J.A.M. van der Weide

Martingales and Financial Mathematics

Introduction

Martingales and financial mathematics. Explore martingales' role in financial mathematics, focusing on arbitrage-free, complete markets, derivative pricing, and the Cox-Ross-Rubinstein model in the discrete case.

Abstract

In this expository paper, we will discuss the role played by martingales in Financial Mathematics. More precisely, we will restrict ourselves to a mathematical formulation of the economical concept of an arbitrage-free, complete market and the pricing of derivatives in such models. For a clear exposition, we only consider the discrete case. We also discuss the Cox-Ross-Rubinstein model which is still one of the most used models in Finance.

Review

This expository paper, titled 'Martingales and Financial Mathematics,' sets out to elucidate the fundamental role of martingales within the context of financial mathematics. Based on the abstract, the paper focuses on crucial theoretical underpinnings, including the mathematical formulation of arbitrage-free and complete markets, and the subsequent application to derivative pricing within these models. A notable choice for enhanced clarity in an expository setting is the exclusive consideration of the discrete case, further concretized by a discussion of the influential Cox-Ross-Rubinstein model. The chosen scope of this paper is highly appropriate for an expository work. By concentrating on the discrete case and featuring the widely recognized Cox-Ross-Rubinstein model, the paper promises to offer an accessible and foundational introduction to these complex yet crucial topics. Martingale theory forms the mathematical backbone of modern financial theory, and a clear, focused exposition on its application to arbitrage-free markets and derivative pricing is of significant pedagogical value. This approach is particularly beneficial for students or researchers new to the intersection of probability theory and finance, providing a structured conceptual foundation. While originality of results is not the primary objective of an expository paper, its success hinges critically on the clarity, rigor, and pedagogical effectiveness of the presentation. For this paper to maximize its impact, the definitions, derivations, and examples – particularly those related to the Cox-Ross-Rubinstein model – must be exceptionally well-articulated and easy to follow. A strong expository paper not only explains concepts but also builds intuition, thereby enabling readers to confidently progress to more advanced literature, such as continuous-time models or stochastic calculus applications in finance. The execution of these elements will ultimately determine the paper's utility as a resource.

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