Uniswap Insights 4 of 6— LP Hedging

LP positions exhibits concavity (also termed negative-convexity) — accelerated falling as price declines and decelerated growth as price increases

Put options exhibit convexity — accelerated climbing as price decreases and decelerated falling as price increases

TLDR:

Primarily two ways of simply hedging an LP position and a third complicated way:

1. Borrowing the underlying asset, but this doesn’t cancel the concavity as one would still retain some degree of divergence loss.
2. Borrowing a similar LP position, here one does get rid off concavity, but no fluid market exists for custom LP positions yet.
3. Combine an LP position with a put option. Here one does temporarily cancel out the concavity, but divergence loss would set in over time. One would require re-rolling options to maintain an LP hedge.

There appears to be no free lunch, but only tradeoffs.

1. Borrow The Underlying Asset

Instead of allocating 100% to an LP position, a strategy is to allocate only a portion of it with the rest being used to short the X asset to reduce but not eliminate divergence loss. The result is the conversion of a payoff structure from a cliff into a hill as shown by Guillaume Lambert:

Guillaume Lambert’s desmos file for combining short + LP: https://www.desmos.com/calculator/669zg1rmvb

Atiselsts.eth also provides an approach of dynamic-hedging, involving the frequent buying and selling as the price moves: https://atise.medium.com/liquidity-provider-strategies-for-uniswap-v3-dynamic-hedging-9e6858bea8fa. This is a very interesting approach, but dynamically-hedging on mainnet would consume a lot in fees so a Layer 2 can actually serve a useful purpose here for LPs, but be aware that lower fees usually come at a cost of centralization, as we have recently seen with Arbitrum’s bug causing the L2 to halt [https://www.coindesk.com/tech/2023/06/07/arbitrum-temporarily-stopped-processing-due-to-software-bug/].

2. Borrow The LP Position — Simpler Approach

The nature of an LP position is that it’s concave. Using a basic interactive example of a full range LP position below:

Uniswap full range LP position value in blue. Change in price in red. Divergence between LP and holding beneath x-axis. Desmos interactive link — https://www.desmos.com/calculator/cntuhfqang

If the price of the main asset moves down, then the losses accelerate. To cancel it out one can borrow such a position and it would be a perfect fit.

“Borrow LP” button enabled in desmos— https://www.desmos.com/calculator/cntuhfqang

Simply borrowing an LP position just by itself converts the payoff from concave to convex, but note that if the price of X *rises*, then the borrowed LP position would actually become more costly.

An increase in price leads to a growth in the borrowed full-range LP value. With concentrated liquidity, this can be reduced.

If one uses concentrated liquidity, one winds up creating linear payoffs past p_a and p_b. Borrowing such positions can exhibit interesting qualities. This is especially evident if one combines borrowing the underlying with borrowing the LP position. One could come up with a spectrum of intriguing payoffs:

1) Long LP. 2) Borrow Underlying + LP. 3) Borrow LP + Long Underlying. 4) Borrow LP + Short Underlying.

I personally call these synthetic combinations CPs — Curved Perpetuals since they don’t expire. When the payoff structure is convex, it’s a— Convex Perpetual — also CP. When the payoff structure is concave — Concave Perpetual — also CP!

Such synthetic DeFi primitives relate to perpetuals through put-call parity. By combining a Long Convex Perpetual with a Long Concave Perpetual one gets a regular Long Perpetual:

This is the exact same concept of combining a Long Call Option with a Short Put Option ATM where one replicated a Long Stock (dashed diagonal) instead of a Long Perpetual.

Theoretically, borrowing and constructing such CPs could be costly and challenging from a transaction cost and engineering perspective, but there’s very smart people working on this problem as I write this, but for now we’re left with using Black-Scholes options to cancel out the curvature of LP positions.

3. Combining Options with LP Positions

In order to understand LP hedging we have to understand “the Greeks” since these derivatives of an option allow us to compare the payoff between an LP position and an option. We can see how all of the Greeks behave visually below for an option as it expires:

The value of a call and put option approaches zero as the option expires at the strike price. The Greeks accentuate. Delta steepens, gamma peaks, and theta accelerates through the floor.

I have created an interactive option calculator visualized above with all of the Greeks that I think will be useful for people beyond just this audience so feel free to share it — https://www.desmos.com/calculator/x5vk8qkweg

The main Greeks for now to know for options are delta, gamma, and theta since we will start comparing them with the LP Greeks and then go onto vega and the other Greeks in part 5.

We will cover only the first three here.

LPs receive theta through fees and option buyers pay theta by hedging.

One is compensated for concavity with fees as an LP.

One pays for the privilege of convexity with theta in options with a premium upfront.

For our put option the important takeaway is that as time approaches the expiration, the value of the option declines in an accelerated fashion captured by theta in purple. It becomes more sensitive to rapid price changes (accentuated delta and gamma) with a steeper shape in red as its value approaches zero.

Put option approaching expiration. Note that delta, gamma, and theta accelerate.

If we were to compare these three option Greeks to LP Greeks when the price barely changes, the price-sensitive Greeks (delta and gamma) are mostly fixed as time progresses, but the theta (strictly positive) expressed in LP fees would vary depending on trading activity in the pool:

The theta is always positive, but it varies for us as an LP depending on trading volume. I kept the price movement fixed for ease of visualization to show that delta and gamma are not impacted by time.

If one were to zoom in, then in order to generate theta through fees, the price itself would have to possess small tick increment movements because if there are no perturbations in the underlying due to swaps in the pool, there can be no fees generated and we would lose out on theta.

If there is no volume >> no swaps >> expected theta goes to 0

The big differences for the three Greeks between an option and an LP position are due to the non-expiring/perpetual property of an LP position causing theta to not match and the nature of the xy=k invariant causing delta and gamma to not align perfectly.

If we were to come as close as possible to match a put option with a concentrated liquidity position below, our deltas and other Greeks approximately do match, but they’re not 100% perfect (see black line), we can also see that the cancellation of the put option in red and the LP position in blue can vary a bit:

The closest put option has a -0.43 delta relative to the LP delta of 0.435. The sum between a put and an LP position should be a perfect black line above, but notice its slight curvature.

What if there’s an option for an asset that has no trading volume on the AMM? Then our deltas would start to diverge over time:

We also will need to factor into account the Greek vega and how implied volatility can alter such a payoff structure in part 5 because such cases lead to an incentive to borrow such an LP position:

A put option whose implied volatility is too costly can make hedging an LP position impractical.

Credit for the desmos file below goes mainly to Guillaume Lambert. I optimized a Black-Scholes option on top of a single LP position to dynamically solve for an approximate hedge here:

Greeks Hedge for Uni v3

But how do we completely match the Greeks for a concentrated LP position? Well, we either need to change the invariant (PrimitiveFi’s approach) or we can use a numerical approach by constructing a set of concentrated LP positions in Uniswap whose sum of Greeks does match the curvature. We will cover this in part 5 with the help of lognormal distributions!

By combining multiple LP positions we can numerically match the Greeks not just for regular European options, but also for Asian options in part 5.

• Part 5 continues here: https://medium.com/@med456789d/uniswap-insights-5-6-lp-hedging-2-the-greeks-467b60f1af32. If you found this educational, feel free to give this hedgehog a follow on twitter: @CK_2049

Disclaimer: This research is for general information purposes only. The Uniswap Foundation was kind enough to sponsor the publication of this research. It does not constitute investment advice or a recommendation or solicitation to buy or sell any investment and should not be used in the evaluation of the merits of making any investment decision. It should not be relied upon for accounting, legal or tax advice or investment recommendations. This post reflects the current opinions of the author. The opinions reflected herein are subject to change without being updated.