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  1. 083A market is either calm or stressed each day. A calm day is followed by another calm day with probability 0.8, and a stressed day is followed by another stressed day with probability 0.6. In the long run, what fraction of days are calm?Markov chains and random walksWarm upDRWLondon · 2026

    Try it first

    What share of days are calm in the long run?

    Show the worked solution

    Two thirds of days are calm. In the long run the fraction of days moving from calm to stressed must equal the fraction moving back, so calm share x 0.2 = stressed share x 0.4. Calm days are therefore twice as common as stressed days, 2/3 against 1/3. The same answer comes from spell lengths: calm spells average 5 days and stressed spells 2.5, and 5 / 7.5 = 2/3.

    Why can you balance flows instead of solving equations?

    Think of two rooms at a party. Every few minutes, one in five people in the kitchen wanders to the lounge, and two in five people in the lounge wander back. Once the crowd settles, the numbers crossing each way must match, or one room would keep filling up. In a two-state chain the long-run shares are fixed by one equation: the share of days leaving calm must equal the share of days leaving stressed. Calm leaves at rate 0.2 and stressed at rate 0.4, so calm must hold twice as many days.

    In the long run the flow out of calm equals the flow back inCalmStressed0.80.60.2 calm to stressed0.4 stressed to calmBalance: (2/3) x 0.2 = (1/3) x 0.4 = 2/15 of all dayscalm 2/3 of daysstressed 1/3average calm spell 1/0.2 = 5 days; average stressed spell 1/0.4 = 2.5 days
    Calm days leave at rate 0.2 and stressed days at rate 0.4, so in the long run calm must hold twice as many days as stressed for the flows to balance: two thirds calm and one third stressed, with each flow equal to 2/15 of all days.
    The relationship
    πC (1−0.8)=πS (1−0.6),πC+πS=1  ⇒  πC=0.40.2+0.4=23\pi_C\,(1-0.8) = \pi_S\,(1-0.6), \quad \pi_C + \pi_S = 1 \;\Rightarrow\; \pi_C = \frac{0.4}{0.2+0.4} = \frac23
    pi_Cthe long-run share of calm days
    pi_Sthe long-run share of stressed days
    1 - 0.8the chance a calm day is followed by a stressed one
    What it says in wordsEach state's long-run share is the rate of leaving the other state, divided by the two leaving rates added together.

    How do you check it a second way?

    Use spell lengths. A calm spell ends each day with probability 0.2, so it lasts 1/0.2 = 5 days on average; a stressed spell ends with probability 0.4, so it lasts 2.5 days. The chain alternates calm spell, stressed spell, calm spell, so the calm share is 5 out of every 7.5 days, which is 2/3. Two methods, one answer, in under a minute.

    How fast does the chain forget where it started?

    The transition matrix has a second eigenvalue of 0.8 + 0.6 - 1 = 0.4, and any gap between today's probabilities and the long-run split shrinks by that factor each day. After five days the starting state explains only 0.4^5, about 1%, of the gap, so the answer does not depend on how the week began. A trader would add the limitation: real regimes are not memoryless, and a stress spell that has already lasted a month is not as likely to end tomorrow as one that started yesterday.

    Where candidates lose it

    The trap is answering 80%, reading the chance of staying calm as the share of calm days. The 0.8 describes one step, and the long-run share depends on how quickly both states are left, not just one of them.

    The second slip is setting up a full eigenvector calculation and running out of time. Say the flow balance in one line, then use the spell lengths as the check.

    What the interviewer asks next

    • Today is stressed. What is the probability that the day after tomorrow is calm?
    • What is the expected number of days until the first stressed day, starting calm?
    • If a desk loses Rs 2 lakh on stressed days and makes Rs 1 lakh on calm days, what is its long-run average daily P&L?

    Asked at DRW, Trader Intern Interview, London, 2026 (Wall Street Oasis): consisted of math, statistics, and probability theory (eg. one question was about markov chains