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Performance Guarantees for Permissionless Lending

Nanqiao Wang, Jia Yuan Yu

arXiv:2610.34774v1stat.APcs.CRstat.ME

TL;DR

The paper addresses opacity and missing risk guarantees in permissionless lending by proposing a marketplace for fixed-term, fixed-rate loans and deriving probability guarantees for liquidation and repayment shortfall. In its Bitcoin-collateral illustration, estimated 30-day repayment shortfall probabilities differ across the fitted return distributions.

  • Problem

    Existing permissionless lending protocols are described as opaque, lacking guarantees on liquidation and repayment shortfall, and exposing users to risks including rate-setting and liquidation manipulation.

    Source: §1 · p. 1

  • Method

    The paper proposes an asynchronous marketplace for over-collateralized, fixed-duration, fixed-rate loans and derives performance guarantees for specified collateral-value trajectories.

    Source: §1 · pp. 1–2

  • Results

    Within 30 days, estimated repayment shortfall probabilities are 0.90% for Gaussian, 4.92% for Student-t, and 9.90% for symmetric stable Paretian returns.

    Source: §4.1 · p. 20

  • Takeaways & Limitations

    Liquidation and top-up actions allow lenders and borrowers to control the probability of their respective rare adverse loan outcomes.

    Source: §1 · p. 2

    The paper does not guide offer-taking or waiting decisions, characterize how quickly offers become loans, or analyze convergence of similar offers’ interest rates.

    Source: §6 · p. 24

Abstract

from arXiv · show

We propose a lending protocol with far fewer risks than existing pool-based lending protocols (e.g., Aave, Compound). In our protocol, lenders and borrowers embody the spirit of decentralization by making competing interest-rate offers, resulting in the discovery of the prevailing interest rate through t\^atonnement-like dynamics as opposed to taking rates partly determined by a central authority (e.g., Aave, Federal Reserve). We eliminate the risk of manipulation by creating loans with a fixed interest rate throughout the loan---like US Treasury bills. In particular, we further derive performance guarantees probabilities of repayment shortfall and liquidation by modeling the value of the collateral as a random walk, thus allowing participants to more efficiently manage their risk. To do so, we use characteristic functions and Fourier inversion to derive exact integral representations of the liquidation and repayment shortfall probabilities under the most general assumptions. We illustrate empirically this method by using Bitcoin as collateral and fitting its historical prices to Gaussian, Student-$t$, and symmetric stable Paretian distributions.

1 Introduction

The paper proposes a permissionless lending marketplace to address existing protocols’ opacity and risks, with fixed-rate loans formed through competing offers. It also develops performance guarantees for liquidation and repayment shortfall under specified collateral-value trajectories.

1 Introduction

  • Existing lending protocols are described as opaque and lacking liquidation and repayment-shortfall guarantees, while also exposing users to arbitrarily set rates and other risks.The listed risks include manipulation of utilization rates, front-running, withdrawal bottlenecks, and conservative collateral valuation.Source: §1 · p. 1
  • Loans use fixed durations and interest rates, and participants asynchronously take up or create offers while competing by updating their terms.Loans are over-collateralized, and the fixed rate is described as impervious to third-party manipulation.Source: §1 · p. 2
  • The protocol offers performance guarantees for loan outcomes under particular collateral-value trajectories, rather than guaranteeing greater efficiency than other DeFi protocols.The guarantees concern repayment with interest being agreeable to both lender and borrower with high probability.Source: §1 · p. 2
  • Lenders seek repayment with interest and borrowers seek their collateral back; liquidation and top-up actions control the probability of rare adverse outcomes, which depends on collateral volatility.The two user groups therefore have distinct desired outcomes and risks.Source: §1 · p. 2
  • The analysis specifies the protocol, derives Gaussian-case probabilities, then uses characteristic-function methods and Fourier inversion for more general return distributions.For liquidation, it computes a running maximum’s characteristic function; for repayment shortfall, it combines a joint characteristic-function recursion with Fourier inversion.Source: §1 · pp. 2–3

2 Proposed Lending Protocol

This section presents a peer-to-peer lending marketplace in which lenders and borrowers compete through offers and loans are overcollateralized and represented by wallet balances and state transitions. It specifies how actions move offers and loans through origination, liquidation, and settlement, and illustrates how collateral choices affect the health factor and liquidation along a historical BTC price path.

2.1 Time, Assets, Wallets, Valuation

  • Unlike lending pools, the protocol represents each offer or loan through its record and associated vault-wallet balances, without credit- or debit-tokens.Collateral value is measured in the liquid asset through a monotone valuation function that can depend on the amount being valued, capturing market-depth effects.Source: §2.1 · pp. 4–5

2.2 Design Principles

  • In lender-first interactions, the lender sets the amount, duration, rate, and repayment probability while the borrower sets the start time and collateral-dependent liquidation probability.In borrower-first interactions, the borrower sets the amount, collateral, duration, and rate, while the lender chooses the start time.Source: §2.2 · p. 5

2.3 Actions, Offers, Loans

  • Offers specify a fixed loan duration and interest rate, and admissible origination requires ρ > c > 0 and collateral value at least ρ times the principal.The protocol focuses on c > 1, so liquidation may become available while collateral still exceeds accrued debt; the action set is described as bare-bones but sufficient for performance guarantees.Source: §2.3 · pp. 6–9
  • Borrowers can repay early or top up collateral, while lenders can liquidate before maturity when collateral value falls below accrued debt times the threshold.At maturity, if liquidation is not triggered, the lender receives collateral worth the repayment amount and the borrower receives the remainder.Source: §2.3 · pp. 7, 9

2.4 State Transitions

  • The state comprises open lending offers, open borrowing offers, ongoing loans, and wallet balances; taking up an offer moves its record into the ongoing-loan collection.The state space is countably infinite-dimensional, and the initial market has no offers with wallets at their initial balances.Source: §2.4 · pp. 9–10
  • Participant actions trigger transitions that track offers, ongoing contracts, wallet and vault balances, and collateral valuation; matching an offer creates a loan only if its collateral and value satisfy the required initial ratio.Borrowers typically choose collateral so its initial value is the collateral ratio ρ times the loan value, subject to rounding and oracle updates.Source: §2.4 · pp. 10–11
  • Collateral volatility and valuation make careful collateral sizing important, while adding collateral later may be too late if liquidation is executed at the first eligible checkpoint.The protocol also permits collateral top-ups; a separate liquidation rule could transfer all collateral to lenders, potentially encouraging lender participation and more conservative health factors.Source: §2.4 · pp. 10, 11, 12, 14
  • A loan is liquidated at an eligible checkpoint when its health factor falls below 1; at maturity, the protocol checks liquidation first and otherwise settles the loan using the collateral-distribution rules.Early borrower repayment pays the lender in liq, whereas maturity settlement pays in col and leaves conversion to the lender; the protocol assumes no conversion costs.Source: §2.4 · p. 12
  • Adding 0.1 BTC at origination kept the health factor above the liquidation boundary throughout the window (minimum approximately 1.09), while baseline and day-5 top-up paths first crossed it on day 4.If liquidation occurs at the first eligible checkpoint, both baseline and top-up loans terminate on day 4, before the scheduled top-up.Source: §2.4 · pp. 12, 14, Figure 2 · p. 13
  • With few participants, the protocol can restrict offer parameters to improve the likelihood of matching, but offer cancellation creates a frontrunning risk when another lender preempts a take-up.One example is a lender canceling a lending offer before taking a borrowing offer, only to be preempted by another lender paying more gas.Source: §2.4 · p. 14

3 Performance Guarantees

The section derives liquidation and repayment-shortfall guarantees under a pessimistic scenario in which borrowers never top up collateral and lenders liquidate as soon as allowed. It models collateral valuation as a geometric random walk and uses the resulting path to quantify both risks.

3 Performance Guarantees

  • Collateral valuation follows a geometric random walk with i.i.d. Gaussian log-valuation increments, while collateral remains fixed after origination.The risk analysis assumes no collateral top-up and immediate lender liquidation whenever allowed.Source: §3 · pp. 14–15

3.1 Borrower Liquidation Probability

  • Under the Gaussian-increment assumption, the section derives borrower liquidation probabilities at time t and cumulatively by time s.The borrower can reduce liquidation probability by lowering the loan amount, duration, interest rate, or liquidation threshold, or by providing more collateral.Source: §3.1 · p. 16
  • Avoiding frequent liquidation or top-ups requires collateral value to grow geometrically faster than accrued loan value, meaning µ > log(1 + r).The liquidation probability is determined by the collateral path over the contractual loan period.Source: §3.1 · p. 16

3.2 Lender repayment shortfall Probability

  • The lender repayment-shortfall probability is quantified for loans that terminate without collateral top-ups or interruption.For fixed admissible collateral, the lender can control this probability through c and ρ.Source: §3.2 · pp. 16–17

4 Generalized Liquidation and Repayment Shortfall Probabilities

The section extends liquidation and repayment-shortfall guarantees from Gaussian returns to arbitrary i.i.d. increments, at the cost of numerical integration. Its BTC example finds Student-t favored by both AIC and BIC and shows that estimated risk depends on the fitted return distribution.

4 Generalized Liquidation and Repayment Shortfall Probabilities

  • Characteristic functions extend the guarantees to arbitrary i.i.d. return increments, allowing distribution families to reflect prior knowledge about collateral valuation dynamics.Unlike the Gaussian-specific results, the generalized integral results require numerical approximation.Source: §4 · pp. 17–18
  • The liquidation result uses the running maximum of valuation changes, with its distribution required to be continuous at the origination threshold.A sufficient continuity condition is that each Y_t has no probability mass at zero.Source: §4 · p. 18
  • The repayment-shortfall guarantee uses joint characteristic functions and requires continuity at both the origination threshold and the repayment threshold.Numerical evaluation of the unbounded integrals introduces truncation and quadrature errors; detailed error analysis is outside the paper's scope.Source: §4 · p. 19

4.1 Empirical Liquidation and Repayment Shortfall Probabilities

  • Both AIC and BIC favor the Student-t distribution among the fitted Gaussian, Student-t, and symmetric stable Paretian models.The models are fitted to daily BTC-USD returns, and the selected distribution is plugged into Theorems 3 and 4 to calculate the probabilities.Source: §4.1 · pp. 19–20
  • 34.55% is the 30-day liquidation probability under the symmetric stable Paretian fit; Gaussian and Student-t estimates are 31.03% and 30.89%, respectively.For short horizons, Student-t and symmetric stable Paretian estimates exceed Gaussian estimates; the stable Paretian estimate is highest throughout the 30-day horizon.Source: §4.1 · p. 20
  • For Gaussian returns, increasing truncation levels makes the numerical liquidation estimates converge to the analytical probabilities.The section also evaluates Gaussian repayment-shortfall approximations against the analytical baseline; the U = Ω = 10 case is omitted because estimates deviate substantially.Source: §4.1 · p. 20
  • 9.90% is the 30-day repayment shortfall probability under the symmetric stable Paretian fit, versus 4.92% for Student-t and 0.90% for Gaussian.Across the 30-day horizon, repayment shortfall probability is lower than liquidation probability; the Gaussian day-30 estimates are 0.90% and 31.04%, respectively.Source: §4.1 · pp. 20–21

5 Related Works

The protocol combines permissionless marketplace lending with fixed-rate, fixed-maturity loans and ongoing competition among participants. Related work distinguishes it from pool- and auction-based lending, while highlighting performance guarantees for both borrowers and lenders as its main distinction from similar fixed-rate markets.

5 Related Works

  • Ongoing loan rates arise through dynamic competition as participants cancel and resubmit offers, with first-come-first-served take-up resembling order books.The protocol also combines permissionless access with fixed rates and fixed maturities.Source: §5 · p. 21
  • The protocol is most similar to Morpho Midnight, but its stated distinction is performance guarantees for both borrowers and lenders.Source: §5 · p. 23
  • Unlike pool-based protocols with variable utilization-linked rates, the protocol fixes rates and avoids utilization-driven interest-rate manipulation.It also contrasts with auction-based lending, which uses fixed rates.Source: §5 · p. 23
  • Prior technical work includes continuous-time valuation models and discrete-time lending models, with slow interest-rate convergence highlighted in.Other cited work analyzes liquidation probability, optimal collateral, and lending-pool models.Source: §5 · p. 23

6 Open Problems and Future Work

The paper frames its guarantees as a first step toward risk management in permissionless DeFi, while leaving several participant and market questions open. Its guarantees apply after a loan is contracted, not to offer choice or market convergence.

6 Open Problems and Future Work

  • The paper describes its performance guarantees as a first step toward rigorous risk management in permissionless DeFi.Source: §6 · p. 23
  • Future work includes extending the analysis to stochastic gas fees and returns that are not i.i.d.The paper also asks how to protect against front-running, quantify participant dynamics, and model the gap between collateral valuation and conversion proceeds.Source: §6 · pp. 23–24
  • The guarantees cover loans after participants contract, not whether to accept, create, or wait for offers, how quickly offers become loans, or whether similar rates converge.The paper leaves these offer and market dynamics beyond its scope.Source: §6 · p. 24

A Figures and Tables

  • Table 2 is identified as notation for the loan contract.Source: Table 2 · p. 29

B.1 Proof of Lemma 1

The proof establishes joint Gaussianity by representing the checkpoint variables as an affine transformation of independent Gaussian increments.

B.1 Proof of Lemma 1

  • The checkpoint vector is an affine transformation of the independent Gaussian increment vector, so it is jointly Gaussian.Source: §B.1 · p. 29
  • The proof refers to the resulting mean vector and covariance matrix as the stated quantities.Source: §B.1 · p. 30

B.2 Proof of Theorem 1

The proof derives liquidation probabilities by partitioning outcomes according to the first checkpoint at which the liquidation boundary is crossed, then summing those disjoint events.

B.2 Proof of Theorem 1

  • At the first checkpoint, liquidation occurs when Y1 > 0; at later checkpoints, it requires no earlier liquidation and a crossing at the current time.Source: §B.2 · p. 30
  • Differences between successive non-liquidation CDFs give the probability of first liquidation at each later checkpoint.Source: §B.2 · p. 30
  • Because first-liquidation events are disjoint, their probabilities sum, and the intermediate CDF terms cancel to yield the result for every s from 1 to d.Source: §B.2 · p. 30

B.3 Proof of Theorem 2

The proof characterizes repayment shortfall through the first-liquidation event and distinguishes it from liquidation itself: a loan can be liquidated without a lender shortfall.

B.3 Proof of Theorem 2

  • At natural ending, the lender is fully repaid, so T = ∞ cannot produce a repayment shortfall.Source: §B.3 · pp. 30–31
  • Under either liquidation rule, a shortfall at liquidation time t occurs exactly when collateral value ϕt(z) is below debt Dt.The rules can produce different recoveries when no shortfall occurs, but they yield the same shortfall event and its conditional probability.Source: §B.3 · p. 31
  • The three regions distinguish no liquidation, liquidation with full repayment, and liquidation with repayment shortfall; the middle region permits liquidation without a shortfall.Source: §B.3 · p. 32
  • The cumulative shortfall probability is built from the finite first-liquidation events, which are mutually disjoint; the proof gives the conditional probability by time s.Source: §B.3 · p. 32
  • At the first checkpoint, crossing the shortfall boundary Y1 > log c also triggers the first liquidation.Source: §B.3 · p. 32

B.4 Proof of Theorem 3

The proof derives the running maximum’s characteristic function from positive-part characteristic functions, then uses Fourier inversion to obtain the probability at the liquidation threshold.

B.4 Proof of Theorem 3

  • Pinelis’s positive-part formula derives each ψ_t from the return characteristic function f for the partial sum Y_t − y_0.The formula applies for t=1,...,d.Source: §B.4 · p. 33
  • Gil-Pelaez inversion evaluates the running maximum’s distribution at −y_0 to obtain the liquidation probability.The proof uses continuity at the evaluation points; a corresponding formula for mass at −y_0 is not considered.Source: §B.4 · pp. 33–35
  • The defining integral for the positive-part characteristic function exists as an improper Riemann integral.The passage attributes existence to Pinelis’s Proposition 1.Source: §B.4 · p. 33
  • Spitzer’s generating-function identity yields a convolution recursion for the running maximum’s characteristic function from the positive-part characteristic functions.The recursion for χ_s uses ψ_t for t=1,...,s.Source: §B.4 · pp. 33–34

B.5 Proof of Theorem 4

The proof computes repayment shortfall by combining survival through earlier periods with the conditional endpoint exceeding the shortfall threshold. The following sections specify a lending-pool model and its state representation.

B.5 Proof of Theorem 4

  • The proof obtains the shortfall probability by conditioning the endpoint on survival, applying Fourier inversion at the shortfall threshold, and summing over liquidation times.The conditional characteristic function retains paths whose past maximum satisfies the survival restriction.Source: §B.5 · pp. 36, 40

C Lending pools

  • The paper introduces a precise lending-pool model to compare against its proposed protocol.The model covers pools such as Aave and Compound.Source: C Lending pools · p. 41

C.1 Tokens, wallets, oracle

  • The pool model uses two base tokens, associated credit and debit tokens, wallet balances, and an external-environment price oracle.The pool controls the credit and debit tokens, which wallets cannot transfer to other wallets.Source: §C.1 · pp. 41–42

C.2 State

  • Pool state includes every participating wallet’s balances, pool reserves, and oracle prices; reserves represent available liquidity not captured by wallet balances.The model assumes a constant number of participating wallets and begins with zero credit and debit balances.Source: §C.2 · p. 42

C.3 Actions

Wallet actions alter token and pool balances, while state transitions also accrue interest based on pool utilization and centrally chosen coefficients.

C.3 Actions

  • Deposits, borrowing, repayment, and redemption exchange base tokens against corresponding pool credit or debit balances, subject to allowed-value checks.The deposit description assumes slippage-free conversion from a token to its credit token.Source: §C.3 · p. 43
  • Liquidation is available when a borrower’s health factor falls below 1, and wallets may also transfer base tokens but not credit or debit tokens.Network transaction fees are omitted for simplicity and can be modeled as additive random variables.Source: §C.3 · p. 44

C.4 State transitions

  • Each state transition accrues interest using a rate based on pool-reserve utilization, with coefficients α and β set by a central decision-maker.The pool then updates wallet credit and debit balances.Source: §C.4 · pp. 44–45
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