Artur Sepp Blog on Quantitative Investment Strategies

Let me introduce our research paper co-authored with Alexander Lipton and Vladimir Lucic for hedging of impermanent loss of liquidity provision (LP) staked at Decentralised Exchanges (DEXes) which employ Uniswap V2 and V3 protocols.

Uniswap V3 protocol allows liquidity providers to concentrate liquidity in specified ranges. As a result, the liquidity of the pool can be increased in certain ranges (typically around the current price) and the potential to generate more trading fees from the LP is increased accordingly. I illustrate the dynamics of staked LP using ETH/USDT pool as an example. A liquidity provider stakes liquidity to a specific range using initial amount of ETH and USDT tokens as specified by Uniswap V3 CFMM. When the price of ETH falls, traders use the pool to swap USDT by depositing ETH, so that the LP accrues more units of ETH. Thus when ETH falls persistently, the liquidity provider ends up holding more units of the depreciating asset, which is similar to being short a put option. In opposite, when ETH price increases, traders will deplete ETH reserves from the pool by depositing
USDT tokens. Thus, the liquidity provider ends up holding less units of the appreciating asset, which is similar to being short a call option. The combined effect of increasing / decreasing the exposure to depreciating / appreciating asset leads to what is known as the impermanent loss in Decentralised Finance (DeFi) applications.

In Figure 1, I show ETH units (left y-axis) and USDT units (right y-axis) for LP on Uniswap V3 with 1m USDT notional and p_{0}=2000, p_{a}=1500, p_{b}=2500. The initial LP units of (ETH, USDT) are (220, 559282). The red bar at p=1500 shows LP units of (543, 0) with LP fully in ETH units when price falls below lower threshold p_{a}. The
green bar at $p=2500$ shows corresponding LP units of (0, 1052020) with LP fully in USDT units when price rises above upper threshold p_{b}. In subplot (B), we show USDT values of 50%/50% ETH/USDT portfolio, Funded LP positions (funded LP involves the purchase of ETH for staking without any delta hedge) and Borrowed LP positions (Borrowed LP is produced by static delta hedge of the initial staked position in ETH).

The value profile of funded LP resembles the profile of a covered call option (long ETH and short out-of-the-money call). The value of the borrowed LP resembles the payoff of a short straddle (short both at-the-money call and put).

Figure 1. The impremanent loss of funded and borrowed LP position

(A) ETH units (left y-axis) and USDT units (right y-axis) for LP on Uniswap V3. (B) USDT value of 50%/50% ETH/USDT portfolio, Funded LP position and Borrowed LP position. Uniswap V3 LP position is constructed using 1m USDT notional with p_{0}=2000, p_{a}=1500, p_{b}=2500.

We define the protection claim against the impermanent loss (IL) as a derivative security whose payoff at time T equals to negative value of the IL.

We develop static model-independent and dynamic model-dependent approaches for hedging of the IL of liquidity provision (LP) staked at Decentralised Exchanges (DEXes) which employ Uniswap V2 and V3 protocols.

For staking of BTC and ETH with liquid options market, the liquidity provider can apply out static model-independent replication to eliminate the IL completely.

In Figure 2, I illustrate the replicating of IL for borrowed Uniswap V3 LP. I use strikes with widths of 50 USDT in alignment with ETH options traded on Deribit exchange (for options with maturity of less than 3 days, Deribit introduces new strikes with widths of $25$). In subplot (A), I show the IL of the borrowed LP position, and the payoffs of replicating calls and puts portfolios (with negative signs to align with the P&L). In subplot (B), we show the residual computed as the difference between the IL and the payoff of the replication portfolios. In Subplot (C), I show the number of put and call option contracts for the replication portfolios. It is clear that the approximation error is zero at
strikes in the grid, which is illustrated in subplot (B). The maximum value of the residual is 0.025% or 2.5 basis points, which is very small. A small approximation error with a similar magnitude will occur in case, p_{0}, p_{a}, p_{b} are not placed exactly at the strike grid.

Figure 2. Replication of IL of borrowed Uniswap V3 LP for allocation of 1m USDT notional, p_{0}=2000 ETH/USDT with p_{a}=1500 and p_{b}=2500. (A) Impermanent loss in USDT and (negative) values of replicating puts and call portfolios; (B) Residual, which is the spread between IL and options replication portfolios; (C) Number of option contracts for put and calls portfolios.

For cryptocurrencies without a liquid options market develop the model-dependent valuation and dynamics hedging of IL protection claims for Uniswap V2 and V3 protocols. Model-based valuation can be employed by a few crypto trading companies that currently sell over-the-counter IL protection claims. When using model-based dynamics delta-hedging for the replication of the payoff of the IL protection claim, the profit-and-loss (P&L) of the dynamic delta-hedging strategy will be primarily driven by the realised variance of the price process. Thus, the total P&L of a trading desk will be the difference between premiums received (from selling IL protection claims) and the variance realised through delta-hedging. Trading desk can employ our results for the analysis of price dynamics and hedging strategies which optimize their total P&L.

The simplest dynamic model is of course the Black-Scholes-Merton model which allows to analyze the sensitivity of the price for IL protection as a function of a single parameter for log-normal volatility

In Figure 3, I show the annualised cost (APR) % for the cost of BSM hedge for the borrowed LP as a function of the range multiple m such that p_{a}(m)=e^{-m}p_{0} and p_{b}(m)=e^{m}p_{0}. I use two weeks to maturity T=14/365 and different values of log-normal volatility \sigma. All being the same, it is more expensive to hedge
narrow ranges.

Figure 3. BSM premium annualised (U^{borrower}(t, p_{t})/T) for borrowed LP with time to maturity of two weeks and notional of 1 USDT as function of the range multiple m such that p_{a}(m)=e^{-m}p_{0} and p_{b}(m)=e^{m}p_{0}.

Further, we consider a wide class of dynamics models with jumps and stochastic volatility for which the moment generating function (MGF) for the log-return  is available in closed-form. The closed-form solution for the MGF is available under a wide class of models including jump-diffusions and diffusions with stochastic volatility. Thus, we can
develop analytic solution for model-dependent valuation of IL protection under various models with analytic MGF.

In particular, we apply the log-normal SV model which can handle positive correlation between returns and volatility observed in price-volatility dynamics of digital assets (see my paper with Parviz Rakhmonov for details).

In Subplot (A) of Figure 4, I show the implied volatilities of the log-normal SV model for a range of volatility of residual volatility with zero volatility beta (which is typical for ETH skews). In Subplot (B), I show the premium APR for IL protection as a function of range multiple for a range of volatility-of-volatility. We see that the model-value of IL protection is is not very sensitive to tails of implied distribution (or, equivalently, to the convexity of the implied volatility). The reason is that the most of the value of IL protection is derived from the center of returns distribution.

Figure 4. (A) BSM volatilities implied by log-normal SV model as function of volatility-of-volatility parameter ; (B) Premiums APR computed using log-normal SV model for borrowed LP as function of the range multiple m such that p_{a}(m)=e^{-m}p_{0} and p_{b}(m)=e^{m}p_{0}.

For liquidity providers, who buy IL protection claims for their LP position, the total P&L will be driven by the difference between accrued fees from LP positions and costs of IL protection claims. The cost of the IL protection claim can be estimated beforehand using either the cost of static options replicating portfolio or costs of buying IL protection from a trading desk. As a result, liquidity providers can focus on selecting DEX pools and liquidity ranges where expected fees could exceed hedging costs. Thus, liquidity providers can apply our analysis optimal allocation to LP pools and for creating static replication portfolios using either traded options or assessing costs quoted by providers of IL protection.

We leave the application of our model-free and model-dependent results for an optimal liquidity provision and optimal design of LP pools for future research.

Enjoy reading the paper available on SSRN https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4887298