American Options in the Hobson-Rogers Model

TL;DR

This paper analyzes American put options within the Hobson-Rogers model, revealing a unique non-monotone separating curve between continuation and stopping regions, influenced by a volatility depending on past asset values.

Contribution

It introduces a Markovian framework for the Hobson-Rogers model and characterizes the non-monotone optimal stopping boundary for American puts.

Findings

The continuation and stopping regions are separated by a non-monotone curve.

The separating curve lies between the two classical BSM curves.

The model captures volatility dependence on historical asset values.

Abstract

In this article, we consider a risky asset $X$ for which evolution follows a model proposed by D.G. Hobson and L.C.G. Rogers\cite{HR98}. We assume that the volatility of $X$ depends on the ratio of the present value and the exponentially weighted average of the past value. Using the Markovian modelling of the enlarged two-dimensional process, we show that, for the American put option with $X$ as the underlying asset, the continuation region and the stopped region are separated a striking curve . This striking curve lies between the two striking curves from the basic BSM model, yet is {\em not} monotone.