Arai, Takuji

写真a

Affiliation

Faculty of Economics (Mita)

Position

Professor

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Research Areas 【 Display / hide 】

  • Natural Science / Mathematical analysis (Mathematical Finance)

  • Natural Science / Applied mathematics and statistics

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Research Keywords 【 Display / hide 】

  • Mathematical Finance

  • Mathematical Finance

  • Probability Theory

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Papers 【 Display / hide 】

  • Monte Carlo simulation for Barndorff–Nielsen and Shephard model under change of measure

    Arai T., Imai Y.

    Mathematics and Computers in Simulation (Mathematics and Computers in Simulation)  218   223 - 234 2024.04

    Lead author, Last author, Corresponding author,  ISSN  03784754

    　View Summary

    The Barndorff–Nielsen and Shephard (BNS) model is a representative jump-type stochastic volatility model. Still, no method exists to compute option prices numerically for the non-martingale case with infinite active jumps. In this paper, selecting the minimal martingale measure (MMM) as a representative martingale measure, we develop two simulation methods for the BNS model under the MMM. The first method simulates the asset price at maturity and the Radon–Nikodym density of the MMM separately. On the other hand, the second method directly computes the asset price distribution under the MMM. In addition, we implement some numerical experiments to evaluate the performance of our simulation methods.

  • Constrained optimal stopping under a regime-switching model

    Takuji Arai, Masahiko Takenaka

    Journal of Applied Probability (Journal of Applied Probability)   2024

    Lead author, Last author, Corresponding author,  ISSN  00219002

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    We investigate an optimal stopping problem for the expected value of a
    discounted payoff on a regime-switching geometric Brownian motion under two
    constraints on the possible stopping times: only at exogenous random times and
    only during a specific regime. The main objectives are to show that an optimal
    stopping time exists as a threshold type under some boundary conditions and to
    derive expressions of the value functions and the optimal threshold. To this
    end, we solve the corresponding variational inequality and show that its
    solution coincides with the value functions. Some numerical results are also
    introduced. Furthermore, we investigate some asymptotic behaviors.

  • A remark on exact simulation of tempered stable Ornstein-Uhlenbeck processes

    Arai T., Imai Y.

    Journal of Applied Probability (Journal of Applied Probability)   2024

    Lead author, Last author, Corresponding author,  ISSN  00219002

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    Qu, Dassios, and Zhao (2021) suggested an exact simulation method for tempered stable Ornstein-Uhlenbeck processes, but their algorithms contain some errors. This short note aims to correct their algorithms and conduct some numerical experiments.

  • APPROXIMATE OPTION PRICING FORMULA FOR BARNDORFF-NIELSEN AND SHEPHARD MODEL

    Takuji Arai

    INTERNATIONAL JOURNAL OF THEORETICAL AND APPLIED FINANCE (WORLD SCIENTIFIC PUBL CO PTE LTD)  25 ( 2 )  2022.03

    ISSN  02190249

    　View Summary

    For the Barndorf-Nielsen and Shephard model, we present approximate expressions of call option prices based on the decomposition formula developed by [T. Arai (2021) Alos type decomposition formula for Barndor-Nielsen and Shephard model, Journal of Stochastic Analysis 2 (2), 3]. Besides, some numerical experiments are also implemented to make sure how effective our approximations are.

  • A Clark-Ocone Type Formula via Itô Calculus and its Application to Finance

    Takuji Arai, Ryoichi Suzuki

    Journal of Stochastic Analysis (Louisiana State University Libraries)  2 ( 4 )  2021.10

    Lead author, Last author, Corresponding author

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Papers, etc., Registered in KOARA 【 Display / hide 】

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Research Projects of Competitive Funds, etc. 【 Display / hide 】

  • ジャンプ型確率ボラティリティモデルに対するボラティリティ・サーフェスの研究

    2022.04
    -
    2025.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Principal investigator

    　View Summary

    代表的なジャンプ型確率ボラティリティ(SV)モデルであるBarndorff-Nielsen and Shephard(BNS)モデルに対し、インプライド・ボラティリティの近似式の導出やボラティリティ・サーフェスの分析を行う。また、これらの成果を発展させ、パラメータのカリブレーション手法の提案も目指す。さらに、BNSモデル以外のジャンプ型SVモデルにも研究を広げたい。
    本研究は数理ファイナンスの主要トピックの一つである金融派生証券の価格付け理論に関するものであり、特に確率ボラティリティモデルに対するボラティリティ・サーフェスの分析を行うことを目的としている。金融派生証券の価格付け理論は、Black-Scholesモデルを拡張させることで発展してきた。ボラティリティ・サーフェス上に現れるスマイルやスキューなどの現象は、Black-Scholesモデルが資産価格モデルとして正しくないことを示している。そこで、これらの現象を説明できるモデルとして、確率ボラティリティモデルが注目されてきた。しかし、これまでボラティリティ・サーフェスの分析が行われきたモデルは、連続なパスを持つものが中心であった。そこで本研究では、代表的なジャンプ型確率ボラティリティモデルであるBarndorff-Nielsen and Shephardモデル(BNSモデル)を中心に、インプライド・ボラティリティの近似式の導出やボラティリティ・サーフェスの分析を行う。さらに、ボラティリティ・サーフェスの導出を、最近発展が著しい深層学習と組み合わせることで、モデルパラメータのキャリブレーションやヘッジ計算など、様々なトピックに応用できることが分かった。令和4年度(2022年度)は、BNSモデルに対して、教師無し深層学習を用いたオプション価格計算の研究に取り組んだ。さらに、深層学習を用いたキャリブレーションに関する研究に着手し、とりわけ、最近注目を集めている2段階アプローチによるキャリブレーション手法に関する研究に取り組んだ。
    研究打ち合わせや国際学会での講演などを目的とした国外出張ができなかったため。
    確率ボラティリティモデルを対象に、深層学習を用いたモデルパラメータのキャリブレーションの研究を推進していく予定である。とりわけ、2段階アプローチと言われる手法に注目している。このアプローチでは、最初のステップでボラティリティ・サーフェスを教師あり学習により導出する。このステップに関してはまだまだ改良の余地があり、ボラティリティ・サーフェスのより詳細な分析を行い、高速かつ高精度なキャリブレーション手法の開発につなげたい。

  • 確率ボラティリティモデルに対する最適ヘッジ戦略の導出と数値計算法の研究

    2018.04
    -
    2021.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Principal investigator

  • Research on mathematical expressions and numerical methods for optimal hedging strategies via Malliavin calculus

    2015.04
    -
    2019.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, ARAI Takuji, Grant-in-Aid for Scientific Research (C), Principal investigator

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    Considering incomplete markets described by a jump type stochastic process, we aimed to derive, using Malliavin calculus, mathematical expressions of two optimal hedging strategies: local risk-minimizing (LRM) and mean-variance hedging (MVH) strategies; and to develop numerical methods for them. We have solved the following problems: (1) mathematical expressions and numerical methods of LRM strategies for BNS models, (2) expressions and numerical methods of MVH strategies for exponential additive models, (3) computation on LRM and MVH for normal inverse Gaussian models, and (4) expressions and numerical methods of LRM strategies for VIX options for BNS models.

  • Research on pricing theory by convex risk measures taking account of hedging, and its related stochastic analysis

    2010.04
    -
    2013.03

    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), ARAI Takuji, Grant-in-Aid for Scientific Research (C), Principal investigator

    　View Summary

    I have studied convex risk measures on stochastic processes in order to deal with shortfall risk measures for American options. In particular, I introduced spaces of stochastic processes whose maximum belongs to an Orlicz space; and obtained representation results for convex risk measures defined on such spaces. Next, I have researched on relationship between convex risk measures and good deal bounds. Supposing the market is a convex cone, I investigated (1) properties of superhedging cost, (2) the equivalence for a convex risk measure between that it represent upper and lower bounds of a good deal bound and that it is given as a risk indifference price, (3) extensions of the fundamental theorem of asset pricing. In addition, I extended the above results to the case where the market is merely convex.

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Courses Taught 【 Display / hide 】

  • SEMINAR: ECONOMETRICS

    2024

  • RESEARCH SEMINAR D

    2024

  • RESEARCH SEMINAR C

    2024

  • RESEARCH SEMINAR B

    2024

  • RESEARCH SEMINAR A

    2024

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Courses Previously Taught 【 Display / hide 】

  • FINANCE THEORY

    Keio University

    2024.04
    -
    2025.03

  • FINANCE THEORY

    Keio University

    2023.04
    -
    2024.03

  • INTRODUCTION TO FINANCE

    Keio University

    2021.04
    -
    2022.03

  • INTRODUCTION TO FINANCE B

    Keio University

    2021.04
    -
    2022.03

  • INTRODUCTION TO FINANCE

    Keio University

    2020.04
    -
    2021.03

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