Impermanent Loss Hedger (ILH)
Impermanent Loss refers to a classic situation under an XYZ model, a spot trading swap system will always result in a loss on the total value of two assets no matter the exchange rate of the currency pair. This is shown in the graph below:
The area between the blue line (holding Eth and USDT) and the yellow line (providing swap liquidity) is the total value lost across all scenarios along the X and Y-axis. The graph assumes a 1 ETH = 400 USDT.
Impermanent loss (IL) is an unavoidable pitfall for traders and liquidity providers, and can even potentially lead to negative yields. This phenomenon has been plaguing the DeFi space, especially for small investors who cannot afford such losses.
Mitigating Impermanent loss to zero has been a constant effort all across the industry, however, Asteria brings the vision and the necessary tech stack to reality with the Impermanent Loss Hedger (ILH)
The principle behind ILH:
Asteria has developed and back-tested an IL portfolio hedging product for the X*Y=K algorithm on DEXs. The principle is as follows:
Mathematical Foundation: Carr-Madan Formula
For any return structure f(ξT) with respect to ξT that expires at time T, it can be realized by constructing a European Option portfolio with ξT as the target and expiration date T, under the condition of f(ξT) is second-order derivable, which is essentially a static investment strategy:
Among them, i0 is a constant determined by the system when constructing the investment portfolio (current period). The portfolio here includes:
By dampening the IL, we can see that the automated spot market maker is equivalent to "free", providing the market with a set of call-options and put-option combinations with different strike prices (that is, the source of IL is the same as using Limit orders traded on centralized exchanges).
For hedging IL, we need to create an options portfolio to generate the following income:
Assuming ξ0=1
We get the curve of f, f’and f’’:
Backtesting - Proving Strategy Effectiveness
The proposed strategy has been battle-tested for efficiency over the years.
We take the options expiration T as 5 days and the annualized volatility σ as 2, by using 40 options with different strike prices. We then substitute it into the Carr-Madan Formula, aiming to obtain an almost perfect portfolio for hedging Impermanent Loss (IL):
Step One: Volatility Estimations
Calculating the annualized average volatility σ of a certain interval using the following method:
Where y_t="log" S_t-"log" S_(t-Δ) as the short-term logarithmic return rate of the underlying (S_t)
We chose to use the volatility when the option is purchased to manage the risk of the entire portfolio and backtested with different interval lengths in the past, such as the volatility of one hour, one day, one week, half a year, and one year, to form a prediction of the volatility of different interval lengths in the future.
Step Two: Option Portfolio Pricing
The classic Black-Scholes-Merton pricing model is applied:
Here,
The exchange rate of the underlying pair of assets would be ξT/ξ0, standardized to ξ0 to 1;
The strike price of the call options is ‘i’
Then,
When ξT / ξ0 > i ，pay ξT/ξ0− i for the buyers of the option,
The price of call options is:
The price of put options is:
Where d1 and d2 satisfy:
N(⋅) is the cumulative distribution function.
The options premiums are evaluated in three different scenarios:
- Scenario 1: Full hedging;
- Scenario 2: Hedging upward IL only;
- Scenario 3: Hedging downward IL only
In Scenario 1: The cost of full hedging IL is only 0.7%. In other terms, the cost to hedge a staking supply of 1000 USDT is only 70 USDT.
In Scenario 2: The cost of hedging the IL downwards only is 0.42%. In other terms, the cost to upward hedge the staked supply of 1000 USDT is 42 USDT.
In Scenario 3: The cost of hedging the IL upwards only is 0.28%. In other terms, the cost to downward hedge a staked supply of 1000 USDT is 28 USDT.
The design fundamentals behind the Impermanent Loss Hedger (ILH)
Based on the back-tested data, Team Asteria designed the options-based Impermanent Loss Hedger (ILH)
In order to interact with the ILH, the users may make three simple selections:
- Hedging Period (Daily/Weekly)
- Hedging Mode (Bidirectional/Only Downside Protection/Only Upside Protection)
- Hedging Level (High/Medium/Low)
Conclusively, a more sophisticated form of hedging based on options, in theory, would result in a greater level of IL being hedged. The higher performance the greater would be the valuation of options employed.
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