Stock direction two sessions later

Question

A stock closes each session either up or down versus the day before, and the next session depends only on the current one:

After an up session, the next is up with probability $0.55$ (and down with probability $0.45$ ).
After a down session, the next is up with probability $0.45$ (and down with probability $0.55$ ).

Today's session closed up.

What is the probability that the stock closes up two sessions from now?

Accepted Answer

Two sessions ahead is two steps. We branch on the session in between and keep the routes that end "up." We start up, the middle session can be either direction, and the final session must be up. - Up, then up, then up: 0.55 	imes 0.55 = 0.3025. - Up, then down, then up: 0.45 	imes 0.45 = 0.2025. Add the two routes that finish up: 0.3025 + 0.2025 = 0.505. So the stock closes up two sessions from now with probability 0.505. Takeaway. For two steps ahead, split on the middle state, multiply the one-step chances along each route, and add every route that ends in the target state.

Stock direction two sessions later

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