Compute negative-price probability after n steps

Q: Compute negative-price probability after n steps

This is a Statistics & Math interview question from Optiver for Software Engineer roles. View the full question and solution on PracHub.

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Question

Random-Walk Price Crossing Probability

Setup

Initial price: S₀ = s (real number).
Step size: x > 0.
Number of days: n (integer ≥ 0).
Each day t = 1, 2, ..., n, the price moves independently:
- Up by +x with probability p.
- Down by −x with probability 1 − p. So S_{t+1} = S_t + X_t, where X_t ∈ {+x, −x}.

Task

After n days, what is the probability that the final price S_n is strictly below 0? Derive:

A closed-form expression (as a binomial CDF/tail or equivalent).
An algorithm to compute it accurately for large n.

Compute negative-price probability after n steps

Random-Walk Price Crossing Probability

Setup

Task

Solution (Locked)

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