#4: Convexity
First I will describe – without formulas – what convexity means to market participants. Then I will explain why this is important in a risk context. Lastly I will explain why areas with high convexity dominate in any long term planning or horizon scanning. In the short term convexity is a trading measure. In the long term you need to understand it to stay on the right side of history.
Convexity and why it matters
Convexity is the rate of change of a quantity with respect to one of the inputs. Consider bonds: if they have fixed coupons and principal repayments, then the bond price has positive convexity with respect to interest rates. This means you gain more money than you expect as interest rates fall and lose less when interest rates rise. Other instruments have negative convexity – you lose more money that you naively expect, or gain less, with movements in the relevant input. Call and put options you sell have this property.
This property is linked to others. When you own an instrument with positive convexity, you tend to pay for this with time decay – as time passes and the market doesn’t change you don’t make money hedging the benefit that positive convexity could give you (the sum of all those extra gains and smaller losses on market moves), and as we expect the market to move a properly priced instrument will have this expected gain priced in. As this value drains drains away with time (time decay), you would expect to see the loss offset with profit from hedging the instrument.
Structurers like to create instruments for customers that fit their requirements in a clear way. Often these include yield enhancement (selling options of multiple types and maturities), plus avoiding of losses in certain regimes (limiting of losses via purchase of options with strikes further from the expected market than those sold). Thus many instruments – and structures – have complex convexities. With sensitivities relative to some inputs that are sometimes positively convex, sometimes negative and sometimes mixed in a complicated way with other inputs.
The value of the convexity may also depend on your model and hence your implied expectations of the market (as these model assumptions will be important at 2nd and 3rd order of effects on values and hence on convexity e.g. think about a standard Black-Scholes option pricing models assumption of a log normal distribution).
If you are structured trader – or run a big option trading hedge fund – convexity is the most important thing you deal with. Your plan is to acquire positive convexity cheaply and sell negative convexity expensively (buying low and selling high is most traders plan after all). This seemingly simple idea is pretty complex in practice. The fixed income markets can give you a sense of this. First you describe your strategy in terms of sets of simple options (swaptions, bond options and caps and floors for instance). If you can do this, then you still need to decide what hedge to buy or sell at which option expiry date, at what strike (difference from the expected value at that date) and what tenor (the term of the instrument you would enter at that point). These things are correlated to each other, and hence depend on your view of the world, and the way you model the world and lastly your view of the ‘actual world effects’. The more complex the product the more assumptions and correlations, and possibly impossible-to-offset risks you take. It's fun in a scary sort of way. If you get it wrong – you miss a negative convexity or mis-price some part of the market, or some element of the hedge does not follow the expected relationships in your pricing model, then you can find this can go wrong very quickly. Giving significant mark to market losses.
What has Convexity got to do with Risk Management
If you mis-price one instrument you have a loss – hopefully in the scale of the expected p/l of the instrument. If you mis-price many you have a risk management problem. This is what ‘model risk’ is, pure and simple.
More basically if you have organised your businesses so they have some type of ‘structural convexity’ in them, then if this is a negative convexity then you may have a big problem when the market and prices move significantly or swiftly. In this sense convexity can add to a crisis when the structural convexity is linked to others via correlations, especially where these correlations only exist in the tail of events. What does this mean in practice? In large market moves, correlations (and anti-correlations) that portfolio managers may have been expecting to reduce risk in their investments can turn out to have related moves (driven by similar positions dominating the market at that point) that are contrary to normal day-to-day volatilities and correlations. This is why stress testing is often testing our portfolios with contradictory-seeming moves in markets; in extreme scenarios these contradictory moves can last longer than traders can hold the mark to market losses. Understanding – or guessing – where these critical points and regimes are in the market becomes a critical skill in risk management.
In an even more basic sense convexity is important in the landscape that our firms operate in. Many technological, economic and societal changes happen more rapidly than we expect – or even notice. This means that all longer term predictions are fraught with uncertainty. Yet that uncertainty may be an upside if planned for in many highly convex situations.
The future and what can be known
The first way of working out what the future holds is through extrapolation. You look for trends whose rate of increase or decrease are changing (hence which are convex); these are the ones which are going to grow and dominate or shrink to nothing. Be aware of the error in calculating the convexity though. This term will dominate the timescale of the change in the medium term, so don’t make too many exact predictions beyond the short term. (Often the error in a prediction will be of the order of the prediction itself).
The second way of predicting the future is to spot paradigm shifts. The ability to spot paradigm shifts will at best be sketchy and with a low success rate. This is a prediction of the future that is hard to trust - generally the 'this time it’s different’ fallacy dominates.
Lastly remember that the unexpected is more common than we think, because we tend not to focussed at the point something new happens. When we plan for a future we assume the future is more mean reverting than it will be. You may not know where the variance from that average will occur, but you know it will arrive.
Making sure you are on the right side of history
If you follow trends based on changes whose drivers appear to be positively convex, assume there will be more volatility that you expect and organise your firm to profit in this world, then you will be on the right side of history.
Don’t be on the wrong side.
I realise I may be able to write a book on convexity, but I will leave it with the notes above. Next I will return to the review of the year, and my predictions for the coming year.
Lewis O'Donald
7th December 2023
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