# 68 | The Physics of On-Chain Lending (I)
Developing Appropriate Risk Pricing Frameworks for Decentralized Lending
The foundation of modern derivative pricing rests upon the realization that, in a so-called complete market, the expected return of an asset is a statistical nuisance. To price a contract derived from the behavior of an underlying asset, we must perform a radical act of subtraction: idiosyncratic expectations must be discarded as mathematically untreatable. If we seek a sound pricing framework, we must first admit that our subjective view is just irrelevant noise. Coherent derivative pricing lives in a ghost world of faceless assets. In this world, value is not linked to expectations, but to the cold assumption that the aggregate market is governed by localized rationality. Rather than attempting to predict the future, we solve for a state of zen-like humility: every instrument, regardless of its own characteristics, drifts at the heartbeat of the risk-free rate.
We know well that every event serves as a violent ex post reminder of the absurdity of those theoretical claims. Yet, we have no superior rational alternative. In order to position ourselves, ex ante, to do the best possible thing, we must embrace the paradox, and admit that the world’s collective rationality is the only plausible assumption—even if we know for a fact this to be a fiction.
Developing Appropriate Risk Pricing Frameworks for Decentralized Lending
Governance, On-Chain Risk Management, and the Vault Curator Concept
Addressable Market, Profitability, and Implications on Token Valuation
A Mathematical Approach to DeFi Lending
This DR is the first of a series on permissionless on-chain lending markets. Lending has been, since DeFi’s inception, one of the most successful novel primitives that blockchain rails have brought to finance. With the rise of Morpho and its isolated market + vault + curator architecture (we have been following the project since the very beginning—see DR #35) the space has reached, together with stablecoins and prediction markets, general attention levels. Lending markets pass the banker friend test (my banker friends now know what a stablecoin is, but also what a prediction market and what a lending market are) even if they don’t pass my mother test—my mother doesn’t yet know about prediction and lending markets even if she knows about stablecoins.
Morpho’s total deposits currently exceed $11b, distribution into widely adopted frontends like Coinbase, Kraken, and others has reached clear PMF, Apollo just announced a cooperation agreement to acquire up to 90m MORPHO tokens (9% of total supply) over 48 months, RWA-based markets are resurgent. The thesis that lending markets will be the de facto primitive of choice for most financial applications is hard to ignore.
But powerful primitives deserve rigorous frameworks. Before we get to incentive alignment, the nature of the lending market maximalist claims, and ultimately also to the question about MORPHO valuation—for disclosure I have indirect personal exposure to MORPHO and a longstanding relationship with the founders, we need to define the mathematical tools required to understand what is actually happening inside these markets. What risks are the participants taking? What is the market pricing? What should it be pricing? There is no black and white—the formulation of appropriate pricing tools should act solely as a map to understand what’s going on in crypto capital markets, and what information about the future we can derive from current prices.
This one is for the nerds.
Merton and (Proper) DeFi Lending
In 1974, Merton introduced a framework to assess credit risk by modeling the behavior of a company’s assets and analyzing certain boundary conditions. The model starts from an extremely simplified capital structure: a firm has total assets worth V, and a single zero-coupon risk-free bond with face value L maturing at time T. If, at maturity, Vt > L, the company will honor the debt and the residual value will accrue to equityholders. If Vt < L, the firm defaults, bondholders take the assets, and equityholders get nothing. Through this simple framework, the model attempts to value a company’s debt based on the return and volatility characteristics of its asset—and of interest rates.
In this simple framework we can reconstruct debt as an equity-linked payout product. More specifically, as a combination of a risk-free bond and a short put option on the company’s assets.
A similar approach could be, in theory, applied to DeFi lending and collateralized debt positions—with caveats we will discuss later on. Lending risk, and the expected return materialized through the required credit spread to hold the position, can be modeled via return and volatility assumptions of the underlying collateral, as well as the defined lending conditions.
Caveat 1: the first-passage problem. Merton’s classic approach assumes that the debt matures (and the assets are observed) only at a specific point in time in the future. This is not what happens in reality and especially in the context of a CDP, where the value of the collateral is continuously observed via oracles and the debt is liquidated as soon as the collateral crosses a predetermined liquidation threshold—Liquidation Loan-to-Value or LLTV in Morpho Blue’s context. This is the first-passage problem Black and Cox addressed in 1976.
In Black-Cox, the collateral follows a stochastic process (geometric Brownian motion or GBM) where the growth of its value C is expected to follow its net long term mean return, impacted by the asset’s estimated volatility.
What we want to estimate is the probability that the collateral C will cross a specific boundary condition B = D / LLTV1 before the end of the time period T. Rather than the probability of default, actually, the model intends to capture the first-passage probability of default—i.e. the probability that C touches B at any time within [0,T]. This has a closed form:
The first term of the formula represents the probability that the collateral value ends up below the boundary condition at the final time T, while the second term represents path correction—i.e. the adjustment for the probability that C touches the barrier but subsequently recovers by the end of the period.
We can now proceed to model what the annualized credit spread should be:
where LGD (or Loss Given Default) is the fraction of debt that is unrecoverable upon liquidation. Within the Morpho context, for very liquid crypto collateral such as BTC or ETH, it is fair to assume that LGD is approximated by the liquidation incentive. In Morpho, the liquidation incentive (LIF) is formulaic and connected to LLTV—for LLTV of 86% (as in Coinbase’s cbBTC/USDC market) the LIF would be I believe 5%.
The other crucial metric to calculate is the distance to default:
This metric allows to evaluate the distance a position has against its LLTV, but expressed in variance units rather than raw LTV. For example, for a position at 70% LTV against an 86% LLTV with ETH at 75% volatility, the risk-neutral drift μ = r − σ²/2 is deeply negative: 0.0425 − 0.28125 ≈ −0.242. The distance to default over a one-year horizon is approximately 0.59, meaning in other words that the barrier is not far away at all in risk-neutral terms: a slightly excessive movement of the asset vs. expected volatility is highly likely to bring the collateral into liquidation. Or, said in even simpler words, that the high volatility consumes the 16 percentage-point buffer quickly.
Caveat 2: the fat tail and jump risk problem. GBM assumes a continuous path in asset prices. Crypto assets violate this routinely, especially those with limited liquidity. Merton (again) was the first one to consider this jump-diffusion problem in 1976. In this revised paradigm, the collateral process is modeled as:
where N is a Poisson process with intensity λ—number of jumps per year, J is the random jump size (lognormally distributed) and k is a compensator3. Under the assumption of jumps, the first-passage probability has no clean closed form and must be approximated:
Caveat 3: zero-coupon spreads and perpetual lending. We have (almost) all the elements to start doing some quantitative analysis. The standard zero-coupon credit spread formula described above, however, answers the question “what spread should a fixed-maturity bond carry", which is not entirely relevant for DeFi perpetual lending—where there is no maturity. A more appropriate metric would consider the forward hazard rate: h(t) = f′(t) / (1 − f(t)), where f(t) = P(τ ≤ t). The hazard rate tells us the instantaneous probability of sudden death given that an asset has survived so far. The average credit spread becomes instead s(T) = −ln(1 − LGD × P(τ ≤ T)) / T, which is the annualized expected loss rate over horizon [0, T]. These two measures are related: s(T) = LGD × (1/T) ∫₀ᵀ h(t) dt.
Interestingly (for the nerds) the term structure so constructed is non-monotonic, peaking at intermediate horizons and declining at longer ones. This reflects the interaction between risk-neutral drifts and first-passage dynamics: in a perfectly risk-neutral world, investors are demanding exactly the expected return of the assets to hold them, meaning that volatility continues to impact asset performance and their proximity to the boundary condition even for appreciating assets. This feature is quite useful especially in perpetual markets. For both ETH and BTC, given the high observed volatility, the drift is significantly negative; surviving positions are not fundamentally safer—they are just lucky, and the drift keeps pushing them towards default. In our simulation, the instantaneous hazard rate rises sharply from inception as diffusion contributes alongside jump risk, peaks around day 7–14, then slowly declines as the surviving population becomes increasingly concentrated in paths that happened to move favorably despite the adverse drift. The average spread s(T) also inherits this shape. We can see the calculations below.
Caveat 4: active rebalancing. The table seems to be telling us that, for an ETH position at 70% LTV, the probability of triggering liquidation is 70-80%, and lenders should require > 400 bps of spread above risk-free considering the liquidation incentive. That’s a lot. This passive-borrower model, however, assumes that collateral is posted and left alone. This is not what happens in practice, where Morpho borrowers (or curators) actively manage positions. If we go to the other extreme, under assumptions of continuous rebalancing with unlimited capital, all diffusion-driven defaults are eliminated: the borrower simply tops up collateral as it drifts toward the barrier. The only surviving defaults are due to jumps large enough to penetrate the buffer in a single instant—before any rebalancing can occur. Formally, default would occur if and only if a single jump J satisfies J ≤ ln(LTV / LLTV). For ETH at LTV = 70% and LLTV = 86%, this requires an instantaneous drop of c. 20%. This, clearly, under the extremely benign scenario of infinite and infinitely reactive capital.
We can now look at revised credit spreads under different rebalancing assumptions, for ETH/USDC markets with LTV = 70% and LLTV = 86%, within a 1-year time horizon:
The pure-jump spread of 45 bps should be the irreducible floor—and under pretty conservative parameters. In reality, borrowers have all sorts of frictions, including finite reserves. With only 20% extra capital available for top-ups, a Monte Carlo simulation gives us a 350 bps average spread; at 100% extra capital the spread would be 130 bps. For discrete rebalancing frequency, moving from per-block to slower rebalancing also progressively reintroduces diffusion risk. For a more realistic borrower behavior—daily monitoring and finite capital, the spread should be 250–400 bps. With observed depositor rates in the range of 0-20 bps for relevant Morpho markets, the mispricing is real across all model specifications.
Overcollateralized lending on liquid crypto-native collateral is Morpho’s bread and butter and represents the vast majority of its TVL. However, we observe consistently narrower spreads (by 5-10x) over risk-free vs. what rational lenders would require based on appropriate mathematical specifications.
So What? It’s Reg-Arb, Baby
Even considering all the effective structural tools that stand in protection of permissionless lending, Morpho depositors are demanding significantly less than what they should from a risk-neutral perspective. The hypothesis that Morpho credit is mispriced is well-supported by the data. The gap could be explained by several compounding factors.
(A) Depositor misperception. Most Morpho USDC depositors (particularly those routed through Coinbase or similar interfaces) are retail allocators treating the product as USDC savings rather than as a credit product. They are implicitly selling puts on crypto collateral without recognizing it as such. The clean UI, the yield headline, and the lack of visible loss events create the perception of a risk-free deposit. In other words, depositors don’t understand the contingent claim they hold.
(B) Regulatory arbitrage. The current interpretation of the Genius Act (and related stablecoin legislation) is making it harder for intermediaries to directly share the underlying yield with depositors. By legally distinguishing decentralized software protocols from financial intermediaries, the Clarity Act would allow Morpho and other similar constructs to benefit from the de facto lack of competition from traditional (regulated and heavily capital cushioned) intermediaries and remain the most convenient, immediate, no-KYC non-custodial savings account of choice for stablecoin holders. Ironically, by forbidding direct risk-free sharing by stablecoin issuers, the regulator is enacting a total pass-through risk transfer onto depositors—one that is not fully compensated by the structural protection mechanisms of decentralized lending. For specific jurisdictions, the well-known tax arbitrage on margin lending vs. selling might be another compelling reason to increase borrowing demand.
(C) Survivorship bias. The major loss events (e.g. Resolv, deUSD) hit specific vaults, not the headline USDC vaults that most depositors track. The flagship vaults curated by Steakhouse and Gauntlet have avoided catastrophic loss so far, creating a track record that reinforces the safe yield narrative and makes rational credit risk pricing less compelling. Credit is a dangerous beast.
(D) Bull market masking. Under the physical measure4, positive expected returns on ETH/BTC dramatically reduce observed liquidation frequency. Depositors extrapolate from a benign period and conclude the risk is low. As we mentioned several times, the risk-neutral spread (the mathematically correct pricing metric) is invariant to collateral drift, but the lived experience of depositors is driven by physical-measure outcomes. The mispricing will become visible when the market turns.
(E) Token subsidies. Token incentives have historically compressed observed rates by subsidizing both sides of the market. Morpho, thanks to its widespread distribution through integration with CeFi venues and UIs, has become the ideal place for protocols and asset issuers to attract adoption through incentive distribution. Being an incentive supermarket, however, has nothing to do with lending.
Leverage Looping: a Different Animal Entirely
So-called leverage looping, where actors deposit asset A and borrow (often instantaneously) asset B, and where the two assets have extremely high correlation and LLTV—like wstETH/wETH, is a different animal altogether. In this case, the borrower is not exposed to the directional price of ETH—both sides of the position move together. The risk is entirely in the basis—the wstETH/ETH exchange rate. A correct application of Merton would be to the basis process, and not the ETH price. The relevant volatility is not ETH’s 75% annualized volatility, but the wstETH/ETH basis volatility, which based on observations runs c. 1–3% annualized in normal conditions and can spike to 5–10% during depegs or liquidity stress.
The correct way to price looping strategies is not, in my opinion, as a credit product, but as a leveraged carry trade on mean-reverting basis. The relevant metrics are the following:
Carry: (staking yield − borrow rate) × leverage turns
Basis volatility: σ_basis, which is regime-dependent (normal vs. stressed)
Mean-reversion speed: how quickly the basis returns to parity after a shock
Liquidation distance: margin in basis-vol units
For a user, the risk-reward might be attractive at moderate leverage (3–5×) where the basis would need to move 15–30% to trigger liquidation—well beyond historical extremes. Above 10×, the strategy becomes a convex bet on basis stability that blows up precisely during liquidity crises, when staking redemption queues lengthen and the basis dislocates most. At 10× leverage, a 5% basis depeg triggers liquidation. In normal conditions, this is relatively safe—the basis simply doesn’t move that much. But in stress, a 5% depeg is plausible over a day or two. At 15× leverage, the margin of safety shrinks even further.
At its peak in mid-2024, a material share of Morpho Blue’s total TVL was driven by a single recursive structure: sUSDe looping. The mechanics were straightforward: deposit Ethena’s sUSDe as collateral, borrow DAI/USDC at low rates, convert back to sUSDe, and loop. At 86% LLTV, borrowers could achieve effective leverage of 7–10× on sUSDe’s 30–60% annualized yield during the funding rate bonanza. Interestingly, MakerDAO (now Sky) was the balance sheet for the trade, providing liquidity through its Direct Deposit Module (D3M) and Spark. The initial allocation was $100m in March 2024, soon increased to $1b. The Maker → Spark → Morpho → Ethena tower, as I described it at the time—see DR #62, was a textbook case of DeFi composability: Maker was LP’ing the trade, Morpho was intermediating it, and Ethena was manufacturing the yield. Each layer earned its spread. Sky has since replicated the pattern with its own product: today sUSDS-collateralized markets on Morpho account for over $300m in deposits—the third largest market, with the same recursive looping (at LLTV of 96.5%) available to anyone willing to lever up on Sky’s savings rate.
Leverage looping strategies can be highly attractive yielding strategies enabled by the structural characteristics of DeFi, and they work pretty well in isolated markets. Bot-based looping can juice up yields in normal conditions, even if losses can be devastating when the trade unwinds and widening frictions amplify spreads further.
Illiquid Collateral: Where All Models Break
Non-crypto-native collateral has been at the center of attention on Morpho for the past several months. The Apollo cooperation agreement, the proliferation of RWA-backed markets, the unmistakable desire of traditional and alternative asset managers to access growing pools of (unsophisticated) retail capital through (unregulated) DeFi integrations. The claim being made, implicitly or explicitly, is that infrastructure designed for deeply liquid crypto-native assets like ETH and BTC can be repurposed for assets that share none of those properties. Affectionate readers of DR will recognize the pattern. We have been here before. When crypto-native yield compresses, the pool of capital sitting on non-custodial rails (capital that cannot easily access other avenues for deployment) starts reaching for off-chain sources of return—the survivors of the first RWA wave will remember. The dream of creating an illiquid credit supermarket with limited supervision and instantly connected to a growing list of financial front-ends across the world is compelling, and not necessarily for the best reasons. After pension funds and insurance companies, private credit is looking for its next escape from reality.
For non-crypto native collateral every assumption in the Merton framework fails simultaneously:
(A) Unobservable volatility. The core input σ is not observable. The price of a (e.g.) private credit position is a mark—updated quarterly, often by the originator. Measured volatility from smoothed marks is artificially suppressed. This is the Scholes-Williams (1977) problem: the true σ is substantially higher than the observed σ, and the distance to default computed from stale marks is dangerously inflated. A position that appears to have DD = 3σ based on smoothed NAVs may be at DD ≤ 1σ in reality.
(B) Discrete monitoring defeats first-passage. The first-passage framework assumes continuous barrier observation. If the oracle updates weekly or monthly, the collateral can be deep below the LLTV before the protocol detects it. Broadie, Glasserman, and Kou (1997) show that for discrete monitoring, the effective barrier shifts by approximately β·σ·√(Δt) where β ≈ 0.5826 and Δt is the monitoring interval. For monthly-monitored illiquid assets with high true volatility, this correction materially increases the implied default probability.
(C) Liquidation is not atomic. Even after detecting a breach, selling private credit takes weeks or months—legal processes, servicer transitions, buyer negotiation. During the liquidation delay Δ, the asset continues to deteriorate. The expected LGD becomes:
where λ is the fire-sale discount (20–50%) and the collateral evolves adversely during the delay. For liquid crypto, Δ ≈ seconds and λ ≈ 5–15%. For private credit, Δ = 30–90 days and λ = 20–50%. The compounding of delay and discount can turn a moderately underwater position into a near-total loss.
(D) Enforceability. Liquidating a tokenized claim doesn’t recover value—it transfers a legal right that still requires enforcement, potentially across jurisdictions. The recovery rate is a function of legal process, servicer quality, and underlying borrower solvency, not of market price.
Given the comments above, continuous-time diffusion models seem unhelpful for pricing RWA risks and spreads. A binary Markov regime-switching model with two states would be more appropriate. The Markov model is honest about what it doesn't know: it doesn't pretend to have a volatility input or a continuous price process. It says: "in normal times, losses are low; in stress, losses are catastrophic; stress happens roughly once every n years and lasts about m months." The implied spread is the minimum compensation for bearing this risk profile with illiquid, hard-to-enforce collateral. I do not believe we have enough information about the current crop of RWA-denominated assets to provide a detailed analysis.
I remain extremely bearish on non-crypto-native collateralized lending, for the reasons described above. My concerns are significantly exacerbated by the temptation of adding leverage via looping strategies where the basis spread remains unobservable until the first impairment event. In addition, considering the hurdle rate of return of crypto capital markets, the adverse selection risk on those assets is high.
The Arrival of Morpho v2
Morpho v2 (which is actually composed of two distinct upgrades, one for Vaults—in production since last September—and one for Markets) is coming online. For Markets, the most ambitious part, the update introduces an intent-based system that matches desired terms on both sides of borrowing. Crypto lending has been surviving on the idea of a pool (isolated as in Morpho or commingled as in Aave) and an arbitrary monotonic-with-discontinuity interest rate curve that is very simple but that does not allow proper interest rate discovery, or the creation of a term structure. Morpho intends to change this.
The bet, as old as DeFi—i.e. not so old, is to finally allow proper institutional capital allocation by introducing predictability in on-chain capital markets. There are still several missing pieces. Intent-based matching requires solver depth that does not yet exist for illiquid pairs. Fixed-rate, fixed-term loans introduce duration risk that Morpho’s current architecture has never had to manage. And the assumption that institutional capital will flow into a system where counterparty matching is delegated to off-chain solvers with no regulatory accountability goes against the certainty guarantees those institutions typically require. Morpho v2 surfaces problems that DeFi has not yet solved—and that may, in fact, run against DeFi’s core comparative advantages. The ambition is grand and justified, and I wish them ultimate success, even if the road will be arduous.
This is a different way of saying when the LTV = D / C reaches the LLTV.
The rate at which the asset grows in a model where risk-aversion doesn't exist and where everybody is fine solely holding risk-free instruments. The risk-neutral drift, in other words, is telling us the behavior we should expect in the underlying assets under the assumption that all investors are risk-neutral and with perfect information.
Parameters for ETH and BTC markets based on 2020-2025 historical data are the following. ETH: annualized realized volatility 75%, jump intensity 1.5, mean jump size -8.3%, jump size volatility 8%. BTC: annualized realized volatility 55%, jump intensity 1.0, mean jump size -6.5%, jump size volatility 6%.
Also referred to as the real-world or statistical measure, represents the probability distribution of asset returns derived from historical data and empirical observation. Unlike the risk-neutral measure, which assumes all assets earn the risk-free rate, the physical measure incorporates the risk premium—the additional return investors demand for bearing market risk. In structural credit models, it is used to estimate the actual probability of default and for stress-testing portfolio survival, whereas the risk-neutral measure is used to derive market-implied credit spreads.
it is intellectually interesting, but is incorrect
1. LGD cannot be approximated by the liquidation incentive
2. It confuses liquidation trigger with credit loss event
great piece. the depositor misperception point is the one that sticks because it's such a common thing for people