Compute negative-price probability after n steps

A price starts at value s. Each day for n independent days, it moves to s + x with probability p and to s − x with probability 1 − p (x > 0). After n days, what is the probability that the final price is strictly below 0? Derive a closed-form expression (in terms of a binomial tail or equivalent) and describe an algorithm to compute it accurately for large n.