I have to use simple no arbitrage arguments to find the price of a barrier option where initial stock price $S_0 = 100$ and barrier/strike $B = K = 80$. Here we don't assume geometric Brownian motion for the stock price. What should be the approach for this problem? The problem I'm facing is at time $T$ the stock price can be above the barrier or below the barrier. But in the second possibility, both the prices can be above the barrier.
You need a couple more assumptions and it becomes doable.
(1) no arbitrage
(2) no interest rates or dividends
(3) spot price moves continuously
Then there is a replication possible. Buy the forward struck at $K$ expiring at the same time as the barrier option. If the barrier is ever hit, the forward is at-the-money so has value zero. In that case sell the forward and have zero value at expiry. If the barrier is never hit, hold the forward until expiry, at which time it has value $S(T)-K$, the same as the barrier option.
In every case the replicating portfolio gives the same final value as the barrier option, so it must have the same initial value as well.
Thus the barrier option value is $S(0)-K $.
Looking back thru the argument, we can actually weaken the assumptions slightly. There can be an interest rate, IF it is exactly equal to the dividend rate. Then the replication succeeds and the price is now $exp (-r T)(S (0)-K) $. And jumps in the spot are allowable, IF it is guaranteed that a jump will not cross the barrier. For instance, upward jumps would be fine.