# What does it mean to long convexity of options?

In this Bloomberg video, Curnett talks about volatility and the convexity of options. Specifically, he says;

"The spread between the VIX sitting there at 20 for a period of time and this realized vol of only 10, that's a big spread. Options market makers will pay something to be long the convexity of options; they like to be long and are willing to pay away some of that negative carry."

I understand what convexity means in the context of bonds, but what exactly mean in the context of options, and how does that apply here (ie, the spread between realized and implied vol)?

First lets understand what convexity means:

convexity refers to non-linearities in a financial model. In other words, if the price of an underlying variable changes, the price of an output does not change linearly, but depends on the second derivative (or, loosely speaking, higher-order terms) of the modeling function. Geometrically, the model is no longer flat but curved, and the degree of curvature is called the convexity.

Okay so for us idiots this means: if the price of ABC (we will call P) is determined by X and Y. Then if X decreases by 5 then the value of P might not necessarily decrease by 5 but instead is also dependent on Y (wtf\$%#! is Y?, who cares, its not important for us to know, we can understand what convexity is without knowing the math behind it). So if we chart this the line would look like a curve.

(clearly this is an over simplification of the math involved but it gives us an idea)

So now in terms of options, convexity is also known as gamma, it will probably be easier to talk about gamma instead of using a confusing word like convexity(gamma is the convexity of options).

So lets define Gamma:

Gamma - The rate of change for delta with respect to the underlying asset's price.

So the gamma of an option indicates how the delta of an option will change relative to a 1 point move in the underlying asset. In other words, the Gamma shows the option delta's sensitivity to market price changes.

or

Gamma shows how volatile an option is relative to movements in the underlying asset.

If we are long gamma (convexity of an option) it simply means we are betting on higher volatility in the underlying asset(in your case the VIX).

Really that simple? Well kinda, to fully understand how this works you really need to understand the math behind it. But yes being long gamma means being long volatility.

An example of being "long gamma" is a "long straddle"

Side Note:

I personally do trade the VIX and it can be very volatile, you can make or lose lots of money very quickly trading VIX options.

Some resources:

What does it mean to be "long gamma" in options trading?

Convexity(finance)

Long Gamma – How to Make a Long Gamma Position Work for You

Delta - Investopedia

• Thanks for your detailed response! I didn't think of convexity as gamma, even though I'm familiar with it. It makes sense then to long a (delta-hedged) straddle position to be long vol. Do you know what the "negative carry" refers to; is it the vol decay? – AK. Jun 20 '12 at 19:55
• @AK Sorry can you link me to where you see "negative carry"? – Kirill Fuchs Jun 20 '12 at 20:06
• Yea, it's the last line of the quote I referenced, "Options market makers will pay something to be long the convexity of options; they like to be long and are willing to pay away some of that negative carry." – AK. Jun 20 '12 at 21:18
• @AK It refers to this bit.ly/eZNHdL options hold a negative carry because they devalue as time progresses(theta). If you want more detail I think it would be best to create a separate question so we don't flood this one. I also added the link into the resources for my answer. – Kirill Fuchs Jun 20 '12 at 23:05

Think about having a positive view of a stock. You think it's undervalued but you're too smart to think that once you've opened up a position the market is suddenly going to understand where it was going wrong and start to price the stock correctly, causing the stock to go up and you to make money. Ideally, what you'd like to do when the stock starts going up is to extend your position to ride the trend of increasing share price. However, you have a life and don't want to be hunched over the terminal all day.

Being long convexity fixes this. Buying long-dated low delta options means that once the market starts to move in the right direction the delta (i.e. exposure to the underlying) of your position starts to increase. If you started with a very out of the money, with a delta of 0.01 you could in theory increase your exposure a hundred times as the share price approaches and then exceeds the strike price of your option.

Obviously, this is an idealized and highly unlikely scenario. You'd need a three or four standard deviation move in the underlying — a veritable black swan event — for things to work out as well as this, but the general principle still stands. A long convexity position automatically increases your exposure as your position starts to make money (and vice versa).

Unfortunately, this favourable behaviour does not come cheap. You have to buy time value which you will see erode your returns for ever day the stock does not move. You can offset this by buying very long dated options but of course these are very expensive. Overall, however, having positive gamma is definitely something to try to achieve, even at the cost of some negative theta because it lets you sleep more soundly at night.

I've explained this in terms of calls and having a bullish outlook. Exactly the same applies if buy puts and have a bearish outlook. The details are left as an exercise for the reader.

Long convexity is achieved by owning long dated low delta options. When a significant move occurs in the underlying the volatility curve will move higher. Instead of a linear relationship between your long position and it's return, you receive a multiple of the linear return.

For example: Share price \$50

Long 1 (equals 100 shares) contract of a 2 year 100 call Assume this is a 5 delta option If the stock price rises to \$70 the delta of the option will rise because it is now closer to the strike. Lets assume it is now a 20 delta option. Then Expected return on a \$20 price move higher, 100 shares(\$20)(.20-.05)=\$300

However what happens is the entire volatility surface rises and causes the 20 delta option to be 30 delta option. Then The return on a \$20 price move higher, 100 shares(\$20)(.30-.05)=\$500

This \$200 extra gain is due to convexity and explains why option traders are willing to pay above the theoretical price for these options.

Convexity is what gives options their L or elbow shape. Gamma is synonymous with convexity. Don't let this term scare you. Do you remember concave and convex in geometry? If a shape has curvature (eg a cup or a lense), then it has convexity. A straight line has no curvature, no convexity.

When a call option is deep in the money, it has a delta or slope of one. When it is deep out of the money, it has a delta or slope of zero. To connect the curve smoothly, you need a bend. This bend is the convexity.

In contrast, an underlying stock has no convexity; its delta or slope is always one (a constant), so the change of delta is zero.

Recall from calculus that the first derivative represents the slope of the curve, whereas the second derivative is the change in the slope. A stock has constant slope and a zero second derivative. It has no convexity.

If you buy an option, you will have positive convexity or a smile shape. If you sell an option, you will have a frown shape or negative convexity.

We can now interpret Cornett's comment. Market-makers are usually short convexity because institutions are buying puts to hedge their downward exposure. MMs are collecting premium in the form of time decay or theta. You can think of this income as negative carry because MMs are being paid to carry this position.

A wide spread between realized past volatility of 10 and a future-looking IV of 20 can be explained by institutions aggressively buying insurance in the form of put options or MMs aggressively buying put options to remove excess negative gamma exposure off their books. Rather than earn the negative carry from a larger book, MMs are giving up some income by aggressively offloading some of that risk.

One last note: bond convexity is also curvature (in the term structure), exactly analogous to the curvature in options, both referring to the second derivative.

I don’t like to revive an old post, but this came up in my search, so maybe this will help someone out someday.

Since the maths are very similar, one can use a physics problem as a metaphor. The idea of convexity can be explained well by comparing it to a motion/ displacement problem in physics.

Let’s equate a few things:

Distance = price (or payout) of option

Time = change in underlying asset price

Speed = [change in distance / time] = {change in option price / change in underlying price} = (Greek: Delta)

Acceleration = [change in speed / time] = {CONVEXITY} = (Greek: Gamma)

Under constant acceleration** a particle’s displacement (Change in distance so, change in option price) versus time is: change D = (S * T) + (1/2) * (A * (T^2))

** in reality the maths are far more complex. For example, an option wouldn’t have constant acceleration, but particle motion is far more complex when A isn’t constant, and we want to keep it simple. (Fun fact, the entire Black-Scholes pricing model for options is derived from the study of a special case of particle motion! It’s called Brownian Motion.)

You can see that A, {convexity}, has a bigger effect on D, {the price of an option}, than S (Delta). — Provided that T [change in underlying asset price] is sufficiently large, of course.

In reality, A and S are both functions of T, as well as historical T values, strike price, expiry date, contract type, and interest rates. So the situation gets very very messy. But comparing it to particle motion, the source, always helped me understand the relationships between the variables better. I hope it helps you too!

Convexity refers to vega. Gamma refers to delta. Negative carry refers to time decay.

• Yeah, but where do I catch a crosstown bus? – keshlam Sep 2 '15 at 1:55