … I'm having enough trouble with you, I could not have handled the both of you at the same time. You know. I couldn't have handled it. I'm fine with this now, but I just coudln't have done it.
Tony Battista: I like it on Jacob. It actually looks good on him.
Thomas Sosnoff: I agree.
Tony Battista: Mine, on the other hand, didn't look so good.
Thomas Sosnoff: You still have the orange thing going on.
Tony Battista: Socks, yeah …
Thomas Sosnoff: You know, the orange pink thing, everything else. But yeah, we're getting there.
Tony Battista: Getting where?
Thomas Sosnoff: I'm not sure I actually don't know where we're getting. The effects on probability of profit.
Jacob P: Yeah.
Thomas Sosnoff: Okay.
Jacob P: The first thing we need to do is talk about how probability of profit is computed. Right.
Thomas Sosnoff: Yeah, how do you find probability of profit?
Jacob P: Most people find it by looking at Toss or Dough, and it tells them what it is.
Thomas Sosnoff: You don't trip over it? Because I always trip over it somehow or whatever it takes.
Jacob P: The actual computation going on under the hood is this standard probabilistic log-normal distribution computation where the important thing you need to figure out is where your break-even points are because once you know that in this range of the underlying you're turning a profit and in this range of the underlying you're turning a loss, you can just do an integral and for the log-normal distribution, it's not necessarily a nice-looking integral, but it's a doable integral. Like any computer program can do it quickly.
Thomas Sosnoff: At what point in the pricing stream does the log-normal distribution get overshadowed by just the pure bubble risk of … I mean at what point of extreme capitulation does the log-normal risk, like the skew kind of reverse? Or does it?
Jacob P: The skew I think don't think ever turns. Right? I mean as far as the models go.
Thomas Sosnoff: Okay, fine.
Jacob P: It's always going to be a log-normal distribution. But the log-normal distribution has a couple parameters, one of which is that the volatility gives it its shape. If you go up to a high volatility which indicates of a very high base price your volatility is going to be multiplied because it's a percentage, you'll get this much more smoothed out, much fatter, closer, uniform-looking version of the log-normal distribution.
Thomas Sosnoff: I'm not sure that's what I'm looking for but whatever. Looking at the profit graph for a trade, it is easy to determine which prices of the underlying at expiration will turn a profit and which a loss, these regions will be separated from each other by break-even points. I think that's pretty …
Jacob P: Right, I mean in general for an open option or a naked option it's just going to be from the strike and then off .. and then translate by the debit or credit as appropriate in the right direction because it's sloped one or zero all the time. It's the math grades.
Thomas Sosnoff: But we're talking, the reasons for using it is because we're talking at expiration.
Jacob P: Right.
Thomas Sosnoff: OK.
Jacob P: We'll get to managing it later but for now we're just gonna pretend we're holding everything till expiration because it makes things …
Thomas Sosnoff: One of the things, Jacob, we're working on right now is trying to determine what are the chances … what are the statistical chances of being able to manage at a certain level of profits, which I think is ultimately going to be the coolest thing about, kind of, the software?
Jacob P: Right, so that's connected to the probability of touch.
Thomas Sosnoff: That's right.
Jacob P: But there's also a certain amount of timing involved.
Thomas Sosnoff: Right.
Jacob P: You need to be looking and strike then.
Thomas Sosnoff: Right, right, right.
Jacob P: And that part is going to be different to model.
Thomas Sosnoff: That's definitely going to be hard to model. It's taken us forever because it's not easy.
Jacob P: Hard to model.
Thomas Sosnoff: Since the log-normal distribution has an explicit density function, the probability that the underlying ends up in between any two values can be computed by integrating the density function. I don't understand that.
Jacob P: Right, so, this is just a like if you log-normal distribution I think is on the next page it has that graph, or the picture for it, and that picture is the graph of a function, and the function is that one and it's a big messy looking function but it's a function and if you want to know what the probability of your end price, the probability of the price is between two particular values, a and b, then you just do the integral. Or you have a computer do the integral for you.
Thomas Sosnoff: Okay. What's the density function?
Jacob P: It's the function when you have the distribution drawn. It's the function whose graph is that curve.
Thomas Sosnoff: I got it, okay.
Jacob P: It's the function that you integrate, right, it is defined by the bottom line. Is the density function. A function is the density function if it obeys the … following the bottom line … and the important thing there is all you need to know is .. you need to know your volatility, your times expiration, right, the a various parameters. But then once you fix an a and b, you can find the probability of any a and b cropping up in that region.
Thomas Sosnoff: If you're new to tastylive, Jacob is a Ph.D. candidate. You're an assistant professor?
Jacob P: No, no, no. I'm a lecturer in the college.
Thomas Sosnoff: You're a lecturer in the college of calculus. He's our resident genius, right, that's a fair way to explain it and help us break down -- not a trader -- but definitely a math expert.
Jacob P: Clear up the math concepts.
Thomas Sosnoff: Clear up the math concepts as we would say, capable of scoring high on the math tests. Jacob's helping us break down the options models because a lot of what we do with respect to derivatives is based around the sensitivity to math.
Jacob P: Right, and the models and the important part here to know here is the a and the b and you'll stick in your break-even points and whichever regions you have profit right her's a picture of the log-normal distribution, and it's important that people look at it kind of often because it's one of those things you really just want to get a good intuition for. It's one of those things when you're working …
Thomas Sosnoff: I'm learning the hard way, by the way.
Jacob P: Underlines come out looking like this …
Thomas Sosnoff: I hate this graph …
Jacob P: They've got this fat tail, and they've got this bulk that sits kind of low.
Thomas Sosnoff: Is it weird when you wake up in the middle of the night and instead of things that you used to think in the middle of the night, you wake up with this log-normal distribution, and you're sweating, and you're thinking, "I cannot stand another minute of this damn log-normal distribution crap."
Jacob P: You'll be fine; you just got to adapt to it.
Thomas Sosnoff: Oh man. What effect does using defined risk trades have on probability of profit?
Jacob P: There's the immediate thing of, oh, well, you know you're decreasing your chance of risk so it's going to decrease your probability of profit, and that's just sort of the fair trade, but you can be more explicit about it because it's sort of easy to see that whatever it is you're paying for your cover, that's just going to slide your break-even point by exactly that much.
Thomas Sosnoff: Whatever it is you're paying for your cover it's going to change … yeah of course, of course.
Jacob P: Whatever data you need to pay to put your cover on, your break-even point moves by that much and so when you go back to your integral, you just move the end point of the integral. You change a or b by the right amount, and so you'll cut off a little bit but the amount you cut off is based off the log-normal distribution, and the important part of it is that when the log-normal distribution is small, when it's low, then moving around your points doesn't change very much because the amount of area under the curve between the two points way out here is very small.
Thomas Sosnoff: So you add the debit, and you subtract the credit.
Jacob P: Right.
Thomas Sosnoff: Okay. Just affecting your break-evens. The premium paid for the long options covering your shorts will move the break-even points by the same amount, but in the wrong direction. In exchange for capping your potential losses, this makes the integrals for the probability of profit over a smaller area and decreases your probability of profit.
Jacob P: The thing to keep in mind is that your payoff graph is got this, it's just still flat, and then it's slope one, and then it's flat …
Thomas Sosnoff: That's the trade-off, you're trading off limited profitability for a higher probability of profit.
Jacob P: Right, but when you used to have this unlimited probability of profit and there's this long, up … most of that was in this very small part of the curve right and this place …
Thomas Sosnoff: Right.
Jacob P: Where the logarithm was reaching was very small and so that's where you're willing to throw it away because that not actually going to cost you that much probability.
Thomas Sosnoff: Right.
Jacob P: Because it was very unlikely to ever get off the ground, anyway.
Thomas Sosnoff: So I said that wrong. You're going to trade … you're going to accept limitations on profit for just … actual … a higher return on capital but a lower amount of required capital.
Jacob P: Right, and that's what gets you the higher return on required capital.
Thomas Sosnoff: Less capital required, higher return on capital, maximum profits.
Jacob P: Right. You know all those very rare events where those underlying is going to move a tremendous amount -- give up on them. Don't let those eat up your capital. Just give up on them. Let them not matter to you.
Thomas Sosnoff: That's the original design behind options and futures. I mean that was the message behind derivatives before they became kind of the strategic play was the whole purpose for creating an option marketplace was us a small … bet a little, win a lot. Okay, which is clearly, God, it's changed, blah blah blah, but that's the way we used to think 30 years ago. Bet a little, win a lot.
Okay. What do we have here.
Jacob P: This is just once you've done, put your cover on, right, how your payoff graph changes. Right from the long call once you sell your short call also you've got this bend down [inaudible 00:10:26] it's hard to find pictures that fit with the tastylive model of always selling premium most pictures that get drawn are using the buyer’s side so all the listeners should just flip all these pictures, should flip this picture upside down in their head, pretend it goes the other way almost all the time.
Thomas Sosnoff: We just did a segment on implied versus actual using a couple hundred thousand data points just to show how consistent, you know, the actual volatility came in inside implied.
Jacob P: I took umbrage with that study because …
Thomas Sosnoff: You took umbrage.
Tony Battista: Umbrage.
Jacob P: Umbrage. Because they didn't look at all the style deviation things in terms of dollars …
Thomas Sosnoff: That's right.
Jacob P: The log-normal distribution is this goofy shape, and it makes all your standard deviations up and down and not symmetric because it's not symmetric and if you worked in standard returns, right, you'd take a log of all the things. Every time you had a change in dollars you'd take a log of the fraction, then every thing becomes a normal distribution and then you're saying standard deviation, up and down are the same thing, right, and the moves up and the moves down should be balanced.
Thomas Sosnoff: Sure.
Jacob P: I mean this study has enough data it's sufficiently backed up by enough stuff the error is going to be small anyway, but it .. I thought it would be nicer if they just would have done it in log space.
Thomas Sosnoff: Linda, delete this last statement by our guest, we don't need to hear this again. Listen, we do … we have limitations of what we're capable of.
Jacob P: I get there's [inaudible 00:11:49] and I was like, "Really guys?"
Thomas Sosnoff: What effect does managing winners have on probability of profit?
Jacob P: Right, so defining your risk …
Thomas Sosnoff: This is a real key point.
Jacob P: Cuts you down this little cut but the real thing is what does managing winners do? And the thing it does best for you is it lets you replace that integral from your break-even point if you're doing short call usually your profit is from zero to your break-even point.
Thomas Sosnoff: Right.
Jacob P: Your probability of ending up there is just whatever is it. But if you're managing your winners, right, you're gonna take it off a little bit past your break-even point, but if you decide you're going to take it off there then instead of waiting for it to end up in that range you're just looking for it to touch and the probability of the touch is twice as high as the probability of getting there, and that's a pretty quick computation to do off of symmetry. Body in motion.
Thomas Sosnoff: That's what we teach people is that the probability of touch is two times the probability of expiring.
Jacob P: That's not one of those things that's an approximate. There will be a lot of shorthand, quick and dirties; this is actually just precisely the case, it is twice.
Thomas Sosnoff: That's weird; I didn't know that because on our software it doesn't always come up as exactly twice.
Jacob P: Really, it should.
Tony Battista: Well, we do probability [inaudible 00:12:52]; it's a little bit off.
Jacob P: So there will be discreet rounding errors and things.
Thomas Sosnoff: Right. But anyway so .. so you agree with that, by the way, the managing winners aspect -- how important the managing winners aspect is of the whole trade kind of management the whole trade mechanics?
Jacob P: Because instead of getting from your break-even point out to you know out to whatever, 0, infinity, as your range where if you ever get there you're in profit, like that's your probability of profit, instead you slide out your break-even point a little bit out to wherever you're going to manage it at and then you get to double the weight under all that of under the curve.
Thomas Sosnoff: If I'm listening to the show right now, my question is you work with Nobel Prize winners. I mean not that you work with them, but you go to the same school with a bunch of Nobel Prize winners, everybody else. Why doesn't this topic ever come up, why isn't the topic of managing winners based on the probability of the touch or whatever you want to call it, why isn't that part of the whole financial discussion? If it ultimately …
Jacob P: You're always asking why people don't do things? I dont'know why people don't do things.
Thomas Sosnoff: I just don't understand. I don't understand. I don't need an answer. I just don't get it; if it's such a big .. would you say of much of the stuff we talk about .. in order to be successful you have to manage your winners.
Jacob P: If you want to control your probability of profits and keep them sort of high, then yes. There's this question about how much you manage them after, but yes.
Thomas Sosnoff: How come is that? Is it because no one talks about derivatives, so when you go back to standard instruments it's not relevant?
Jacob P: No because you can manage your …
Thomas Sosnoff: Well, of course, you can, but then you would theoretically end up with the same amount of winners and losers almost.
Jacob P: Theoretically, it's all fair. Your managing is going to increase your probability of profits .. and in exchange …
Thomas Sosnoff: We're not talking about P&L, it's going to manage your number winners, that's all we care about. If you're managing your winners, then there will be a point past your break-even point where if the underlying reaches that value, you'll take your trade-off That's our whole philosophy. Again, that doesn't mean you're going to have 8 million winners and therefore turn that into some dollar amount. It just means if you're expected to win 70% of the time, you're going to win 70% of the time plus. Okay. It doesn't mean that's going to be a positive P&L number, that's the point I wanted to make clear. For any value of the probability that the underlying touches that value between now and expiration us twice the probability that the underlying is past that value at expiration. This is something that we built into software in 2001, which is cool as hell because if you think about the rest of the industry, they're still trying to figure out what we're talking about today. Rather than our probability of profit being the integral out from the break-even point, it will be twice the integral out of our manage point, which can be significantly higher if the manage point isn't too far past the break-even point.
Jacob P: Right, so this is the thing, if you decide you're going to manage at you know 80% well then you're not actually going to gain that much because you're going to move your manage point so that even doubling this is not as big as just keeping your original. You're still going to help your probability of profit but the thing that is ignored and maybe the other reason that the software will is display not double, the probability of touch is double, but you've got a little bit of extra ability to win because they're the ones where it never touched but still ended up past your break-even, right, the ones where you were managing it and it still rode until expiration but still turned a profit.
Thomas Sosnoff: This will even underestimate the probability of profit for a managed trade, since it ignores the times when the underlying finishes on the right side of the break-even point but never hit your target. Which is really fair because that happens based on the statistics we looked at earlier a lot more than you think it does.
Jacob P: Well it depends how often how far your manage point is. If your managed point is close to your break-even point …
Thomas Sosnoff: Our manage point is 50% of max profits so that still leaves a little bit of room, but of all the research we've done, it's the most applicable for all the research we've done. The earlier you are willing to take your trades off, the better your probability of profit and the lower the contribution from this effect. We have done at least 20 or 30 studies on this and you can see it all the time and the difference is you need to manage your transaction costs your bid as differential and along with what number are you willing to take.
Jacob P: Right.
Thomas Sosnoff: But again we've always argued if you want to be 90% be 90% if you want to be 80%, be 80%, you know it doesn't matter to us.
Tony Battista: That's the beauty of it. You pick what you want.
Jacob P: Right, but then there's the real world problems, like your transaction costs and things which …
Thomas Sosnoff: That's right your [inaudible 00:17:04] but then the theory is you trade more you get lower rates, you trade liquid products you get tighter markets.
Tony Battista: Good job out of you, Jacob.
Thomas Sosnoff: Jacob is our, again, our resident math expert.
Tony Battista: Guest genius.
Thomas Sosnoff: Guest genius. You can also watch a lot of his videos on tastylive describing lots of things. I like that little wink you do at the end of the things.
Jacob P: Aw, that is my least favorite part of all this.
Thomas Sosnoff: Really?
Jacob P: Well, I felt good the first couple, but eventually I was like, "Well, can't you use the old ones?"
Thomas Sosnoff: All right. Jacob, thank you so much; we'll see you next week; we're looking forward to it. Again, thanks a lot.
Tony Battista: We got a bootstrapper next on tastylive Live, we'll be back in 90 seconds.