Some applications of first-passage ideas to finance

TL;DR

This paper explores first-passage time applications in finance, including goodness-of-fit tests, optimal asset selling times, and trading strategies with transaction costs, using random walk models and boundary problems.

Contribution

It introduces novel first-passage approaches to three key financial problems, enhancing understanding of tail events, optimal timing, and trading thresholds.

Findings

Modified goodness-of-fit test emphasizing tail events

Optimal sell timing near maximum asset value

Threshold-based trading strategy with boundary mapping

Abstract

Many problems in finance are related to first passage times. Among all of them, we chose three on which we contributed personally. Our first example relates Kolmogorov-Smirnov like goodness-of-fit tests, modified in such a way that tail events and core events contribute equally to the test (in the standard Kolmogorov-Smirnov, the tails contribute very little to the measure of goodness-of-fit). We show that this problem can be mapped onto that of a random walk inside moving walls. The second example is the optimal time to sell an asset (modelled as a random walk with drift) such that the sell time is as close as possible to the time at which the asset reaches its maximum value. The last example concerns optimal trading in the presence of transaction costs. In this case, the optimal strategy is to wait until the predictor reaches (plus or minus) a threshold value before buying or selling. The value of this threshold is found by mapping the problem onto that of a random walk between two walls.