The timing of the settlement calls are automated using Chainlink Automation. At expiration, the settleVault() method is called, and the vault obtains the settlement price from GMX's on-chain oracle and stores it in the vault contract. The settleVault() method also rolls the vault into the next epoch, beginning the new deposit period.
After settleVault() is called, option holders can call settleOption() to claim proceeds if the options expired ITM. Depositors can claim their stablecoin premiums from the short positions, along with their collateral by calling claim(). Depositors have the option to automatically rollover their collateral into the next epoch when claim() is called.