A study of no arbitrage conditions in financial markets and its application

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K13737
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 07060:Money and finance-related
• Research Institution: Shinshu University
• Principal Investigator: Tsuzuki Yukihiro 信州大学, 学術研究院社会科学系, 准教授 (00801599)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: Option pricing / financial bubble / Black-Scholes equations / perpetuity / executive stock option / funding

Outline of Final Research Achievements

This study achieves two contributions.
First, the Laplace transforms of some perpetuities of the three-dimensional Bessel process are computed, where a perpetuity is an integral functional of a diffusion whose integral range is the whole of positive real numbers. Some new results are obtained, and several established results, such as Dufresne’s perpetuity and a particular case of its translated version, are recovered.
Second, the prices of derivatives for a fundraiser, who buys or sells stock as funding activities, are derived. As an application, pricing executive stock options are considered and a new numerical scheme is proposed for call option prices in a market with a bubble.

Free Research Field

数理ファイナンス

Academic Significance and Societal Importance of the Research Achievements

本研究において学術的・社会的意義が最も大きい成果は、バブル・モデルにおけるデリバティブ価格の新しい数値計算方法の確立である。バブル・モデルは学術的にも金融実務的にも関心が高くデリバティブの価格式は数式として導出されているものの、数値計算方法は確立されていなかった。これは対応する偏微分方程式の解に一意性がなく、有限差分法で意図する解を得るためには境界条件に特別な注意が必要であるためである。先行研究では素朴な境界条件しか考えられておらず、整合性が保てない点があったが、本研究ではこの点を改良した。