# Exchange rate

At its core, Lido for Solana (“Solido” for short) enables converting SOL into stSOL and back. This conversion involves an exchange rate. This page describes how Solido sets that exchange rate, and the rationale behind that.

To programmatically access the exchange rate, see the price oracle page.

## Definition#

The exchange rate used by Solido is x SOL = y stSOL, where:

• $x$ is Solido’s SOL balance at the start of the current epoch.
• $y$ is the amount of stSOL in existence at the start of the current epoch.

These variables are updated at the start of the epoch, and remain fixed for the duration of the epoch. This means that the exchange rate changes at most once per epoch. We will go into more detail about this in the next sections.

Solido’s SOL balance, $x$, is the total amount of SOL managed by the instance. Solido holds this SOL in two places:

• In its reserve account (unstaked).
• In validator stake accounts.

Validation rewards that are paid by the Solana runtime into vote accounts, are not counted as part of the Solido balance, until Solido withdraws them into its reserve.

The amount of stSOL in existence consists of:

• Minted stSOL, as recorded in the SPL token mint account.
• Unclaimed validation fees. For technical reasons, validation fees are not paid directly. Instead, the the amount is recorded in the Solido instance, and the fees are minted when the validator claims them.

When the Solido instance is first created, the SOL balance $x$ and stSOL amount $y$ are both zero, so the exchange rate is not well-defined. For this case, we choose to define the rate as 1 SOL = 1 stSOL.

## Deposits and donations#

To explain why the exchange rate in Solido is fixed per epoch, we need to introduce two terms:

• A deposit is a change in $x$ — Solido’s SOL balance — that keeps the exchange rate constant. In other words, it is paired with a corresponding change in $y$ — the stSOL supply — such that $x/y$ remains constant.

• A donation is a change in $x$ where $y$ remains constant.

Deposits do not affect the exchange rate, whereas donations do. If the change in $x$ is positive, then a donation benefits stSOL holders by making the SOL value of stSOL go up.

Depositing with Solido is a deposit in the above sense: we adjust the SOL balance by a given amount, and then we need to mint a corresponding amount of stSOL to keep the exchange rate fixed. (Withdrawing from Solido works in the same way, but the change in $x$ is negative.) A donation happens when new funds appear in one of Solido’s accounts, without going through the deposit or withdrawal instruction. In practice this happens when the Solana runtime pays validation rewards, but in theory any user can transfer funds to one of Solido’s accounts at any time.

Naively, we can define the $\textup{deposit}$ and $\textup{donate}$ functions as follows:

\begin{align*} \textup{deposit} : \textup{Sol} &\to \textup{Solido} \to \textup{Solido} \\ a &\mapsto (x, y) \mapsto (x + a,\, y + a \cdot y/x) \\ \\ \textup{donate} : \textup{Sol} &\to \textup{Solido} \to \textup{Solido} \\ a &\mapsto (x, y) \mapsto (x + a,\, y) \end{align*}

Here $\textup{Solido} = \textup{Sol} \times \textup{StSol}$ is the set of possible Solido states. Its elements are tuples $(x, y)$ of the SOL balance and stSOL supply, that together determine the exchange rate. $\textup{Sol}$ and $\textup{StSol}$ are the sets of SOL and stSOL balances, which for this analysis we assume to be equal to $\mathbb{Q}$. (In practice, they do not have unlimited precision, and small rounding errors do occur.)

A few properties follow from this definition:

Deposit commutes with itself:

$\forall a, b \in \textup{Sol}: \textup{deposit}(a) \circ \textup{deposit}(b) = \textup{deposit}(a + b) = \textup{deposit}(b) \circ \textup{deposit}(a)$

Donate commutes with itself:

$\forall a, b \in \textup{Sol}: \textup{donate}(a) \circ \textup{donate}(b) = \textup{donate}(a + b) = \textup{donate}(b) \circ \textup{donate}(a)$

Deposit and donate do not commute in general:

$\exists a, b \in \textup{Sol}: \textup{deposit}(a) \circ \textup{donate}(b) \neq \textup{donate}(b) \circ \textup{deposit}(a)$

## The ordering challenge#

The non-commutativity of $\textup{donate}$ and $\textup{deposit}$ presents a challenge when processing validation rewards. We would like to process a validation reward as follows:

1. Inspect the validator vote account, and see if there are any new funds in there that we can withdraw. If not, we are done.
2. Split the reward in a fee part, and an stSOL appreciation part.
3. $\textup{donate}$ the stSOL appreciation part.
4. $\textup{deposit}$ the fee part, and distribute the resulting stSOL to fee recipients.

Solido aims to support many validators, but because Solana has a fairly low upper bound on the number of accounts that a transaction can reference, it is not feasible to inspect all validator vote accounts in a single transaction. We need to visit validators one by one. This means that for validation rewards $v_1, v_2, \ldots$, and a fee percentage $f$, we get a sequence of donations and deposits:

$\cdots \circ \textup{deposit}(f \cdot v_2) \circ \textup{donate}((1 - f) \cdot v_2) \circ \textup{deposit}(f \cdot v_1) \circ \textup{donate}((1 - f) \cdot v_1)$

Note that this depends on the order in which we visit the validators! If we collect validation rewards in a different order, the fees will be different, and the final exchange rate will also be different! This ordering dependence is undesirable, especially with the eventual goal of permissionlessness in mind, where any user should be able to trigger Solido to collect validation rewards.

## Fixing the exchange rate#

To remove the ordering dependence, Solido fixes the exchange rate for the duration of the epoch. Effectively, $\textup{deposit}$ becomes indexed by the epoch:

\begin{align*} \textup{deposit}_i : \textup{Sol} &\to \textup{Solido} \to \textup{Solido} \\ a &\mapsto (x, y) \mapsto (x + a,\, y + a \cdot y_i / x_i) \\ \end{align*}

where $(x_i, y_i) \in \textup{Solido}$ is the fixed exchange rate for epoch $i$. Within a given epoch, this version of $\textup{deposit}$ does commute with $\textup{donate}$:

$\forall a, b \in \textup{Sol}, i \in \textup{Epoch}: \textup{deposit}_i(a) \circ \textup{donate}(b) = \textup{donate}(b) \circ \textup{deposit}_i(a)$

This removes the ordering dependence: we can now collect validation rewards in any order, and the net result will be the same.

An alternative way of looking at the fixed exchange rate per epoch, is to say that the order of any $\textup{deposit}$ and $\textup{donate}$ operations within an epoch is no longer relevant, and therefore the time at which they happened within the epoch is no longer relevant. Time for Solido moves in discrete ticks, one per epoch.

## Exchange rate update#

Solido stores the exchange rate that is used throughout the epoch in the Solido instance. Once per epoch, the maintenance deamon calls the UpdateExchangeRate instruction, which updates the variables to the latest values according to the definition above. The on-chain program disallows collecting validation rewards (which also distributes fees) if the exchange rate is outdated, but withdrawals and deposits are never blocked. This means that users who deposit in epoch $k$, might still get the exchange rate for epoch $k - 1$, if they manage to execute their deposit before UpdateExchangeRate executes. This is not a problem: users could have deposited in epoch $k - 1$ anyway. For Solido, UpdateExchangeRate effectively defines the start of the new epoch.