Curve Pool Parameters

We describe the curvature parameters a user should choose when deploying a Curve Crypto Pool and provide a batch of default parameters that have proven to be efficient.

Whenever a user decides to deploy a Principal Token (PT), they will need to specify the parameters of the corresponding Curve Crypto Pool against which the PT will be tradable with the interest-bearing asset it is backed by. Specifically, the user must carefully choose the parameters $A$ and $\gamma$, as these directly influence the curvature of the trading function and the geometry of the curve, consequently shaping how the market will behave, as well as the payoff of its agents.

Henceforth, we will use the abbreviations PT for Principal Token and IBT for Interest-Bearing Token.

What does the PT/IBT market represent? This market determines the pricing of the expected interest rate generated over the entire duration of the PT. Therefore, when discussing volatility in the PT/IBT market, we are referring to the volatility of the expected cumulated interest rates over the entire duration of the PT. Parameters should be selected based on this volatility.

To comprehend the impact of parameters on market invariants, we have developed market models and devised metrics and risk measures. Specifically, we have established a connection between the volatility of cumulated interest rates and optimal parameters, where "optimal" denotes minimisers of a loss function. The underlying principle can be stated as follows: the less volatile the market in question, the flatter the curve should be. This minimizes price impact by mitigating potential impermanent loss. In a non-volatile market, impermanent loss becomes a lesser concern, since the exchange rate is not allowed to vary, therefore preserving more the value of the liquidity provider's portfolio.

On the other hand, in a more volatile market, additional curvature must be incorporated into the trading function to minimise the impact of impermanent loss while simultaneously increasing potential price impact.

One must always find the equilibrium point of adversarial agents that are the traders and liquidity providers. The traders are exposed to price impact, while liquidity providers to impermanent loss. The curve whose shape minimises price impact increases impermanent loss, while if it minimizes the effect of impermanent loss, it increases price impact. Given the implied volatility of the market, one finds the equilibrium parameters as minimizers of a loss.

Hereinafter, we present a guide for choosing the optimal curvature parameters. The deployer is strongly advised to follow our heuristics regarding the pool parameters and is encouraged to perform further research if needed. In order to understand how the aforementioned parameters influence the shape of the curve, one can use the following visualisation: Desmos Calculator.

This parameters have been found by fixing an average impermanent loss threshold of 0.1% and then by searching over the parameters that minimise price impact, given a PT/IBT market with a given volatility parameter.

Volatility	Not Volatile	Moderately Volatile	Volatile
Curvature Parameters	$(4 \times 10^9,2 \times 10^{16})$	$(2.2 \times 10^9,2 \times 10^{16})$	$(8.9 \times 10^8, 1.5\times 10^{16})$

For further reference on the other parameters of the Curve Pool (fee parameters and the underlying fee mechanism, internal moving average parameters, etc...), please refer to the Curve documentation: https://docs.curve.fi/cryptoswap_exchange/pools/overview/. Note that we will provide the deployer with default values for the remaining parameters.