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# Impermanent Loss Hedger (ILH)

Hedging, in general, refers to combining an original position with another in order to render the values of users’ combined portfolio less sensitive to the risk (impermanent loss) being analyzed.
Options hedging is a dynamic new way of making positions both delta and gamma neutral, in addition to buying or selling the underlying, the options with opposite risk exposure to LP position. This process is extremely unfeasible in terms of cost and operational efficiency when done manually.
Beaver uses a concept developed by Asteria Finance Labs - employing a European Options Portfolio to hedge impermanent loss.

## How does Beaver’s Impermanent Loss Hedger work?

Beaver employs Carr-Maden Formula as a solution to hedge against impermanent loss. 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 the second-order derivable, which essentially is a static investment strategy: We create a portfolio of call options and put options with different strike prices and expirations dates to generate the following revenue volume which stands in exact contrast with the estimated impermanent loss: Assuming ξ0=1 We get the curve of f, f’’ and f’’’ By decomposing 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).

## The measure of Efficiency through backtesting practices:

#### 1. 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. #### 2. Option Portfolio Pricing:

We apply the classic Black-Sholes-Merton option pricing model. Since cryptocurrency does not generate dividends, the price of European-style call options is the same as that of American-style call options, so Asteria uses American-style call options to construct the investment portfolio (The VIX calculation of CBOE index also refers to a similar method)
We take the option expiration T as 7 days and the annualized volatility σ as 2, by using 40 options with different strike prices, and substituting it into the Carr-Madan Formula to obtain one almost perfect portfolio for hedging IL: As shown in the graph above, the cost of completely hedging IL is a minimal of 0.7%.

#### 3. Hedging and Pay off testing:

The key to the success of ILH as a product-market fit is its capability to hedge on behalf of option market makers/sellers to gain the profit to pay off at the end of each cycle.
So we stimulate the hedging (assume every 10 mins) for products on High level and Bidirectional ILK and on both periods of Daily/Weekly to identify the safety spectrum that the ILH provides users as well as to determine the liquidity pool’s return rate.
The results were extremely promising. • Graph 1: illustrates the payoff at each expiration.
• Graph 2: is histogram after hedging, the x-axis is the daily income in USDT, and the y-axis is the number of statistics.
• Graph 3: illustrates the payoff results. Blue points represent the payoff promised (to compensate the IL). Yellow points represent the actual return of option hedging. The x-axis is the rate of change between two tokens.
• Graph 4: is the scatter chart of the income promised to LP with a 45-degree straight line and the income after hedging. The X-axis and Y-axis are USDT, and the blue straight-line scatter chart with a 45-degree angle is the income promised to LP.
The yellow scatter chart shows the hedged return corresponding to the return promised to the LP. For example, in a certain period, if the promised payoff is 50 and the hedging payoff is 75, the blue dot of the promised payoff will be marked at the coordinates (50,50), and the hedging payoff will be marked at the coordinates (50,75) as a yellow dot.