Derivatives Foundation puzzles, solved step by step

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  1. 083A price starts at 100 and moves up or down 1 each day with equal chance for 10 days. What is the probability that it touches 104 at some point in those 10 days?Random walks and Markov chainsHardExotics tradingQuant trading

    Try it first

    Which quantity is easier to count, and how does it relate to touching?

    Show the worked solution

    232 of the 1,024 paths touch 104, about 22.7%. A path ending at or above 104 must have touched it: that needs at least 7 up days, 176 paths. A path that touches 104 and ends below it can be reflected in the 104 line after its first touch, and the reflection ends above 104, needing at least 8 up days: 56 paths. The pairing is one to one, so the count is 176 + 56 = 232.

    Why is the reflection a fair way to count?

    Imagine a walker who reaches a wall at some point and then wanders on. Flip every step after the first touch, up for down, and you get another walker who also touched the wall at the same moment and is now the same distance on the other side of it. Each flipped step is as likely as the original, so the two walks have equal probability. Reflection pairs every path that touches 104 and finishes below it with exactly one path that finishes above 104, so the hard count becomes an easy count of endpoints. Paths that finish at exactly 104 are their own partners and are counted once.

    Every path that touches 104 and ends below pairs with one that ends aboveday 0day 2day 4day 6day 8day 109698100102104106108barrier 104first touch, day 4ends 100ends 108actual pathmirror imageCount by the endpointend at or above 1041767, 8, 9 or 10 up daystouched 104, ended below56= mirrors of paths ending above 104paths that touch 104232out of 1,024: 22.7%the counting idea behind barrier options
    A path that climbs to 104 by day 4 and ends at 100 is mirrored in the 104 line after the first touch into a path that ends at 108, and counting by endpoint gives 176 paths ending at or above 104 plus 56 mirrors of paths ending above it, 232 of 1,024 paths, which is 22.7%.

    How do the binomial counts fall out?

    After 10 days of plus or minus 1, the position is 100 plus ups minus downs, which is 100 + 2 x ups - 10. Ending at or above 104 needs ups of at least 7: C(10,7) + C(10,8) + C(10,9) + C(10,10) = 120 + 45 + 10 + 1 = 176. Ending strictly above 104 means at or above 106, since the endpoint is always even, so ups of at least 8: 45 + 10 + 1 = 56. The probability of touching is (176 + 56)/1,024 = 232/1,024, about 22.7%, against only 176/1,024 = 17.2% for ending at or above 104. The gap is the 56 paths that visited the barrier and came back.

    The relationship
    P(max⁡S≥4)=P(S10≥4)+P(S10>4)=176+561024=2321024P(\max S \ge 4) = P(S_{10} \ge 4) + P(S_{10} > 4) = \frac{176 + 56}{1024} = \frac{232}{1024}
    S_10the net move after ten days
    P(S_10 >= 4)paths ending at or above the barrier, 176
    P(S_10 > 4)paths ending strictly above, which are the reflections of touch-and-return paths, 56
    What it says in wordsThe chance of touching a level equals the chance of ending at or past it plus the chance of ending strictly past it.

    Say why a derivatives desk cares. A knock-out or one-touch option pays on exactly this event, and the reflection principle is how the closed-form barrier prices are derived in continuous time. The limitation is in the symmetry: reflection needs the up and down steps to be equally likely and the barrier to sit on the lattice. With a drift, or in continuous time with a non-zero rate, the reflected path is not equally likely and a correction factor appears, which is why barrier formulas carry that extra power of the barrier over the spot.

    Where candidates lose it

    The usual loss is answering the probability of ending at or above 104, 17.2%, and forgetting the paths that touched and came back. The interviewer asked about touching, and the difference is a quarter of the answer.

    The second loss is trying to enumerate paths that stay below 104 by hand. There are 792 of them and no clean way to list them in the room. Reflection exists so you do not have to.

    What the interviewer asks next

    • What is the probability the price touches 104 and ends at 100?
    • Change the odds to 55% up. Why does reflection stop working as stated?
    • How does this count turn into the price of a one-touch option?