Days until a market regime crashes

Question

A simplified market model has three daily regimes. From Calm, tomorrow is Calm with probability $1/2$ and Volatile with probability $1/2$ . From Volatile, tomorrow is Calm with probability $1/3$ , Volatile with probability $1/3$ , and Crash with probability $1/3$ . Crash is the end state (once it crashes, the process stops). Today is Calm.

On average, how many days until the crash?

Accepted Answer

Let c and v be the expected number of days to reach Crash starting from Calm and from Volatile. Crash itself has value 0. Each day is one step, then you move according to the transition probabilities: c = 1 + 	frac12 c + 	frac12 v, \qquad v = 1 + 	frac13 c + 	frac13 v + 	frac13 \cdot 0. Simplify each. From the Calm equation, c - 	frac12 c = 1 + 	frac12 v, so 	frac12 c = 1 + 	frac12 v, giving c = 2 + v. From the Volatile equation, v - 	frac13 v = 1 + 	frac13 c, so 	frac23 v = 1 + 	frac13 c, i.e. 2v = 3 + c. Substitute c = 2 + v: 2v = 3 + (2 + v) = 5 + v \;\Longrightarrow\; v = 5, …

Days until a market regime crashes

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