AMM-Based LTV Model for Cross-Reserve Lending

Introduction

This mathematical model for LTV in an AMM-based lending system takes into account:

• Constant Product AMM: Using the AMM's invariant formula.
• Realized Volatility: Factoring in price movement sensitivity.
• Price Impact of Liquidation: Estimating liquidation feasibility based on the available liquidity curve.
• Dynamic LTV: Adjusting based on existing borrows and slippage constraints.

Constant Product AMM Dynamics and Price Impact

AMM Invariant

We model the pool with reserves \(X\) of asset A and \(Y\) of asset B, obeying the constant product invariant \(x \cdot y = k\). The current price of asset A in terms of B is \(P_A = \frac{Y}{X}\).

When a liquidation sells a quantity \(\Delta X\) of A (collateral) for B, the reserves shift to \(X' = X + \Delta X\) and \(Y' = \frac{k}{X'}\). The amount of B obtained from liquidating \(\Delta X\) is:

This formula captures price impact: selling more A (larger \(\Delta X\)) yields diminishing returns of B because the price moves against the seller as reserves rebalance.

The execution price for the liquidation trade is \(P_{\text{exec}} = \frac{B_{\text{out}}}{\Delta X} = \frac{Y}{X + \Delta X}\), which is lower than the initial market price \(P_A = \frac{Y}{X}\).

The price impact (slippage) can be quantified as the relative drop:

Incorporating Realized Volatility into LTV

Volatility Risk

Crypto asset prices can swing rapidly. We introduce a volatility haircut to adjust LTV for price fluctuations. Let \(\sigma\) be the realized volatility of the collateral asset (e.g. annualized standard deviation).

Over a short liquidation timeframe \(t\) (e.g. 1 day or a few hours), an approximate worst-case price drop fraction can be estimated (for example, using a multiple of \(\sigma\sqrt{t}\) or historical VaR). We denote this anticipated worst-case drop as \(\delta\) (e.g. \(\delta = 0.20\) for a 20% price drop).

To ensure the loan remains covered after a potential drop, the effective collateral value is scaled by \( (1-\delta)\). In practice, this means the volatility-adjusted LTV must satisfy:

Equivalently, the maximum volatility-adjusted LTV ratio is \(LTV_{\text{vol}} = 1-\delta\).

We treat \(\delta(\sigma)\) as a function of realized volatility (e.g. higher \(\sigma\) → larger \(\delta\)). In a simple model, one could set \(\delta = z \sigma \sqrt{t}\) for some confidence factor \(z\).

The key idea is that as realized volatility rises, the maximum allowable LTV falls to mitigate liquidation risk.

LTV and Liquidation Price Impact

Combining Slippage and Volatility

When liquidating collateral A for debt B, the actual recoverable value in B is reduced by AMM price impact and by any adverse price movement from volatility.

Using Equation (1), before considering volatility we require \(B_{\text{out}} \ge \text{Debt}\) to safely cover the loan. Substituting the debt \(\text{Debt} = \text{Loan} = LTV \times (\text{Collateral Value}) = LTV \times \Delta X \times P_A\), the condition becomes:

Canceling \(\Delta X\) and \(Y\), this simplifies to \( \frac{1}{1 + \Delta X/X} \ge LTV\). In other words, ignoring volatility, maximum LTV from AMM depth is:

where \(\frac{\Delta X}{X}\) is the fraction of the A-pool being sold.

Now include the volatility haircut \(1-\delta\). We require \(B_{\text{out}} \ge \text{Debt} \times (1+\epsilon)\) (perhaps \(\epsilon\) as a small auction incentive or buffer). For simplicity, set \(\epsilon=0\).

If the price drops by \(\delta\) before or during liquidation, effectively the pool's price starts \((1-\delta)\) lower. We can approximate this by multiplying the recoverable B by \((1-\delta)\). The safe liquidation condition becomes:

Following the same algebra, this yields a combined LTV limit:

This formula encapsulates both sources of risk: a volatility buffer \((1-\delta)\) reducing usable collateral value, and an AMM depth factor \(1/(1+\Delta X/X)\) reducing value from price impact. For a given collateral size \(\Delta X\), the allowed LTV must be below this curve.

Dynamic LTV Adjustment for Utilization and Liquidity

Reserve Utilization

As users borrow asset B against A, the pool's composition shifts. If many loans are already drawn, the available B liquidity (reserve) is reduced and A reserve is increased (the pool becomes heavy in A).

This high utilization of B means any new liquidation would face even worse slippage (since \(Y\) is smaller and \(X\) larger). In our model, the current state of the pool (current \(X, Y\)) dynamically affects LTV for new loans.

We can formalize utilization U of asset B as the fraction of B liquidity taken:

with \(Y_{\text{initial}}\) the pool B reserve when no loans are taken. As \(U_B \to 1\) (most B drained), the pool becomes very imbalanced (price of A drops significantly in the AMM), and any further liquidation will yield drastically less B.

To protect the system, the model reduces allowable LTV as \(U_B\) rises. One simple relation could be a linear scale, e.g.:

meaning if half of B liquidity is already utilized (50% drained), new loans can only use up to 50% of the normal LTV limit.

Preventing Excessive Drain

The goal is to ensure new borrows do not overly deplete available liquidity or put existing loans at risk. For example, the protocol might forbid any single loan from consuming more than a certain percentage of remaining B reserves.

In terms of our formula, this could manifest by capping \(\Delta X/X\) for new loans or enforcing a minimum \(Y'\) after the trade. Essentially, as \(Y_{\text{current}}\) falls, the factor \(1/(1+\Delta X/X)\) in Eq. (2) becomes the dominant limiter on LTV.

The system may also increase fees or interest rates at high utilization to discourage reaching such a state (outside the scope of LTV but part of holistic risk management).

Concluding Insights