What does it mean to long the convexity of options?

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

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.

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.

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!

Let me give this a try:

1. WHAT IS CONVEXITY

The change can be explained in many ways mathematically, one way is the Taylor Series. People who uses math in the Financial industry use the term Duration to refer to first order derivative and use the word Convexity to refer to the second order derivative.

Change in Price = -Duration * Delta + 0.5 * Convexity * Delta^2 + ...

In "normal" days, you won't care about the rest of the series as they are negligible and very rarely people care about Convexity even.

It is easy to treat convexity as positive only, but in Finance, there are always two sides, so sometimes convexity can be negative like mortgage backed securities.

(In the US, most home owners can prepay their fixed rate mortgage, like with an embedded call option. When interest rate goes up, the prepayment declines, duration increase, and become more sensitive, when rate goes down, prepayment increases, shortening duration, and less sensitive to decline, sucks both ways)

2. WHY I NEED CONTEXITY

However, when yield curve changes in a non-parallel fashion, things become interesting and high convexity become a safe haven that people pursue as effect is ALWAYS positive. If you have high convexity, hell yeah! You outperform the ones with the same duration when yield goes high or goes low. There is no free meal, for those who knows the yield curve will be volatile but unsure of the direction, convexity is like an insurance that comes with a price. The investors forgive some of the gain and incur losses only when the yield curve stay the same, but should have been any change one way or the other, the insurance pays back.

3. HOW DO I GET CONVEXITY

Bonds with longer duration tend to come with higher convexity, but for the people who try to maintain the same duration, that is where derivatives or options comes in. You can either reduce convexity by selling bonds with embedded options like callable bonds, mortgage backed securities and vice versa. For those who are eligible to buy derivatives without constraint (lots of fixed-income managers are not allowed to touch derivatives), they can purchase future contracts. Future contracts in nature is an EXTREMELY highly leveraged position, the only required investment is the margin to maintain the position.

4. Examples

To give you a sense, a US 2 Year might have a duration close to 2 with an effective convexity of 0.05 while a US 30 year with duration of 22 and convexity of 6 that priced closed to par say $100. However, for a future contract, the price could be only $4 with a convexity of 800 and effective duration of 400!