Why Loss Versus Rebalancing (LVR) and Impermanent Loss (IL) are essentially the same
PLEASE ALSO SEE THIS POST FOR AN IMPORTANT AMENDMENT
I recently wrote about the AMM Triangle that establishes a relationship between what I’ve call Gamma Income, Fee Income, and Option Profile. In this post I want to related those items to well-known measures in the AMM space, notably Impermanent Loss (IL) and the recently popularised Loss Versus Rebalancing. TLDR: they are the same. If you want to know more, read on.
The Impermanent Loss (that should really be called Divergence Loss) is well known and described in more detail for example here (page 30). The key formula is
Above, xi is the normalised price ratio of the two assets (it is unity to begin with), the fraction on the left is the HODL term when you hold the 50% in each asset, and the interesting term is the square-root term on the left: that is the actual value of a CFMM portfolio, and this is where the optionality of AMMs comes from. This optionality is of course linked to the fact that the AMM operates a replication strategy aka a delta hedge, and it can be shown that the strategy “at every time hold 50% of your assets in either token” does indeed replicated the square root profile. See here (chapter 4, page 19) for a more thorough discussion of this.
The concept of Loss Versus Rebalancing (LVR) AMMs has been proposed in this paper and explained for example in this thread. The key formula here is the definition of LVR which is as follows
The definition above is that of the instantaneous LVR, and to obtain the macroscopic figure one needs to perform the usual stochastic integration process outlined in the paper that is usual to Brownian-motion based finance.
The term above is of course well known — it is the operative “Gamma” term of the Black Scholes equation (see here, page 19)
As a reminder the two terms to the left of this are the cost of borrowing for financing the delta hedge and the interest income on the option premium respectively, and whilst important in the real world, we don’t really care about them if we only want to study optionality. Below we show the same formula rewritten with the so-called “Greeks” which are explained in more detail here (page 19–20)
People often think the the groundbreaking part of Black, Scholes and Merton’s work was about pricing options, but this is not entirely true: Bachelier already knew how to price options with essentially the Black Scholes formula, and even the ancient Greeks had some ideas about it. Their ground breaking idea was instead the hedging: they showed that, in a Brownian motion context, options could be perfectly replicated via delta hedging, because Ito’s Lemma says essentially that dW²=dt is deterministic and that therefore if you take care of the first order terms (the Delta) via hedging, then you pay a cost for this (the Gamma bleed; the term in red in the equation above) because you buy high sell low, but this term is deterministic.
To summarize what we’ve seen so far: the LVR is really the well-known Gamma term of Black Scholes. We know that we integrate that Gamma term we get a profile, and it turns that for a CFMM that profile is the square root component (see here, chapter 4.4, page 24) that is the key component of the IL. Here is therefore our update AMM triangle
In other words
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