The risk-free rate (US T-bills) anchors all pricing theory; however, DeFi lacks a formal equivalent, despite having protocol-guaranteed staking rewards. We formalize blockchain staking as a protocol-conditional risk-free rate and construct the first academically rigorous numéraire framework for pricing decentralized finance (DeFi) derivatives.

Staking basics

The two dominant types of consensus mechanisms on blockchains are proof of work (PoW) and proof of stake (PoS). The latter relies on the idea that new blocks are added to the blockchain by validators, each of whom has deposited coins as a staking account. If validators behave dishonestly, such as by attempting to falsify transaction records, their staked assets are destroyed, a process known as slashing. Conversely, honest validators will receive a portion of the block reward, which depends on how much they stake.

PoS is used by an increasing number of blockchains, most notably Ethereum. Because honest validators yield a deterministic reward, guaranteed by the protocol, at the end of each block, we construct the staking rate as the risk-free rate on such blockchains.

The Validator Account

Similarly to the money market account (MMA) in TradFi, we construct the validator account $\Pi_B$ for a given blockchain, which represents the staked assets, with $r$ denoting the staking rate.

$$d\Pi_B(t) = r(t)\Pi_B(t)dt$$

For simplicity and coherence with traditional finance (TradFi), we assume continuous compounding. This is not true for all blockchains, though.

Concept	Traditional Finance	DeFi (VA Framework)
Numéraire	Money Market Account	Validator Account
Risk-free rate	T-bill rate	staking rate
Measure	$\mathbb{Q}$ (risk-neutral)	$\mathbb{B}$ (blockchain measure)
Discounting	$\exp\left(-\int r dt\right)$	$\exp\left(-\int r dt\right)$
Martingale property	S/MMA is $\mathbb{Q}$-martingale	S/$\Pi_B$ is $\mathbb{B}$-martingale

By construction of $\Pi_B$ and the properties of the staking rate $r$, we can demonstrate that the validator account serves as a valid numéraire.

Why I excluded the MEV

We note that validators have the privilege of ordering transactions within the same block, which can allow them to position some transactions favourably and receive an opportunistic reward called the maximum extractable value (MEV).

Constructing the $\mathbb{B}$-measure

We can then apply the fundamental theorem of asset pricing (FTAP) and construct a risk-neutral measure $\mathbb{B}$ on the blockchain. Under this measure, all future cash flows are discounted by the validator account.

Under canonical notations, we can write the following.

$$ \begin{align*} \forall t\leq T,\quad S^\mathbb{B}(t) &= \frac{S(t)}{\Pi_B(t)}\\ &= \mathbf{E}_\mathbb{B}\left[\frac{S(T)}{\Pi_B(T)}\middle|\mathcal{F}t\right] \end{align*} $$

with $S^\mathbb{B}$ a $\mathbb{B}$-martingale. This is a powerful result that enables us to extend most of TradFi financial mathematics to DeFi.

Empirical validation

We validated our observation over 3 months of data on Ethereum, which amounts to 92 days, or 656,197 blocks. Our results confirmed the following.

The MEV coefficient of variation is above 40% and is only present in 92% of the blocks, indicating its unsuitability for constructing the staking rate.
The staking rate (excluding MEV) has a coefficient of variation of 7.18%, which is in the same order of magnitude as SOFR (1.88% during the same period), the benchmark in TradFi. The stakign rate does behave as expected as a risk-free rate.
We could fit a Vasicek model, a common representation of the interest rate, to the daily staking rate and obtained a KS p-value of around 0.65, indicating a good fit.

Pricing on-chain

In the paper, we provide pricing examples on-chain for stablecoins and LSTs. For LSTs in particular, we used the Lido sETH rate and evidenced a liquidity premium of 0.14% annualized.

Methodology

This post refers to my paper The Validator Account: A New Numéraire for Assets on Blockchains. A public version is available on SSRN.

All data on the Ethereum blockchain is public and can be accessed, either by setting up one's own node or via open-source APIs. I personally chose BeaconCha.In API because it is relatively simple, has no added proprietary information, and I was more familiar with it. Our data sample spans from July 1st to September 30th, 2025. This covers 656,197 blocks on Ethereum, which is a large enough sample. We note also no major protocol update during that time.

Arbitrage pricing theory is a fundamental concept in quantitative finance. An essential foundation is Harrison & Pliska, who introduced the modern form of the FTAP in their 1981 paper.

While many books have been written on this topic, I recommend Shreve's classic Stochastic Calculus for Finance series. Interest Rate Models by Brigo and Mercurio is also a must-read, providing more detailed insights into TradFi pricing for interest rates.

You can view the other references I quote directly from my paper.

On-Chain NFT

This paper is represented on-chain by NFTs on the following chains. The metadata is compatible with Schema.org JSON-LD standards for agentic AI discovery, parsing, and citation.