Risk Management · Lesson 10 · Advanced
Convexity
You distinguish convex from concave payoffs and recognize why leverage is structurally concave.
With a free account: interactive chart exercises, quizzes with answers, progress and XP.
Convex versus concave
Convex means: capped loss, open gain, dispersion helps you. Concave means: capped gain, open loss, dispersion hurts you.
Leverage creates the deadly concavity. A 5 percent move against a 10x position is not a 5 percent loss, it is 50 percent.
Double the move and see whether the loss rises by more than double.
Interactive exercise: here you learn right on the chart, with feedback on every click. Sign up freeto start it.
Test yourself
What is the practical conclusion from the concavity of leverage?
- Leverage up, then more gain
- Keep the loss side linear and capped, keep the gain side convex
- Does not matter, on average it evens out
Sign up free for the answers with an explanation for each option.
The key points at a glance
- Concave: capped gain, open loss, dispersion hurts you.
- More than double the loss for double the move means the position is fragile.
- With leverage you do not experience the average, you experience the worst path.
Deep dive
How to spot a concave payoff: the doubling test
You only need to measure the acceleration of your losses. Mentally double the move against you: if the loss rises proportionally, the payoff is linear. If it rises more than double, it is concave and your leverage is too high.
A 5 percent move against a 10x position is a 50 percent loss, and the next step toward liquidation comes faster than the one before. Taleb: whoever trades leveraged without a stop has written a short option on volatility.
Why leverage makes you live the worst path
Taleb's dice example: the square of the average payoff is 12.25, the average of the squared payoffs is 15.17. Convexity and dispersion create a hidden edge of about 24 percent. With concave payoffs this flips into damage.
For the leveraged trader: you do not experience the average, you experience the worst path. You can be right over the month and still get liquidated, because that one extreme move catches you first.
From concave to convex: what you actually change
You can actively shape the form of your payoff. A trade becomes convex when the loss is hard-capped and the upside stays open.
- A fixed stop-loss caps the loss at your known 1R.
- A trailing stop lets the winner run uncapped to the upside.
- Less leverage pushes the liquidation threshold far away.
- On the demo exchange, run the same trade with high and low leverage against a vol spike.
Sources: Taleb, Murphy, Burniske
Make this lesson interactive
Sign up for free and learn with click exercises right on the chart, quizzes with explanations and saved progress. Then you practice everything risk free on the demo exchange.
100% free, no payment details.