Optionality as a binary operation

Peter Carr; Doug Costa

doi:10.3934/fmf.2023015

Article Contents

2023, Volume 2, Issue 3: 313-339. Doi: 10.3934/fmf.2023015 Free Access

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Optionality as a binary operation

Peter Carr^1, and
Doug Costa^1,2, ,

1.
New York University, USA
2.
Susquehanna International Group, USA

^*Corresponding author: Doug Costa
^*Corresponding author: Doug Costa

Received: January 2022

Revised: March 2023

Early access: April 4, 2023

Published online: April 4, 2023

Tragically, Peter Carr passed away on March 1, 2022.

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Abstract

In finance, optionality is a possible property of a financial contract giving the owner a choice between two or more assets. For example, a convertible bond has optionality because its owner must choose between having a bond or having some shares of stock. In mathematics, a binary operation acts on two elements in a set to produce a third element in that set. When a financial contract such as a convertible bond enjoys optionality between exactly two assets, then the arbitrage-free current value of the contract can potentially be treated as the outcome of a binary operation acting on the two current asset values. In this paper, we treat one of the two assets as riskless and demand that the binary operation linking the two current asset values always produces an arbitrage-free option price. In this context, we focus on the interplay between the properties of the risk-neutral density of the risky asset and the algebraic properties of the binary operation.

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Mathematics Subject Classification: Primary: 60E05, 91G20; Secondary: 08A02, 20M99, 60G99.

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