This paper provides model‐independent lower bounds for prices of arithmetic Asian options expressed through prices of European call options on the same underlying that are assumed to be observable in the market, and the corresponding subreplicating strategy is identified. The first bound relies on the no‐arbitrage assumption only and turns out to perform satisfactorily in various situations. It is shown how the bound can be tightened under mild additional assumptions on the underlying market model. This considerably generalizes lower bounds in the literature, which are only available in the Black–Scholes world. Furthermore, it is illustrated how to adapt the procedure to the case where only a finite number of strikes is available in the market. As a by‐product, the finite strike upper bound on the Asian call price of Hobson et al. (2005a.), who considered basket options, is rederived. Numerical illustrations of the bounds are given together with comparisons to bounds resulting from model specifications.