Origins
The word exchange hides a rare linguistic journey. Most English words from French trace back to Latin or Greek. Exchange traces back to Celtic. The Gauls used *cambion to mean 'barter' or 'exchange', and the Romans, lacking a satisfying word of their own, borrowed it as cambiāre.
From cambiāre, Vulgar Latin created *excambiāre — 'to exchange out' — which entered Old French as eschangier. English picked it up in the 14th century. The prefix shifted from es- to ex- under Latin influence, giving the modern spelling.
The Celtic root colonised the Romance languages thoroughly. Spanish cambiar, Italian cambiare, Portuguese cambiar, and French changer all descend from it. In Italy, a cambio is a bureau de change. In Spain, cambio appears on every currency exchange sign.
French Influence
The word exchequer shares part of this history. The medieval Exchequer — the royal treasury — took its name from the chequered cloth on which accounts were settled, itself from Old French eschequier ('chessboard'), related to the counting and exchanging of money.
Even botany adopted the root. The cambium layer in a tree trunk is where cells transform and exchange — old wood on one side, new bark on the other. Growth, in a tree as in a market, happens at the point of exchange.