Holder-extendible European option: corrections and extensions

TL;DR

This paper extends the pricing formulas for holder-extendible European options to cases with time-dependent drift and volatility, correcting previous errors and introducing new solution types for negative dividend yields.

Contribution

It derives closed-form solutions for these options under more realistic, time-varying market conditions, including cases with negative dividend yields, and corrects existing formula errors.

Findings

Derived formulas for time-dependent drift and volatility.

Introduced new solution type for negative dividend yields.

Corrected common typographical errors in existing formulas.

Abstract

Financial contracts with options that allow the holder to extend the contract maturity by paying an additional fixed amount found many applications in finance. Closed-form solutions for the price of these options have appeared in the literature for the case when the contract underlying asset follows a geometric Brownian motion with the constant interest rate, volatility, and non-negative "dividend" yield. In this paper, the option price is derived for the case of the underlying asset that follows a geometric Brownian motion with the time-dependent drift and volatility which is important to use the solutions in real life applications. The formulas are derived for the drift that may include non-negative or negative "dividend" yield. The latter case results in a new solution type that has not been studied in the literature. Several typographical errors in the formula for the holder-extendible put, typically repeated in textbooks and software, are corrected.