Probability of hitting a barrier

[adamcox]

How is the probability of exceeding a barrier (minimum % + or - return) in a minimum number of time steps, from today's date and stock price calculated?

thanks.

 

[Mad_Mac]

I'd personally use a bootstrap simulation. Find a time period that you feel is similar to the one in the time frame of interest - then take random draws of share price returns from that period and run a simulation.of how the share price could move. If you hit the barrier - then add 1 to a variable that counts number of successes. If not - don't add anything. Repeat this exercise a couple of million times - and then divide the total number of successes by the number of iterations you ran. The output will give you an stimate of the probability you are looking for. / Mac

 

[David Harper CFA FRM]

In the context of a binomial, I may be underthinking (under-encapsulating) this but it seems straightforward. Let S = current asset price; u = up jump magnitude; and h = barrier up, then:

• S*u^n = H; i.e., barrier H is reached in minimum steps (n) if the up-magnitude compounds (n) times, so this is just jumping up the tree as quickly as possible
• Then n = log[base u](H/S) is the minimum number of steps
• The probability is then p^n; i.e., the probability of (n) consecutive up jumps.

 

[adamcox]

Thanks for the input.