Split the Yield, Share the Risk: Pricing, Hedging and Fixed rates in DeFi

The gist The authors present a formal treatment of yield tokenization, proposing a model to characterize yield token dynamics using stochastic differential equations and deriving a no-arbitrage pricing framework for these tokens, which facilitates risk transfer and price discovery in decentralized finance (DeFi).

Yield Tokenization Model

Definition 1 (Yield Bearing Token)

The paper defines a yield bearing token as a fungible token A where the holder accrues a reward paid either in A, or in some underlying numeraire N, or in a mix of tokens Ai.

Definition 3 (Yield Tokenizer)

A yield tokenizer is defined as a smart contract that allows a holder to deposit A into the contract for a time-to-maturity T in exchange for minting a principal token P and a yield token Y.

Definition 5 (Yield Token)

A yield token is defined as the right to collect all future yield accrued on the underlying token A, until such a time when either the holder decides to sell or send Y to another holder, or the time-to-maturity T is reached.

The model assumes that asset prices change stochastically via the vector stochastic differential equation dX¯t = ¯µt(X¯t)dt + Σt(X¯t)dW¯t.

Fair Pricing and No Arbitrage

The fair price of a yield token is derived by applying the canonical no arbitrage condition, resulting in Theorem 1, which states that the price of a yield token Y with maturity T is given by P T Y(t, X¯t) = E[Z T t e - R s t r(u)duY¯ (s, X¯s) · X¯sds].

Corollary 1 claims that a portfolio of yield tokens with varying maturities, priced according to Theorem 1, cannot have an arbitrage opportunity.

Risk Management in Lending Pools

Lenders and borrowers can use yield tokens to hedge against interest rate uncertainty; specifically, a borrower might buy the yield token to lock in the interest amount they pay in a specific time period.

Theorem 1 shows that All lenders and borrowers can completely hedge their positions till time T, after selling/buying the yield token at a price equal to the fair price E[Y].

The presence of a yield tokenizer leads to increased welfare for lending pool participants, as shown in Theorem 2, where the welfare of the lending pool participants increases in presence of a yield tokenizer.

Market Maker Design and Liquidity Aggregation

The paper designs an automated market maker (AMM) that incorporates a menu of bonding curves to aggregate liquidity from participants with heterogeneous risk preferences.

Corollary 4 defines the optimally efficient menu as a tuple Ψ(x0, y0, UP, Y1, · · ·, Yn) = (ψ1, ψ2, · · ·, ψn) where each curve ψj in the menu is optimally efficient.

The liquidity provider can improve revenue by taking into account the risk preferences of traders by designing a menu of bonding curves with fees to earn premia from higher risk aversion.

Fixed Rate Lending Protocol

A modular fixed-rate lending protocol is proposed that synthesizes on-chain yield token markets and lending pools, enabling robust interest rate discovery and enhancing capital efficiency.

The protocol allows a borrower to have an effectively fixed interest rate by simulating the selling of yield futures in the AMMs corresponding to the asset.

Slashing Insurance Markets

Yield tokenization for liquid staking tokens opens up hedging opportunities for users, including gas fee hedging and slashing risk insurance.

Lemma 9 provides the amount of native token N a liquid staker should spend to buy insurance against slashing, given the probability pS that a slashing event will occur.

The analysis shows that holding the principal token can act as a form of insurance, where the value of the principal token would rise from 1 to 1−p in case of slashing.

The paper concludes that by modeling agents with heterogeneous beliefs and designing market makers that aggregate liquidity across diverse risk preferences, programmable tokenization can replicate and extend core financial infrastructure within a permissionless setting.

1. Bold price discovery in yield markets using a menu of bonding curves: The system can design a menu of bonding curves to aggregate liquidity from participants with heterogeneous risk preferences and use these to create an efficient and incentive-compatible mechanism for trading yield tokens and yield futures.

2. Fixed rate lending optimization: The protocol can synthesize on-chain yield token markets and lending pools, enabling robust interest rate discovery, leading to a modular fixed-rate lending protocol that synthesizes on-chain yield token markets and lending pools, enabling robust interest rate discovery and enhancing capital efficiency.

3. Risk hedging for borrowers and lenders: The system can enable participants to hedge against interest rate uncertainty by allowing borrowers to buy the yield token to lock in the interest amount they pay in a specific time period, which helps both parties hedge against interest rate manipulation by an adversary.

4. Optimized liquidity provision via risk-aware bonding curves: The Automated Market Makers (AMMs) can be designed so that liquidity providers can specify their bonding curve as per their risk appetite, allowing them to earn premia from higher risk aversion, which leads to a more efficient transfer of interest rate risk.

5. Slashing insurance markets for liquid staking: The system can provide a mechanism where a liquid staker can buy principal tokens to insure against slashing, with the required amount being determined by the formula: pI = pS / (1 − p + pS E[Qn i=1(1 + Yi)]).

We present the first formal treatment of yield tokenization, a mechanism that decomposes yield-bearing assets into principal and yield components to facilitate risk transfer and price discovery in decentralized finance (DeFi). We propose a model that characterizes yield token dynamics using stochastic differential equations. We derive a no-arbitrage pricing framework for yield tokens, enabling their use in hedging future yield volatility and managing interest rate risk in decentralized lending pools. Taking DeFi lending as our focus, we show how both borrowers and lenders can use yield tokens to achieve optimal hedging outcomes and mitigate exposure to adversarial interest rate manipulation. Furthermore, we design automated market makers (AMMs) that incorporate a menu of bonding curves to aggregate liquidity from participants with heterogeneous risk preferences. This leads to an efficient and incentive-compatible mechanism for trading yield tokens and yield futures. Building on these foundations, we propose a modular fixed-rate lending protocol that synthesizes on-chain yield token markets and lending pools, enabling robust interest rate discovery and enhancing capital efficiency. Our work provides the theoretical underpinnings for risk management and fixed-income infrastructure in DeFi, offering practical mechanisms for stable and sustainable yield markets.