Renewal equations for option pricing

TL;DR

This paper introduces a renewal equation-based methodology for pricing financial derivatives within a CTRW market model, enabling analysis of standard and exotic options, and connecting to classical Wiener process results.

Contribution

It develops a novel renewal equation approach for option pricing in CTRW models, extending to exotic derivatives and linking to traditional models.

Findings

Derived pricing formulas for standard options using renewal equations.

Demonstrated how to price exotic derivatives within the CTRW framework.

Showed the connection between CTRW models and classical Wiener process results.

Abstract

In this paper we will develop a methodology for obtaining pricing expressions for financial instruments whose underlying asset can be described through a simple continuous-time random walk (CTRW) market model. Our approach is very natural to the issue because it is based in the use of renewal equations, and therefore it enhances the potential use of CTRW techniques in finance. We solve these equations for typical contract specifications, in a particular but exemplifying case. We also show how a formal general solution can be found for more exotic derivatives, and we compare prices for alternative models of the underlying. Finally, we recover the celebrated results for the Wiener process under certain limits.