This question relates to the frequency with which a discretely observed brownian motion exceeds its all time high.
Assume my retirement portfolio's value can be modelled as a process with normally distributed returns, upwards drift of μ per year and standard deviation of δ per year.
Today my portfolio hits an all time high, and I take my family for dinner. I promise to take them out for dinner at every subsequent all time high.
How many days per year do I expect to take them out?
Is the single parameter needed σ/μ?
Is the probability biased upwards because I'm starting at an all time high? What is the steady state probability?
A crude monte carlo simulation shows that I take them out for roughly 11% of days if σ=μ. Clearly if σ=0 I take them every day.
Is there a closed form solution?
I also read that since 1950, the US stockmarket has reached all time highs on about 7% of days. Empirical returns are fat-tailed not normally distributed, does this make much difference?
This is not an interview question or class homework, I just make it up, because I was actually thinking of doing the restaurant thing!