A stock that rises or falls each day for a week

Question

A stock is worth \100 today. Each day, independently, it rises 20\% (multiplies by 1.2) or falls 10\% (multiplies by 0.9), each with probability 	frac12. It trades for five days. What value should you expect after five days?

Accepted Answer

Each day multiplies the price by a random factor, and the days are independent. Condition on the price before each day with the tower rule. Step 1, the average daily factor. E[	ext{factor}] = 	frac12 	imes 1.2 + 	frac12 	imes 0.9 = 0.6 + 0.45 = 1.05. Step 2, the recursion. Let Pt be the price after t days. Given P{t-1}, the next day multiplies it by an independent factor averaging 1.05, so E[Pt] = 1.05 \, E[P{t-1}]. Step 3, iterate from P0 = 100 over five days: E[P5] = 100 	imes 1.05^5 = 100 	imes 1.2762815625 \approx 127.63. So you expect about \127.63. The average daily factor of 1.0…

A stock that rises or falls each day for a week

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