A stock ticking to ten dollars or zero first

Question

A thinly traded share sits at 7 dollars. Model its price as moving in whole dollars: each tick it goes up 1 with probability 	frac12 or down 1 with probability 	frac12. Trading in it halts the moment the price reaches 10 dollars or falls to 0. What is the probability the price reaches 10 dollars before hitting 0?

Accepted Answer

Let pk be the probability of reaching 10 before 0 starting from price k, with p0 = 0 and p{10} = 1. Because each tick is an equal up or down move, first-step analysis gives pk = 	frac12 p{k-1} + 	frac12 p{k+1}. This forces equal consecutive gaps, so pk is linear: pk = k/10. At k = 7, p7 = \frac{7}{10} = 0.7. Takeaway. A fair price walk between an upper target and zero reaches the target first with probability equal to its starting fraction of the distance, here 7/10.

A stock ticking to ten dollars or zero first

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