Impermanent Loss & Options

Smilee Genesis: Impermanent Loss from bug to Feature
The rise of Decentralized Exchanges (DEXs) have attracted tens of billions of dollars into liquidity pools.
However, liquidity providers (LPs) in DeFi are exposed to a now infamous risk: when markets are volatile, they significantly underperform an equal-weighted portfolio (EW portfolio), a problem known as Impermanent Loss (IL).
Mathematically, the EW portfolio payoff is:
EW portfolio payoff formula.
The LP payoff formula is:
LP payoff formula.
The difference is the Impermanent Loss, which can be written as:
Impermanent loss formula.
Where we have that r is the return of the token x in terms of the token y and V0 is the initial capital.
Graphically, the payoffs of the LP and the EW portfolios are:
A comparison between providing liquidity and holding an equal-weight portfolio of ETH / USDC worth $10,000. Initial ETH price = $1,000
If we plot their difference, we obtain the chart of the Impermanent Loss:
As we can see, Impermanent Loss is a concave function, which means the more the pair moves, the larger the Impermanent Loss is, which is equivalent to having a short volatility position (short gamma).
In exchange for such risk, Liquidity Providers receive a yield (APY) that depends on the fees generated on the DEX (long theta).
The combination of a short gamma exposure and a long theta one makes the liquidity provider payoff that of an options seller.
But who is the options buyer?
No one.
People can go LONG on liquidity (selling options), but they can’t go SHORT (buying options). This happens by design: automated market makers (AMMs) are built this way.
At Smilee we realized this is not a bug, it’s a feature.
Starting with the Impermanent Loss payoff, we derived the function to replicate it by selling a portfolio of options. The resulting formula is:
Options Weights Vector formula.
Where P is the matrix of the Payoffs of every option in each point of our replication grid, w is the vector of weights that we are computing and L is the vector containing the IL in each point of the grid.
We can get a progressively better approximation of the IL by selling more option strikes:
The relation between Impermanent Loss and options, never fully understood so far, allows us to create a brand new primitive whereby we decompose Impermanent Loss into options and recompose them to generate any type of volatility payoff.
This means that as long as there is enough liquidity on Smilee, we can mint and sell any option, on any strike, without having someone selling that option in the first place.
Again, this solves chicken and egg problem for options AMMs and positions Smilee as a primitive that other protocols can use to hedge risks fully on-chain in a fully customizable way.