Optimal Time to Sell a Stock in Black-Scholes Model: Comment on "Thou shall buy and hold", by A. Shiryaev, Z. Xu and X.Y. Zhou

TL;DR

This paper reexamines the optimal timing to sell a stock within the Black-Scholes framework, confirming previous results and extending analysis to broader parameter ranges using path integral methods, also deriving the distribution of the maximum price time.

Contribution

It introduces an alternative path integral approach to validate and extend prior findings on optimal stock selling times in the Black-Scholes model.

Findings

Confirmed previous results on optimal selling time

Extended analysis to new parameter regions

Derived the distribution of the maximum price time

Abstract

We reconsider the problem of optimal time to sell a stock studied recently by Shiryaev, Xu and Zhou using path integral methods. This method allows us to confirm the results obtained by these authors and extend them to a parameter region inaccessible to the method used by Shiryaev et. al. We also obtain the full distribution of the time t_m at which the maximum of the price is reached for arbitrary values of the drift.