モンテカルロ法に関連する諸問題と金融工学への応用

• Project/Area Number: 20740059
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Osaka University
• Principal Investigator: 河合 玲一郎 Osaka University, 金融・保険教育研究センター, 特任助教 (20464258)
• Project Period (FY): 2008 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2009: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2008: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: モンテカルロ法 / レビー過程 / マリアバン解析 / 分散減少法 / 数値解析 / 確率数値解析 / 確率論 / モンテカルロ分散減少法 / 確率微分方程式弱近似 / 金融工学 / 感応度分析 / 無限分解可能分布

Research Abstract

1. レビー過程のサンプルパス発生には、レビー過程を構成する全ジャンプを発生させる確率数列表現法を用いる方法があります。まずレビー過程の固定時刻分布である無限分解可能分布の確率数列表現法と準モンテカルロ法との相性についての研究を遂行し、"Quasi-Monte Carlo method for infinitely divisible random vectors via series representations"という題名の論文を完成し投稿しました。さらにレビー過程、ジャンプ型確率微分方程式における確率数列表現法への発展を現在模索しており、これら一連の研究成果はレビー過程の実用に貢献するものと期待できます。
2. ジャンプ型確率微分方程式に関連する期待値の計算においては、サンプルパスを近似的に発生させるオイラー法が主流となっているが、生成作用素とディンキン公式をもとに数理計画法を用いて期待値の上界下界を算出する方法を昨年度提案しました。今年度はさらに確率微分方程式の固定時刻分布が安定分布のような裾の厚い分布である場合にもexponential temperingを施すことで対応できることを示し、"A weak approximation of stochastic differential equations with jumps through tempered polynomial optimization"という題名の論文を完成し投稿しました。さらに、これらの研究成果をもとに、中間時点や別の初期地点に関する期待値算出、また初期地点が確定的でない場合においても当該手法が適用可能であることが判明し、研究成果を論文として現在執筆中です。

Research Products

• [Journal Article] Sensitivity analysis and density estimation on the Hobson-Rogers stochastic volatility model (2009)
  • Author(s): Reiichiro Kawai
  • Journal Title: International Journal of Theoretical and Applied Finance 12 Pages: 283-295
  • Peer Reviewed
• [Journal Article] Adaptive Monte Carlo variance reduction for L\' evy processes with two-time-scale stochastic approximation (2008)
  • Author(s): Reiichiro Kawai
  • Journal Title: Methodology and Computing in Applied Probability 10(2) Pages: 199-223
  • Peer Reviewed
• [Journal Article] Optimal importance sampling parameter search for L'evy processes viastochastic approximation (2008)
  • Author(s): Reiichiro Kawai
  • Journal Title: SIAM Journal on Numerical Analysis 47(1) Pages: 293-307
  • Peer Reviewed
• [Journal Article] A multivariate Levy process model with linear correlation
  • Author(s): Reiichiro Kawai
  • Journal Title: Quantitative Finance (In press)
  • Peer Reviewed
• [Journal Article] Asymptotically optimal allocation of stratified sampling with adaptive variance reduction by strata
  • Author(s): Reiichiro Kawai
  • Journal Title: ACM Transactions on Modeling and Computer Simulation (In press)
  • Peer Reviewed
• [Journal Article] Sensitivity analysis and density estimation on the Hobson-Rogers stochastic volatility model
  • Author(s): Reiichiro Kawai
  • Journal Title: International Journal of Theoretical and Applied Finance (forthcoming)
  • Peer Reviewed
• [Journal Article] A multivariate L'evy process model with linear correlation
  • Author(s): Reiichiro Kawai
  • Journal Title: Quantitative Finance (forthcoming)
  • Peer Reviewed
• [Presentation] Recent Developments on Financial Greeks Computation for Models with Pure-Jumu Levy Processes (2009)
  • Author(s): Reiichiro Kawai
  • Organizer: Third Conference on Numerical Methods in Finance
  • Place of Presentation: Ecole des Ponts Paris Tech, France
  • Year and Date: 2009-04-17
• [Presentation] A Optimization Approach to Weak Approximation of Stochastic Differential Equations (2009)
  • Author(s): Reiichiro Kawai
  • Organizer: Workshop on "Mathematical Finance and Related Topics"
  • Place of Presentation: 福岡市
  • Year and Date: 2009-01-22
• [Presentation] Recent Developments on Financial Greeks Computation for Models with Pure-Jump L\'evy Processes (2008)
  • Author(s): Reiichiro Kawai
  • Organizer: INFORMS Annual Meeting
  • Place of Presentation: Washington DC, USA
  • Year and Date: 2008-10-12
• [Presentation] Recent Developments on Financial Greeks Computation for Models with Pure-Jump L\'evy Processes (2008)
  • Author(s): Reiichiro Kawai
  • Organizer: 日本オペレーションズリサーチ学会秋季研究発表会
  • Place of Presentation: 札幌市
  • Year and Date: 2008-09-10
• [Presentation] Recent Developments on Financial Greeks Computation for Models with Pure-Jump L\'evy Processes (2008)
  • Author(s): Reiichiro Kawai
  • Organizer: Workshop on "Finance and Related Mathematical and Statistical Issues"
  • Place of Presentation: 京都市
  • Year and Date: 2008-09-04
• [Presentation] Greeks formulae for an asset price dynamics model with gamma processes (2008)
  • Author(s): Reiichiro Kawai
  • Organizer: 7th World Congress in Probability and Statistics
  • Place of Presentation: Singapore
  • Year and Date: 2008-07-18
• [Presentation] Optimal importance sampling parameter search for L\'evy processes viastochastic approximation (2008)
  • Author(s): Reiichiro Kawai
  • Organizer: Eighth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (MCQMC2008)
  • Place of Presentation: Montreal, Canada
  • Year and Date: 2008-07-10
• [Remarks]
  • URL: http://www.geocities.jp/reiichiro_kawai/index.html
• [Remarks]
  • URL: http://www.geocities.jp/reiichiro_kawai/index.html