Each day the market is in one regime: Bull, Bear or Flat. From Bull: Bull 1/2, Bear 1/4, Flat 1/4. From Bear: Bull 1/4, Bear 1/2, Flat 1/4. From Flat: Bull 1/2, Bear 1/2, Flat 0. All numbers are next-day probabilities. What is the long-run fraction of Flat days? Give a fraction a/b in lowest terms.

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$3$ -regime long run

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Each day the market is in one regime: Bull, Bear or Flat. From Bull: Bull $\frac{1}{2}$ , Bear $\frac{1}{4}$ , Flat $\frac{1}{4}$ . From Bear: Bull $\frac{1}{4}$ , Bear $\frac{1}{2}$ , Flat $\frac{1}{4}$ . From Flat: Bull $\frac{1}{2}$ , Bear $\frac{1}{2}$ , Flat $0$ . All numbers are next-day probabilities. What is the long-run fraction of Flat days? Give a fraction a/b in lowest terms.

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Hint

Bull and Bear are symmetric in how they feed into each other and into Flat, so look for equal stationary values.

Solution

Let the stationary probabilities be $u$ for Bull, $v$ for Bear and $w$ for Flat.
The Bull equation gives $\frac{u}{2} = \frac{v}{4} + \frac{w}{2}$ and the Bear equation gives $\frac{v}{2} = \frac{u}{4} + \frac{w}{2}$ .
Subtracting them gives $\frac{u - v}{2} = \frac{v - u}{4}$ , which forces $u = v$ .
The Flat equation is $w = \frac{u}{4} + \frac{v}{4} = \frac{u}{2}$ .
Normalising $u + v + w = 1$ gives $2 u + \frac{u}{2} = 1$ , so $u = \frac{2}{5}$ and $w = \frac{1}{5}$ .

Answer: $\frac{1}{5}$

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