Before revision:  Via a natural assumption of limited opportunities for unlimited resulting wealth from trading, coined the "No Unbounded Profit with Bounded Risk" condition ,  we establish that  asset-prices have to be semimartingales.
After revision:  Via a natural assumption of limited opportunities for unlimited resulting wealth from trading, coined the "No Unbounded Profit with Bounded Risk" condition , namely, absence of arbitrages of the first kind, we establish that asset-prices have to be semimartingales.