This paper studies optimal trading and statistical arbitrage in constant product markets (CPMs), the most common type of automated market maker (AMM) in decentralised finance. The authors first empirically show that execution costs in CPMs are accurately approximated by convexity costs, which are linear in trade size and depend on the pool depth and exchange rate. Using Uniswap v3 data, they confirm that convexity costs capture the curvature of the trading function. They then formulate three stochastic control models for a liquidity taker who trades in a CPM: (1) when exchange rates form in a centralised exchange (CEX), (2) when rates form in the DEX, and (3) when rates form in both venues with cointegration. The optimal strategies balance exchange rate risk and execution costs, and are characterised by semilinear PDEs. To overcome computational challenges, the authors propose a piecewise constant approximation of convexity costs that yields closed-form strategies and converges to the optimal continuous strategy. Out-of-sample tests on Uniswap v3 data from 2021–2023 show that the proposed liquidation strategy significantly outperforms TWAP and single-trade benchmarks, and that statistical arbitrage strategies exploiting CEX/DEX discrepancies are profitable. The work provides a rigorous framework for algorithmic trading in AMMs and highlights the importance of accounting for stochastic convexity costs.