The Skinny On Options Math

# What is vega and what does it tell us?

| Apr 3, 2014
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The Skinny On Options Math

# What is vega and what does it tell us?

Apr 3, 2014

Jacob, tastylive’s resident mathematician, joins the show to discuss options prices and how volatility affects them. Options greeks are derivatives of options pricing models that measure how sensitive these prices are to changes in other factors. Vega measures price sensitivity to changes in volatility.

Jacob gives the formula for vega, showing that it is always a positive number that denotes how much value is gained or lost on an option position as volatility changes. Since directionless (neutral) positions (like iron condors and straddles) are almost entirely dependent on vega in order to be profitable, Jacob also reviews the mean reverting tendency of volatility.

Since volatility is mean reverting and vega is always positive, we try to sell premium in times of high implied volatility (IV), not low IV. This is because we expect IV to return to its mean, and so being short volatility in times of high implied volatility may allow us to earn the credit from selling options as volatility moves back to its mean. Due to positive vega, the option becomes less expensive.

Jacob also gives the formula for vomma, the second derivative of an option’s price with respect to volatility. This metric looks at the acceleration of price movements with respect to changes in volatility. He asserts that volatility tends to revert faster to its mean the further out it is from its mean, and finishes up recapping why selling premium in low volatility environment does not work when considering vega and vomma.

Transcription: Tony: [music] Thomas. I'm back, my friend. The skinny on options math with Jacob in the house.

Tom: I like it. Listen man, I'm sick and tired of you stealing his boots.

Tony: (laughs)

Tom: I'm tired of it. I can't, I don't like it one bit, OK?

Jacob: Just because we have the same boots, doesn't mean I stole his boots.

Tom: I think the…

Tony: Those are not my boots, those are my galoshes. (laughs)

Jacob: It's an important distinction.

Tom: Let me tell you right now. You go walk about Brooklyn, there's 7,000 pairs of those things.

Jacob: I know, they're great.

Tony: Today's Brooklyn, perhaps. (laughs)

Tom: Can we see this, Linda? Can we get a picture of this? This is great.

Jacob: Behind the desk.

Tony: They're behind the desk. You can't, but they're adorable. They're mine, we have the same size.

Tom: Can you hold your foot up for a second?

Jacob: I can hold my foot up for a second.

Tom: You couldn't see it?

Tony: Probably not. We'll find out, I don't know.

Tom: When you leave here today, we're going to have the guys come in, they're going to hold you down. They're going to take the boots off and give them back to the bat where they rightfully deserve.

Tony: You ride your bike in, right?

Jacob: Not in this weather.

Tony: OK.

Tom: Still raining?

Jacob: It's pouring.

Tom: Give you a hundred dollars right now, you walk around the rest of the day with those boots on.

Jacob: (laughs)

Tom: You don't even have to leave the building.

Tony: You don't have enough money. (laughs)

Tom: 200

Tony: You don't have enough money. You don't have enough money.

Tom: That's a bold statement. (laughs)

Tom: Everybody else…

Tony: You don't have enough money to spend.

Tom: Everybody else in this office will do it for 200.

Tony: I've traded with him for a long time.

Tom: 200, I can get every other person.

Tony: 200? 20. The research team? You kidding me? They're hungry over there.

Tom: 20 bucks, anybody else in this office does it. 200 for you.

Jacob: The research team will do it on a dare.

Tom: How much would it take me? To have you wear around those boots all day? Really, how much would it take? I could get like three hours of footage. How much is it worth? It's probably worth thousands to us. All the filming I could get from that. (laughs) For years it'll hold up.

Jacob: (laughs)

Tony: It ain't going to happen.

Tom: I just want to send one picture to your mom. (laughs) Look what happened to Tony. Can you believe this is your son?

Tony: It ain't going to happen.

Jacob: Respectable. You could try, though.

Tony: Listen, Mark, maybe have a piece called 12, instead of called two, then maybe we can talk. (laughs)

Tom: 300? (laughs)

Tony: Keep going. (laughs)

Tom: Can I get to it, or do I just have to go to four digits?

Tony: You have to go to four digits for me to be there.

Jacob: I can take five. (laughs) We're narrowing it down. We've got orders of magnitude.

Tom: Dammit. I'm not going to four digits for you.

Tony: Told ya.

Tom: Mr. Jacob, are you ready?

Jacob: I think so.

Tom: This is the segment of tastylive where we do some, this is where we get a little funky.

Tony: We let Jacob run!

Tom: We let Jacob run with a little math. With crazy math here. I think you're going to have some fun with today, we haven't covered it. I don't believe we've ever covered this. What's Vega? We don't usually cover the Greeks, but I think for your segment it's kind of fun to look at the math. You are all drenched.

Jacob: It is pouring out.

Tom: Really? OK.

Tony: Man.

Jacob: It is just pouring. I had an umbrella and a rain jacket and I'm still soaked.

Tony: Obnoxious.

Tom: We'll have the helicopter take you back.

Jacob: That would be much better.

The Greeks in general are derivatives of the costs of options with respect to the various inputs that you use to gather the price of an option. If you want to price an option, you need to know the risk-free rate, and the volatility, and the strike and the current spot price. All these things. If you change any of them, the price of the option will change. The Greeks measure how much it changes by. Vega is the sensitivity to volatility, which is, for a lot of purposes, probably the most important thing. Especially for a premium seller. Anytime you're going to do a directionless trade, you're really only trading off of your Vega.

Tom: The easiest way to say it is Vega is the sensitivity index of your position to implied volatility.

Jacob: Right. To volatility.

Tom: To volatility.

Jacob: That's philosophical. (laughs)

Tom: Right.

Tony: Philosophical now.

Jacob: Vega is going to tell you how much value you lose or gain as volatility changes.

Tom: Vega written with the Greek letter?

Jacob: Nu.

Tom: Nu.

Jacob: Is the derivative of price.

Tony: Wait, Nu?

Tom: With respect to volatility. Approximately how much will the value of a position, how much the value of a position will change if volatility changes by a single percentage. There's lots of different places inside of TOS and you can obviously play around with this. You can go to the analysis page. You can play around with volatility multiple ways. You can even do it right from the trade page. We're not going to show you that right now, but you can even do it right from the trade page, and you can adjust volatility.

Jacob: It's one of the things that I really like about the TOS Platform where you can just pull up your Greeks right alongside your trade page.

Tom: That's right. Even on the top of the trade page, you can pull up, you can change the volatility level to see what happens to the current mark price, and then the theoretical value of an option. We built that so that you can see what happens if volatility changes by single percentage or by ten percent. How would volatility change by ten percent? A binary event. Or 20%, or 30%. Maybe a binary event.

Jacob: After earnings, it's going to tank right?

Tom: That's right. Or, two weeks before earnings, it may expand. Directionless positions such as straddles, they're directionless because when you put them on, you're selling both the put and the call, or buying both the put and the call? Both have a 50 delta, which, although there's risks in the position, they're considered directionless.

Iron condors, which replace the usual bullish or bearish assumptions on the future with assumptions about how volatility will change, and are almost wholly dependent on Vega in order to be profitable.

Jacob: They pick up some theta to K, but really, the actual reason to put on a straddler nine condor is because you haven't made an assumption about whether or not the stocks are going to go up or down, but you are making an assumption about what the volatility is going to do.

Tom: Right.

Jacob: That's entirely about Vega. Vega's the thing that tells you what your assumptions are.

Tom: Everything we do here at tastylive has a, I shouldn't say everything, but a lot of the stuff we do. Not our positions that have embedded direction, but most of the stuff we do with IV rank. We're always, our bet is that volatility is a mean reverting…

Tony: Animal.

Jacob: Process.

Tom: Process. Is better. I was looking for the right word. I was going to say number, but I don't think that's right. Volatility is a mean reverting process. Everything we do, or at least things that we do that seem to work better is when you sell something that's rich, it mean reverts to some level of normality.

Jacob: I think that's the great trick of sort of what you do here. Everything else in the market is this sort of inherently fair Markovian process, [martingale 00:06:44]. You don't know if it's going up or down. Betting up or down is sort of just taking these fair bets all the time. Volatility has this mean reverting drift to it. It's always trying to get back to, or at least it appears to be that it's always trying to get back to some sort of stable levels. That means that you can know to do with your Vega. You know what vega you want.

When you decide that you want a positive delta or a negative delta, you have to do that based off a lot of assumptions. When you want to decide if you want a negative or a positive vega, that's based off of just the mean reversion assumption. That's a much more strong and stable assumption.

Tom: Got it. In the black shoals model, the vega for calls and puts is identical. By linearity, the vega for a more complicated position is just the sum of the vegas for its component options. Since the formula is strictly positive, a premium seller is inevitably going to have negative vega.

Jacob: Every call and every put has a positive vega. Of course right, calls and puts have their profit. Their payoff graphs look like this, and so if underlying moves more, they're more profitable. If volatility goes up, they're better.

Tom: If you're [inaudible 00:07:51], right.

Jacob: That's what the call and the put is, so if you're selling calls and puts, you want the opposite of that.

Tom: Of course. I think everybody gets this, but it's nice to see it kind of explained out. It's nice to see with the math and kind of explained out.

Jacob: You can just see the formula is necessarily positive. 2 pi's a number.

Tony: That was definitely a formula I saw there.

Jacob: 2 pi's a number, square root of that's a number, it's all positive. K is your strike price, that's a number. Tau is time times expiration, that's a positive number, square root of that's a positive number. Exponential of anything.

Tony: That's a positive number.

Jacob: Regardless of what's in there, is a positive number.

Tom: In case this is a newsflash for you, most people in the world, I think the world's a smart, I think people are smart. I think most people are smart, but Jacob, most people a little rich for us.

Jacob: The key trick is, all you need to know.

Tom: No, hat's a little rich for us.

Jacob: Know exponentials are always positive. You can take out that whole messy junk and go, "That has to be positive." (laughs)

Tom: (laughs) Saying to review the formula, is a little bit, it's rough. I'm just protecting my boy. That's all.

How does the IV rank inform our ideal vega? This is a great question.

Jacob: This is the mean reversion trick. Really, if we're doing mean reversion, we maybe would rather base off IV percentile than rank, but it's hard to tell, because there's just not enough data collected on either of these two. Certainly if you're in an extreme IV rank, you're also going to be in an extreme IV percentile in any sort of reasonable data set, so you can use one as a reasonable proxy for the other.

If your IV ranks really high, mean reversion means you're expecting IV to come down. You're going to want a negative vega. When IV ranks very low, you're expecting volatility to go up, and you either want a positive vega, or if you're sticking to a strict premium selling philosophy, you want to get out.

You just don't want to be trading then. You don't want to be selling premium then, because you're expecting… there are other reasons you could be doing it. You might have some directional assumptions you're willing to make, you might have a good theta, but the vega's going to be counting against you any time you're in a low IV. That's why low IV is a bad time to be selling premium.

Tom: It's a bad time to be doing mostly anything. We're in that environment right now, which makes it incredibly difficult. If we believe that volatility is mean reverting, then when it is significantly above historical values, we expect it to come back down and should intentionally take on negative vega.

We're not historical vol freaks here at all. We don't go back into the five year history, and things like that.

Jacob: Here I actually mean historical IV.

Tom: Right. Exactly. We set some very short term.

Jacob: [inaudible 00:10:31] For mean.

Tom: Right. We hope to make this all actually user defined, so you can, hey, three months for you, six months for you, year for you. We want to keep it relative. On the other hand, when volatility's low, not only is selling options less attractive because they're cheaper, but we actually expect volatility to start increasing in order to return to its mean. Which means holding negative vega could be very costly.

That's the whole argument. That's why you don't sell stuff when it's cheap. Listen. You don't sell vol at 13%. Yesterday people were saying, hey, should I be buying the Vicks futures here at 14.5? I don't know if you should be buying Vicks futures here, but I sure wouldn't be selling them. (laughs) That's the difference.

Tony: Right.

Jacob: Correct.

Tom: Then who's selling them? People make markets. That's their job. We call those liquidity providers. You know who those are? Those are the people you think have an advantage. Would you want to sell here? Somebody's got to provide the liquidity.

Tony: Correct.

Tom: If you look more closely at mean reversion, almost every model that includes volatility mean reversion does so through an elastic term in the volatility process. This means that beyond weighting volatility to return to its average values, the models predict that it will do so faster the further it is from its mean. The further we're away from the mean, the faster that we will revert to the mean.

Jacob: At least in the models that get used. These are notes, the non-constant [inaudible 00:12:01] like Black Shoals, the minomial price model, these are very well understood, very thoroughly studied. Except for the fact that constant volatility is a bad assumption, are pretty well vetted.

The non constant volatility models that get used are all sort of ad hoc and a little bit more in the experimental end of financial mathematics. All of the ones that there are, use this elastic. It's like a rubber band. That's how they force it. That's how they try to drift it back to its mean. The further it gets away, the stronger the pull back towards the mean gets. At least as far as all the models go.

Tom: You understand that the world of higher education, meaning university level, and also graduate school finance, whether it's from Morton to Harvard to everybody else. The traditional line of thinking, which has never changed, which still exists today is, you buy premium when it's cheap, or you buy opportunity all the time. There's never a mention, if everyone understands the math here, you, as a math professor. It would be almost impossible for you to buy premium.

Jacob: In extremely low vol, I would be tempted.

Tom: If it's at record lows.

Jacob: Right. If we're down in a 5% IV rank or percentile.

Tom: Or lower.

Jacob: Right. I would be tempted. That probably, I'm expecting that to return up.

Tom: You only have duration work against you. You'd be tempted. If you're waiting for, let's just say. You're a smart guy. You're waiting for something to happen between zero and 5%.

Jacob: It's a rare event.

Tom: It's a rare event, and also, that should not the basis of an investment strategy.

Jacob: It shouldn't be, but also, we don't want to say that too loudly, or else the liquidity will stop being there.

Tom: We can say it as loud as we want, because you realize, there are only 50,000 people watching us. We can scream it at the top of our lungs. We can get up on the chair. Just do what you want to do. That's never, don't worry about that.

Jacob: This elastic term, because it gives you this high speed when you're, means that oftentimes when you get something that's going to change a lot, you don't just want to pay attention to the first derivative. The first derivative is sort of this linear approximation. You start wanting to care about your accelerations. Your second derivatives. The second derivative price with respect to volatility is called vomma. More silly names.

In theory, you'd have to pay attention to vomma the same way you'd pay attention to gamma when you're expecting a large shift in the underline. There's something sort of mathematically very convenient about vega and vomma, which is vomma's just a straight multiple of vega. If you have the vega you want, you'll also have the vomma you want.

Tom: On an institutional level, if you're doing vol RB, or if you're managing volatility, like we've had Shelly Nayburg on the show. If your job is to manage prop volatility, which is proprietary trading volatility funds, whatever else it is, all that's relative. There's your vomma.

We're individual investors. We're not worried about vomma. Let's be fair. We don't go to sleep at night going, "Damn that vomma."

Jacob: That's arguing some sort of matter of scale difference.

Tom: It's real.

Jacob: You should be trying to treat your thousand dollars the exact same way they treat their million dollars.

Tom: I think that is the approach that we take. In fact, we actually take it a step further and say, we can be far more flexible. We're not stuck having to do what they do. What you wrote here is, "In times of extreme IV rank, it may even be wise to put in a trade for the sake of its vomma. The derivative of vega with respect to volatility or the the acceleration of a price of an option with respect to volatility.

As vomma as directly proportional to vega, it will already point in the right direction with appropriate magnitude as long as we mind our vegases. (laughs)

Jacob: This just means that there's even more reason to be extremely averse to selling premium in extreme low volatility.

Tom: Yes!

Jacob: In addition to it going up, which will cost you in you vega, it's likely to go up quickly. Vomma's going to take over, and your vomma's also going to be negative.

Tom: There's a reason we don't sell cheap vol. It doesn't work.

Tony: Correct.

Tom: That's why we don't sell cheap vol. There's a reason we sell high volatility, because it works.

Jacob: Every time?

Tony: It's crazy you say that, but it just about does work every time.

Tom: It's crazy that we're still options with respect to, it's crazy that people are still out there going, "Volatility schmolatily. I don't like Facebook here."

Tony: (laughs) Doesn't matter.

Tom: Dude, I quit. That's why I want the exchanges to stop teaching. That's what I want the [unknown 00:16:50] to stop teaching. It's time. Let's move on. It has to, we have to evolve.

Tom: Good job out of you, Jacob.

Tony: That's a good job out of you. We've got a bootstrapper next. You're listening to tastylive live.

Tom: Hold on, before you go. I've got to tell people this. There is a bunch of Jacob tapes now on the Internet. (laughs) They're the good kind.

Tony: What are you talking about, like cassette tapes?

Tom: Yes. Not like the Kardashian tapes.

Jacob: I made mixtapes for everyone I know.

Tony: Lindsay Lohan.

Jacob: I thought I was important.

Tom: There are, I should restate this.

Tom: There are a bunch of segments that we have recorded with Jacob on our new network, Dough TV. Hopefully we'll show them on tastylive pretty soon. That go over and describe some of the basics. You've done, what have you covered so far?

Jacob: We did Black Shoals model, binomial model, the conversion of one to the other. Volatility and explaining what the vix is. These are all very short, one to two minute pieces. If you…

Tom: They're great too.

Jacob: Want to know something real fast.

Tom: They're awesome. They're already up, they're on, you can check them out through Doe. On Doe TV. Again, we'll start to play some of them through the tastylive network as well. They're up there, they're great. 90 seconds to basically, they're a minute and a half to two and a half minutes long.

Jacob: I don't think any of them are over two yet. (laughs)

Tom: They're excellent You want more of Jacob, they're all up there. There's five or six of them right now, and I hope we're adding much more.

Jacob: There's more coming.

Tom: Awesome. Thanks.

Tony: Perfect. We'll take a quick break and come back/ Bootstrapping next. This is tastylive live.

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