# Discounted optimal stopping of a Brownian bridge, with application to American options under pinning

## TL;DR

This paper models the optimal stopping problem for American options under the pinning effect using a Brownian bridge, incorporating discounting, and provides numerical methods and real data analysis to improve exercise strategies.

## Contribution

It introduces a novel approach to optimal stopping with future information modeled as a Brownian bridge, including discount factors, and compares it with traditional models using real market data.

## Key findings

- The solution to the discounted optimal stopping problem is characterized mathematically.
- Numerical methods for computing the optimal stopping boundary are developed.
- The approach is validated with real market data exhibiting pinning effects.

## Abstract

Mathematically, the execution of an American-style financial derivative is commonly reduced to solving an optimal stopping problem. Breaking the general assumption that the knowledge of the holder is restricted to the price history of the underlying asset, we allow for the disclosure of future information about the terminal price of the asset by modeling it as a Brownian bridge. This model may be used under special market conditions, in particular we focus on what in the literature is known as the "pinning effect", that is, when the price of the asset approaches the strike price of a highly-traded option close to its expiration date. Our main mathematical contribution is in characterizing the solution to the optimal stopping problem when the gain function includes the discount factor. We show how to numerically compute the solution and we analyze the effect of the volatility estimation on the strategy by computing the confidence curves around the optimal stopping boundary. Finally, we compare our method with the optimal exercise time based on a geometric Brownian motion by using real data exhibiting pinning.