Research on mathematical expressions and numerical methods of optimal hedging strategies for stochastic volatility models

• Project/Area Number: 18K03422
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Keio University
• Principal Investigator: Arai Takuji 慶應義塾大学, 経済学部(三田), 教授 (20349830)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 数理ファイナンス / 確率論 / 数値計算

Outline of Final Research Achievements

This project aimed to derive an expression of the mean-variance hedging for the Barndorff-Nielsen and Shephard(BNS) model and develop its numerical scheme. The results obtained are (1) derivation of local risk-minimizing for digital options, (2) derivation of decomposition and approximation formulas for option prices for the BNS model, and (3) computation of option prices for the BNS model using unsupervised deep learning.

Academic Significance and Societal Importance of the Research Achievements

ジャンプ型モデルに対するmean-variance hedgingの導出は難しく、本研究ではその準備段階しかできなかった。しかし、伊藤解析を用いた分解公式の研究や深層学習を用いたアプローチなど新たな試みに挑戦したことで、研究手法の幅を拡げることに成功した。様々な手法を組み合わせることにより、ファイナンスにおける数学的知見を金融実務へ還元する礎を築けたものと確信している。

Research Products

• [Journal Article] Deep learning-based option pricing for Barndorff-Nielsen and Shephard model (2023)
  • Author(s): Takuji Arai
  • Journal Title: International Journal of Financial Engineering Volume: -
  • Peer Reviewed
• [Journal Article] Approximate option pricing formula for Barndorff-Nielsen and Shephard model (2022)
  • Author(s): Takuji Arai
  • Journal Title: International Journal of Theoretical and Applied Finance Volume: Vol. 25 No. 2 Issue: 02 Pages: 2250008-2250008
  • DOI: 10.1142/s021902492250008x
  • Peer Reviewed
• [Journal Article] Alos Type Decomposition Formula for Barndorff-Nielsen and Shephard Model (2021)
  • Author(s): Takuji Arai
  • Journal Title: Journal of Stochastic Analysis Volume: Vol. 2 No. 2 Issue: 2
  • DOI: 10.31390/josa.2.2.03
  • Peer Reviewed
• [Journal Article] A Clark-Ocone Type Formula via Ito Calculus and its Application to Finance (2021)
  • Author(s): Takuji Arai and Ryoichi Suzuki
  • Journal Title: Journal of Stochastic Analysis Volume: Vol. 2 No. 4 Issue: 4
  • DOI: 10.31390/josa.2.4.05
  • Peer Reviewed
• [Journal Article] Pricing and hedging of VIX options for Barndorff-Nielsen and Shephard models (2019)
  • Author(s): Takuji Arai
  • Journal Title: International Journal of Theoretical and Applied Finance Volume: 22 Issue: 08 Pages: 1950043-1950043
  • DOI: 10.1142/s0219024919500432
  • Peer Reviewed
• [Journal Article] Optimal Initial Capital Induced by the Optimized Certainty Equivalent (2019)
  • Author(s): Takuji Arai, Takao Asano, and Katsumasa Nishide
  • Journal Title: Insurance: Mathematics and Economics Volume: 85 Pages: 115-125
  • DOI: 10.1016/j.insmatheco.2019.01.006
  • Peer Reviewed
• [Journal Article] A numerically efficient closed-form representation of mean-variance hedging for exponential additive processes based on Malliavin calculus (2018)
  • Author(s): Arai Takuji、Imai Yuto
  • Journal Title: Applied Mathematical Finance Volume: 25 Issue: 3 Pages: 247-267
  • DOI: 10.1080/1350486x.2018.1506259
  • Peer Reviewed
• [Presentation] Deep Learning-Based Option Pricing for Barndorff-Nielsen and Shephard Model (2023)
  • Author(s): 新井拓児
  • Organizer: 第58回冬季JAFEE大会(2022年度冬季大会)
• [Presentation] Decomposition and Approximation for Barndorff-Nielsen and Shephard model (2021)
  • Author(s): 新井拓児
  • Organizer: 2021年度夏季JAFEE大会
• [Presentation] Decomposition and Approximation for Barndorff-Nielsen and Shephard model (2020)
  • Author(s): 新井拓児
  • Organizer: 2020年度中之島ワークショップ「金融工学・数理計量ファイナンスの諸問題 2020」
  • Invited
• [Presentation] Pricing and hedging of VIX options for Barndorff-Nielsen and Shephard models (2019)
  • Author(s): Takuji Arai
  • Organizer: Vienna Congress on Mathematical Finance - VCMF 2019
  • Int'l Joint Research
• [Presentation] A closed form representation of mean-variance hedging for additive processes via Malliavin calculus (2018)
  • Author(s): Takuji Arai
  • Organizer: 40th Conference on Stochastic Processes and Their Applications
  • Int'l Joint Research
• [Presentation] A closed form representation of mean-variance hedging for additive processes via Malliavin calculus (2018)
  • Author(s): Takuji Arai
  • Organizer: the 10th World Congress of the Bachelier Finance Society
  • Int'l Joint Research