Why AMM Gamma is not so different from derivatives Gamma after all; or is it? (after-fees edition)

Note: the associated Jupyter notebook is on Github and on Binder.

No fees (recap part 1)

In the previous post, I have shown that — for fee levels of zero — the AMM profile looks very different from a residual profile after Delta hedging: whilst the latter is in first order quadratic (because the hedge takes care of level and slope), the AMM profile has a cusp

This has important implications for the value transfer that is induced by that “Gamma”: when hedging options, the Gamma bleed is relatively independent of the rebalancing frequency: for a perfect square profile it is flat (green dotted line), and for another profile it only moves to the extent that higher order terms become important (blue line). An AMM however behaves at every point in time like a straddle at maturity, ie it has infinite Gamma. As a consequence, for instantaneous rebalancing the value transfer diverges, ie it becomes infinite.

With fees

We have seen above that for instantaneous rebalancing the value transfer diverges because of the extremely violent behaviour of the Brownian motion when looked at over short time scales. We remind ourselves however that this was in the case where fees=0. I will show here that fees effectively operate as a high-frequency cutoff and that with fees the AMM case is similar to — albeit not quite the same as — the option hedging case.

What happens when fees are introduced is that the straddle-like cusp of the AMM profile is pulled apart and transformed into a straddle-like profile, as shown below. I also show a square profile that approximates this particular straddle reasonably well

Here the same chart zoomed in where the straddle-like nature becomes even more clear.

We also see from the chart that there are three distinct regimes with respect to the AMM profile and its square approximation

• In the central area (around x=1), the square profile dominates the AMM profile, ie the square profile is above the AMM profile
• In the intermediate area, the AMM profile dominates, and finally
• In the wings area, the square profile dominates again.

When we look again at the chart drawing rebalancing P&L vs rebalancing frequency (or rather, rebalancing time) then we recognise those three regimes clearly: the red curve starts below the green one (central area), rises above it (intermediate area), and finals falls below it again (wings area)

Comparing this chart to the fees=0 chart (dashed red line) we see that indeed the pole at t=0 is removed, and instead the rebalancing P&L goes to zero:

Below I zoom in on the interesting area around the instantaneous rebalancing point (t=0). We see that the P&L starts are zero flat (corresponding to the flat region of the strangle), then increases quickly when the rebalancing period starts covering the two strikes of the strangle. Finally reverses course and falls again when the rebalancing time is so long that the dominance of the square profile in the wings starts becoming important.

In conclusion — we have seen that the AMM Gamma is not that different from derivatives Gamma in that once one introduces fees (or any other type of transaction cost for that matter) this introduces a high-frequency cut-off on the rebalancing, and entirely removes the divergence. However, whilst the pole is removed, we overshot the target: instead of landing at a nice flat’ish curve, we ended up with a curve where the rebalancing P&L goes to zero when rebalancing frequency goes up. This implications of this are profound, and they actually solve the optimal-AMM-fee conundrum. But this is the topic of an upcoming post.