Calm and Storm

Question

Each day the market is either calm or volatile. A calm day is followed by a volatile day with probability 0.1; a volatile day is followed by a calm day with probability 0.3. Over a very long horizon, what fraction of days are volatile?

Accepted Answer

In steady state, the daily flow of probability from calm into volatile must equal the flow from volatile back into calm (otherwise the occupancy of one state would drift). With pi_C and pi_V the long-run fractions: pi_C cdot 0.1 = pi_V cdot 0.3 quadLongrightarrowquad pi_V = frac0.10.1 + 0.3 = frac14. So 25% of days are volatile in the long run, regardless of today's regime. The general two-state answer is worth memorizing: with switching probabilities a (into state 2) and b (back into state 1), pi_2 = fracaa+b. A useful cross-check via mean run lengths: calm spells last frac10.1 = 10 days on average, volatile spells frac10.3 approx 3.33 days, so the volatile fraction is frac3.3310 + 3.33 = frac14, the same answer from a renewal argument. Note the trap: the answer is not 0.1 or frac0.10.3. Persistence matters, not just entry probability. Regime models in volatility forecasting are built on exactly this balance equation, and the run-length intuition (sticky regimes occupy more of the calendar than their entry rates suggest) is the first sanity check on any fitted regime model.

Calm and Storm

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