A three-year autocallable on the index pays principal plus 9% per year elapsed if the index is at or above its start on a yearly observation. If never called, principal comes back unless the index ends below 60% of its start, when the loss is one for one. Work three paths and say what the investor is short.

Path A pays 109 after one year, path B pays 118 after two, path C repays only 55, and path D repays 100 with no coupon. The coupon is earned only when the index is at or above its start on an observation date, and the note then ends. If it survives to year three below the 60% barrier, the investor takes the full index loss. The investor is short a down-and-in put struck at the start level and has sold away the index's upside beyond the coupon.

Walk the observation dates in order and ask one question at each: is the index at or above where it started? If yes, the note ends there with 100 plus 9 for each year. Only if all three answers are no does the barrier come into play, and then only the year-three level matters. Path A is at 105% in year one: called, 109. Path B is at 92%, then 102%: called in year two, 118. Path C is at 90%, 80% and 55%: never called, and 55% is below the 60% barrier, so the investor gets 55. Path D is at 90%, 80% and 65%: never called, above the barrier, so principal comes back, 100, and no coupon at all.

An insurer can pay a generous bonus because it collects premiums for disasters it might have to cover. The autocall investor is the insurer: the coupon is paid for by selling a down-and-in putA put that only comes into existence if the underlying falls through a barrier. Here it is struck at the starting level and activated if the index ends below 60%. struck at 100 with a barrier at 60, plus giving up everything the index earns above the coupon. On path C the put comes alive and pays the bank the full 45% fall; that is why the investor's loss is one for one from the start level, not from the barrier. The early call is the bank's protection too: in the paths where the index does well, the note ends early and the investor's 9% stops, so the upside is capped twice, by the coupon and by the call.

Three things in plain words. First, the 9% arrives only if the market is at or above today's level on one of the dates; in a sideways or falling market the note pays nothing extra, like path D. Second, the barrier is a cliff, not a cushion: once the index ends below 60%, the investor loses the whole fall from 100, not just the part below 60. Third, the note is an unsecured promise by the issuing bank, so its credit matters on every path. The limit of this exercise is that four paths are not a probability distribution; a desk prices the note by simulating thousands of paths, and the answer depends heavily on volatility and on the correlation between the index falling and markets panicking. Higher volatility makes the put more valuable, which is exactly why autocalls are sold with fatter coupons after turbulent periods.