# Main Strategy for Hedging Impermanent Loss

### 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 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).

Assuming ξ0=1

We get the curve of f, f’ and f’’：

### Impermanent Loss Hedging using Carr-Madan Formula

From the graph:

The orange line is the impermanent loss ratio with respect to the price of the underlying asset.

The blue line is the payoffs of a portfolio of put and call options, and the number of options is calculated by the Carr-Madan Formula.

The payoff of the options closely matches the impermanent loss graph.

The option premiums are evaluated in three different scenarios: full hedging, hedging upward IL only, and hedging downward IL only：

The cost of full hedging IL is only 0.2%, or the cost to hedge 10000 USDT staked is 20 USDT.

The cost of hedging downward IL is only 0.12%, or the cost to hedge 10000 USDT staked is 12 USDT.

### Volatility Estimation

Calculating the annualized average volatility σ of a certain interval using the following method:

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.

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