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Applications of Levy Processes to Mathematical Finance

Research Project

Project/Area Number 13640131
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionNagoya City University

Principal Investigator

MIYAHARA Yoshio  Nagoya City University, Graduate School of Economics, Professor, 大学院・経済学研究科, 教授 (20106256)

Co-Investigator(Kenkyū-buntansha) NOTOCHORD Morihiro  Nagoya City University, Graduate School of Natural Sciences, Associate Professor, 大学院・システム自然科学研究科, 助教授 (30347421)
SHIMIZU Akinobu  Nagoya City University, Graduate School of Natural Sciences, Professor, 大学院・システム自然科学研究科, 教授 (10015547)
MISAWA Tetsuya  Nagoya City University, Graduate School of Economics, Professor, 大学院・経済学研究科, 教授 (10190620)
FUJIWARA Tukasa  Hyogo University of Teacher Education, Department of Mathematics, Associate Professor, 学校教育学部, 助教授 (30199385)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2003: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2002: ¥500,000 (Direct Cost: ¥500,000)
Keywordsmathematical finance / option pricing / geometric Levy process / martingale measure / relative entropy / 平滑近似法 / Poisson random measure / オプション価格 / 確率数値近解析 / Markov chain / 確率数値近似法 / 時系列データ平滑化 / subordination
Research Abstract

We have studied the option pricing problems in the incomplete asset market, which is one of the important problems in the field of mathematical finance. Our goal is the construction of the [Geometric Levy process & MEMM] pricing model, in which the geometric Levy processes are adopted as the underlying asset price processes and the MEMM (=minimal entropy martingale measure) is adopted as the martingale measure. And we have investigated the fundamental theories for the construction of this model and the applications of this mode to the option pricing.
We first established the existence theorem of MEMM for the geometric Levy processes, and we nest investigated the properties of MEMM and the properties of the [Geometric Levy process & MEMM] pricing model. Especially we have studied the relations between the MEMM and the Esscher martingale measure comparing each other. We investigated the methods for the application of this model, for example the method for the estimation of Levy processes. We also investigated the calibration problems of our model.

Report

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

    (42 results)

All Other

All Publications (42 results)

  • [Publications] Y.Miyahara: "[Geometric Levy Process & MEMM] Pricing Model and Related Estimation Problems."Asia-Pacific Financial Markets. Vol.8,No.1. 45-60 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 宮原 孝夫: "レヴィ過程の推定"統計数理研究所共同研究リポート. Vol.146. 38-50 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Miyahara: "Estimation of Levy Processes"Discussion Papers in Economics, Nagoya City University. No.318. 1-36 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Miyahara, A.Novikov: "Geometric Levy Process Pricing Model"Proceeding of Steklov Mathematical Institute. Vol.237. 176-191 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Miyahara: "A Note on Esscher Transformed Martingale Measures for Geometric Levy Processes"Discussion Papers in Economics, Nagoya City University. No.379. 1-14 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Fujiwara, Y.Miyahara: "The Minimal Entropy Martingale Measures for Geometric Levy Processes"Finance and Stochastics. Vol.7. 509-531 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Tetsuya Misawa: "A Lie Algebraic Approach to Numerical Integration of Stochastic Differential Equations"SIAM Journal on Scientific Computing. Vol.23. 866-890 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Asada, T.Inaba, T.Misawa: "An Integration Dynamic Model : the Case of Fixed Exchange Rates"Studies in Regional Science. Vol.31,No.2. 29-42 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 森 隆一, 三澤哲也: "ウェーブレット関数系による時系列データの平滑化"計算機統計学. 第14巻. 1-14 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Mori, T.Misawa: "Analysis of Time Series Data by Smooth Fitting with Meyer Wavelets"Discussion Papers in Economics, Nagoya City University. Vol.344. 1-27 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 宮内肇, 竜口玄太, 三澤哲也: "カリフォルニア電力市場価格の回帰分析"電気学会論文誌B. 第124巻. 199-206 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Tokuzo Shiga, Akinobu Shimizu, Takahiro Soshi: "Passage-time moments for positively recurrent Markov chains"Nagoya Mathematical Journal. Vol.162. 169-185 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Sano, A.Shimizu, M.Iizuka: "Coalescent process with fluctuating population size and its effective size"Theoretical Population Biology. Vol.65. 39-48 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 宮原 孝夫: "株価モデルとレヴィ過程"朝倉書店. 118 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Miyahara: "[Geometric Levy Process & MEMM] Pricing Model and Related Estimation Problems"Asia-Pacific Financial Markets. 8(2001), No.1. 45-60

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Miyahara, A.Novikov: "Geometric Levy Process Pricing Model"Proceedings of Steklov Mathematical Institute. Vol.237. 176-191 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Miyahara: "Estimation of Levy Processes"Discussion Papers in Economics, Nagoya City University. No.318. 1-36 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Fujiwara, Y.Miyahara: "The Minimal Entropy Martingale Measures for Geometric Levy Processes Finance and Stochastics"7. 509-531 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Miyahara: "A Note on Esscher Transformed Martingale Measures for Geometric Levy Processes"Discussion Papers in Economics, Nagoya City University. No.379. 1-14 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Misawa: "A Lie algebraic approach to numerical integration of stochastic differential equations"SIAM Journal on Scientific Computing. Vol.23. 866-890 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Asada, T.Inaba, T.Misawa: "An Interregional Dynamic Model : the Case of Fixed Exchange Rates"Studies on Regional Science. Vol.31, No.2. 29-42 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Shiga, A.Shimizu, T.Soshi: "Passage-time moments for positively recurrent Markov chains"Nagoya Mathematical Journal. 162. 169-185 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Shimizu, T.Soshi: "Positively recurrent Markov chains and the stepping stone model as a Fleming-Viot process"Yokohama Mathematical Journal. 49. 89-103 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Sano, A.Shimizu, M.Iizuka: "Coalescent process with fluctuating population size and its effective size"Theoretical Population Biology. 65. 39-48 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Miyahara: "A Note on Esscher Transformed Martingale Measures for Geometric Levy Processes"Discussion Papers in Economics, Nagoya City University. No.379. 1-14 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Fujiwara, Y.Miyahara: "The Minimal Entropy Martingale Measures for Geometric Levy Processes"Finance and Stochastics. Vol.7. 509-531 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Mori, T.Misawa: "Analysis of Time Series Data by Smooth Fitting with Meyer Wavelets"Discussion Papers in Economics, Nagoya City University. Vol.344. 1-27 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 宮内肇, 竜口玄太, 三澤哲也: "カリフォルニア電力市場価格の回帰分析"電気学会論文誌B. 第124巻. 199-206 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.Sano, A.Shimizu, M.Iizuka: "Coalescent process with fluctuating population size and its effective size"Theoretical Population Biology. Vol.65. 39-48 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] 宮原 孝夫: "株価モデルとレヴィ過程"朝倉書店. 118 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Fujiwara, Y.Miyahara: "The Minimal Entropy Martingale Measures for Geometric Levy Processes"Finance and Stochastics. (to appear). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Miyahara: "Estimation of Levy Processes"Discussion Papers in Economics, Nagoya City University. No.318. 1-36 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Miyahara, A.Novikov: "Geometric Levy Process Pricing Model"Proceeding of Steklov Mathematical Institute. Vol.237. 176-191 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Asada, T.Inaba, T.Misawa: "An Integration Dynamic Model : the Case of Fixed Exchange Rates"Studies in Regional Science. Vol.31,No.2. 29-42 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 清水 昭信: "A generalized Ewens' sampling formula for normalized subordinations"統計数理研究所共同研究リポート. 157. 61-72 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 藤原 司: "On the optimal strategy for the exponential utility maximization in the geometric Levy process model"統計数理研究所共同研究リポート. 157. 12-17 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Miyahara: "[Geometric Levy Process & MEMM] Pricing Model and Related Estimation Problems"Asia-Pacific Financial Markets. Vol.8, No.1. 45-60 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 宮原 孝夫: "レヴィ過程の推定"統計数理研究所共同研究リポート. Vol.146. 38-50 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Tokuzo Shiga, Akinobu Shimizu, Takahiro Soshi: "Passage-time moments for positively recurrent Markov chains"Nagoya Mathematical Jounal. Vol.162. 169-185 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 清水 昭信: "Random discrete probability distributions derived from subordinators"統計数理研究所共同研究リポート. Vol.146. 60-72 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Tetsuya Misawa: "A Lie Algebraic Approach to Numerical Integration of Stochastic Differential Equations"SIAM Journal on Scientific Computing. Vol.23. 866-890 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 森 隆一, 三澤哲也: "ウェーブレット関数系による時系列データの平滑化"計算機統計学. 第14巻. 1-14 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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