Valuing a Digital Cash-or-Nothing Option on Gold

This interview question asks you to value a digital cash-or-nothing option written on a commodity whose price follows an arithmetic Brownian motion. The payoff depends only on whether the underlying price exceeds a fixed threshold at a given maturity, not on how far above it ends up. Candidates must translate the stochastic description of the gold price into a distribution for its future level, then use that to compute the expected payoff and present value. This style of question is common in quantitative finance roles that focus on derivatives pricing, especially when an interviewer wants to see comfort with nonstandard underlying processes beyond the textbook geometric Brownian motion.

The solution relies on understanding normal distributions for terminal prices under arithmetic Brownian motion, as opposed to lognormal distributions. It leans on identifying the mean and variance of the future price, mapping the strike condition into a tail probability, and evaluating that via the normal cumulative distribution function. With a zero risk-free rate, discounting becomes trivial, so the emphasis is on correct probabilistic reasoning. Interviewers watch for clear articulation of the modeling assumption, clean parameter scaling over time, and an ability to separate payoff structure from underlying dynamics.

What it tests

When pricing digital options on assets following arithmetic Brownian motion, the key structure is that the terminal asset price is normally distributed, not lognormally as in geometric Brownian motion. The probability of the payout is determined by the cumulative distribution function (CDF) of the normal distribution, centered at the current spot (if drift is zero) and scaled by the volatility over the relevant time. The present value of the option is simply the payout multiplied by this probability, since the risk-free rate is zero and there is no discounting. This approach generalizes to any threshold and payout, as long as the underlying follows a process where the terminal value is a shifted and scaled normal variable. The intuition is that the option's value is just the chance of being in-the-money, times the fixed payout, because the underlying's distribution is symmetric and tractable.