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RESEARCH on PRICING DERIVATIVES BASED ON RISK MEASURES

Research Project

Project/Area Number 13440029
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionTHE UNIVERSITY OF TOKYO

Principal Investigator

KUSUOKA Shigeo  The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (00114463)

Co-Investigator(Kenkyū-buntansha) YOSHIDA Nakahiro  The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (90210707)
TAKAHASHI Akihiko  The University of Tokyo, Graduate School of Economics, Associate Professor, 大学院・経済学研究科, 助教授 (50313226)
SEKINE Jun  Osaka University, Graduate School of Engineering Sciences, Associate Professor, 大学院・基礎工学研究科, 助教授 (50314399)
Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥11,100,000 (Direct Cost: ¥11,100,000)
Fiscal Year 2004: ¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2003: ¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2002: ¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2001: ¥3,200,000 (Direct Cost: ¥3,200,000)
KeywordsMathematical Finance / Derivatives / Risk measures / Numerical Analysis / Malliavin Calculus / Free Lie Algebra / Stochastic Differential Equations / Runge-KUtta Method / アクチュアリー / 生命保険 / 数値解析 / ファイナンス / リスク / 法則不変 / 多期間モデル / 連続極限 / フィルトレーション / ルンゲ・クッタ / リー環 / 拡散過程 / モンテカルロ法 / アメリカンオプション / ヨーロピアンオプション / 加法過程 / 特性関数
Research Abstract

This research focused on the pricing of American or European derivatives in the case where the market is not complete or where there exist frictions such as transaction cost, short sale constraint, tax etc., from view point of Asset Liability Management by using risk measures.
First, we did research on coherent risk measures proposed by Artzner, Delbaen, Eber and Heath, which is quite practical for banks. We introduced a new notion, law invariance, and characterized law invariant coherent risk measures.
Then we did research on a new effective numerical computation method for pricing derivatives.
We gave a rigorous proof of the effectiveness of the method proposed by Longstaff-Schwartz such that the value function is approximated with polynomials by the least square method. We also discussed the bound of this method.
We also introduced a new numerical computation method for pricing of European derivatives based on Malliavin calculus applied to stochastic differential equations and on free Lie algebra, and we developed this method. The details are the following. We introduced new notions, m-similar Markov operators and m-L similar random variables, and then we gave flexibility of the approximation methods and studied the algebraic structure of iterated stochastic integrals. Also, we used the Runge-Kutta method which is effective in numerical computation in ordinary differential equations, to construct definite m-similar Markov operators and m-L-similar random variables, and we did research on practical algorithm.

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (31 results)

All 2004 2003 2001 Other

All Journal Article (14 results) Publications (17 results)

  • [Journal Article] Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus2004

    • Author(s)
      Shigeo Kusuoka
    • Journal Title

      Advances in Mathematical Economics 6

      Pages: 69-83

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Stochastic Newton Equation with reflecting boundary condition2004

    • Author(s)
      Shigeo Kusuoka
    • Journal Title

      Advanced Studies in Pure Mathematics 41

      Pages: 233-246

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus2004

    • Author(s)
      S.Kusuoka
    • Journal Title

      Advances in Mathematical Economics vol.6

      Pages: 69-83

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Stochastic Newton Equation with reflecting boundary condition2004

    • Author(s)
      S.Kusuoka
    • Journal Title

      Advanced Studies in Pure Mathematics 41

      Pages: 233-246

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Stochastic Newton Equation with reflecting bounadary conditio2004

    • Author(s)
      Kusuoka, Shigeo
    • Journal Title

      Advanced Studies in Pure Mathematics 41

      Pages: 223-246

    • Related Report
      2004 Annual Research Report
  • [Journal Article] An Asymptotic Expansion Scheme for Optimal Investment Problems2004

    • Author(s)
      Takahashi Akihiko
    • Journal Title

      Statist.Infer.Stochast.Process 7

      Pages: 153-188

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Malliavin Calculus revisited2003

    • Author(s)
      Shigeo Kusuoka
    • Journal Title

      Journal of Mathematical Sciences, the University of Tokyo 10

      Pages: 261-277

    • NAID

      110000071225

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Monte Carlo Method for Pricing of Bermuda type derivatives2003

    • Author(s)
      Shigeo Kusuoka
    • Journal Title

      Advances in Mathematical Economics 5

      Pages: 153-166

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Nonlinear transformation containing rotation and Gaussian measure2003

    • Author(s)
      Shigeo Kusuoka
    • Journal Title

      Journal of Mathematical Sciences, the University of Tokyo 10

      Pages: 1-40

    • NAID

      110000071216

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Malliavin Calculus revisited2003

    • Author(s)
      S.Kusuoka
    • Journal Title

      Journal of Mathematical Sciences, (University of Tokyo) vol.10

      Pages: 261-277

    • NAID

      110000071225

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Monte Carlo Method for Pricing of Bermude type derivatives2003

    • Author(s)
      S.Kusuoka
    • Journal Title

      Advances in Mathematical Economics vol.5

      Pages: 153-166

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Nonlinear transformation containing rotation and Gaussian measure2003

    • Author(s)
      S.Kusuoka
    • Journal Title

      Journal of Mathematical Sciences, (University of Tokyo) vol.10

      Pages: 1-40

    • NAID

      110000071216

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Law Invariant Coherent Risk Measure2001

    • Author(s)
      Shigeo Kusuoka
    • Journal Title

      Advances in Mathematical Economics 3

      Pages: 83-95

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Law Invariant Coherent Risk Measure2001

    • Author(s)
      S.Kusuoka
    • Journal Title

      Advances in Mathematical Economics vol.3

      Pages: 83-95

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] Kusuoka, Shigeo: "Nonlinear transformation containing rotation and Gaussian measure"Journal of Mathenatical Sciences University Tokyo. 10. 1-40 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Kusuoka, Shigeo: "Malliavin Calculus revisited"Journal of Mathenatical Sciences University Tokyo. 10. 261-277 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Kusuoka, Shigeo: "On an Ergodic Property of Diffusion Semigroup on Euclidean Space"Journal of Mathenatical Sciences University Tokyo. 10. 537-553 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Kusuoka, Shigeo: "Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus"Advances in Mathematical Economics. 6. 69-83 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Kusuoka, Shigeo: "Stochastic Newton Equation with reflecting bounadary condition"Advanced Studies in Pure Mathematics. (掲載予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Yoshida, Nakahiro: "Conditional expansions and their applications"Stochastic Processes and their Applications. 107. 53-81 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Kusuoka, Shigeo: "Malliavin Calculus revisited"Journal of Mathenatical Sciences University Tokyo. (掲載予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kusuoka, Shigeo: "Nonlinear transformation containing rotation and Gaussian measure"Journal of Mathenatical Sciences University Tokyo. (掲載予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kusuoka, Shigeo: "Mote Carlo Method for Pricingof Bermuda type derivatives"Advances in Mathematical Economics. 5. 153-166 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kunitomo, Naoto: "On Validity of the Asymptotic Expansion Approach in Contingent Claim Analysis,"The Annals of Applied Probability. (掲載予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 高橋明彦: "モンテカルロフィルタを用いた金利モデルの推定"統計数理. 50・2. (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kusuoka, Shigeo: "On a certain Metric on the Space of Pairs of a Random Variable and a Probability Measure"Journal of Mathenatical Sciences University Tokyo. 8. 343-356 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kusuoka, Shigeo: "Laplace Approximations for Diffusion Processes on Torus : Nondegenerate Case"Journal of Mathenatical Sciences University Tokyo. 8. 43-70 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kusuoka, Shigeo: "Law Invariant Coherent Risk Measure"Advances in Mathematical Economics. 3. 83-95 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kusuoka, Shigeo: "Approximation of Expectation of diffusion processes and Mathematical Finance"Advanced Studies in Pure Mathematics. 31. 147-165 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Uchida, Masayuki: "Information criteria in model selection for mixing processes"Statistical Inference and Stochastic Processes. 4. 73-98 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Yoshida, Nakahiro: "Malliavin calculus and martingale expansions"Bull. Sci. math.. 125. 431-456 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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