papers/lux-liquid-staking/sections/08-exchange-rate-dynamics.tex

\section{Exchange Rate Dynamics}
\label{sec:dynamics}

\subsection{Rate Evolution}

The sLUX/LUX exchange rate evolves as:
\begin{equation}
    \rho(t+1) = \rho(t) \cdot \frac{\mathsf{totalPooled}(t) + (1-f) \cdot R_t - \text{slashLoss}_t}{\mathsf{totalPooled}(t)}
\end{equation}
where $f = 0.10$ is the fee rate, $R_t$ is gross rewards, and $\text{slashLoss}_t$ is the net loss after insurance.

\begin{theorem}[Rate Lower Bound]
The exchange rate satisfies $\rho(t) \geq 1 - p_{\text{max}}$ where $p_{\text{max}} = 0.033$ is the maximum single-epoch loss rate (10\% delegation cap $\times$ 33\% critical slash).
\end{theorem}

\begin{proof}
In any epoch, at most one validator can be critically slashed (33\%). With delegation cap of 10\%, the maximum pool loss is $0.10 \times 0.33 = 0.033$ of total pooled value. If the insurance pool is empty, $\rho$ decreases by at most $0.033$. Over multiple epochs, subsequent slashing events are independent (different validators), and the insurance pool rebuilds between events.
\end{proof}

\subsection{Peg Stability}

The sLUX/LUX market price on AMMs may deviate from the protocol rate $\rho(t)$:

\begin{definition}[Peg Deviation]
Let $P_{\text{market}}$ be the AMM price and $\rho$ the protocol rate:
\begin{equation}
    \text{deviation} = \frac{P_{\text{market}} - \rho}{\rho}
\end{equation}
\end{definition}

Arbitrage maintains peg stability:
\begin{itemize}
    \item If $P_{\text{market}} < \rho$: buy sLUX on AMM, redeem via protocol at $\rho$ (profit after unbonding).
    \item If $P_{\text{market}} > \rho$: mint sLUX via protocol at $\rho$, sell on AMM (instant profit).
\end{itemize}

\begin{proposition}[Peg Convergence]
Under arbitrage, the market price converges to $P_{\text{market}} \in [\rho - \delta, \rho]$ where $\delta$ is the AMM swap fee plus the time value of the unbonding period.
\end{proposition}

For a 0.3\% AMM fee and 14-day unbonding at 7\% annual opportunity cost: $\delta \approx 0.003 + 14/365 \times 0.07 \approx 0.57\%$. The peg should hold within 0.6\% of protocol rate.