Asymptotic distribution theory for mathematical finance

• Project/Area Number: 24684006
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Partial Multi-year Fund
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Osaka University
• Principal Investigator: Fukasawa Masaaki 大阪大学, 基礎工学研究科, 教授 (70506451)
• Project Period (FY): 2012-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥8,840,000 (Direct Cost: ¥6,800,000、Indirect Cost: ¥2,040,000); Fiscal Year 2015: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2014: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2013: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000); Fiscal Year 2012: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
• Keywords: 数理ファイナンス / インプライド・ボラティリティ / 漸近展開 / 確率解析 / 中心極限定理 / 安定収束 / 取引費用 / 確率制御 / ボラティリティ / デルタヘッジ / 最適ヘッジ戦略 / 同質化 / 国際研究者交流（フランス） / ヘッジ戦略 / 国際研究者交流 / フランス：イギリス / 確率積分 / 強近似 / 離散ヘッジ / 凸リスク測度 / オイラー丸山近似 / 停止時刻

Outline of Final Research Achievements

We extend asymptotic distribution theory of stochastic processes to find explicit solutions to problems in mathematical finance. They include optimal hedging problem under transaction costs, for which we constructed explicit hedging strategies minimizing relative asymptotic hedging error or asymptotic error variance. These results can be seen as developments in approximation theory of stochastic integrals. We studied also small-time asymptotics of stochastic processes which are models for financial asset prices. We derived and validated an asymptotic expansion of marginal distribution to show in particular that models with fractional Brownian motion explain the term structure of option price data.

Research Products

• [Int'l Joint Research] エコール・ポリテクニーク/パリ第6大学(フランス)
• [Int'l Joint Research] ニューヨーク市立大学(米国)
• [Int'l Joint Research] パリ第6大学/パリ第7大学/エコール・ポリテクニーク(フランス)
• [Int'l Joint Research] スイス連邦工科大学チューリッヒ校(スイス)
• [Int'l Joint Research] ウルム大学(ドイツ)
• [Journal Article] Short-time at-the-money skew and rough fractional volatility (2017)
  • Author(s): Masaaki Fukasawa
  • Journal Title: Quantitative Finance Volume: 17 Issue: 2 Pages: 189-198
  • DOI: 10.1080/14697688.2016.1197410
  • Peer Reviewed
• [Journal Article] Asymptotic replication with modified volatility under small transaction costs (2016)
  • Author(s): Jiatu Cai and Masaaki Fukasawa
  • Journal Title: Finance and Stochastics Volume: 20 Issue: 2 Pages: 381-431
  • DOI: 10.1007/s00780-016-0294-2
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Optimal Discretization of Hedging Strategies with Directional Views (2016)
  • Author(s): Jiatu Cai; Masaaki Fukasawa; Mathieu Rosenbaum; Peter Tankov
  • Journal Title: SIAM Journal on Financial Mathematics Volume: 7 Issue: 1 Pages: 34-69
  • DOI: 10.1137/151004306
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Efficient discretization of stochastic integrals (2014)
  • Author(s): Masaaki Fukasawa
  • Journal Title: Finance and Stochastics Volume: 18 Issue: 1 Pages: 175-208
  • DOI: 10.1007/s00780-013-0215-6
  • Peer Reviewed
• [Journal Article] Volatility derivatives and model-free implied leverage (2014)
  • Author(s): Masaaki Fukasawa
  • Journal Title: International journal of Theoretical and Applied Finance Volume: 17 Issue: 01 Pages: 1450002-1450002
  • DOI: 10.1142/s0219024914500022
  • NAID: 120006937137
  • Peer Reviewed
• [Journal Article] The YUIMA Project: A Computational Framework for Simulation and Inference of Stochastic Differential Equations (2014)
  • Author(s): A. Brouste, M. Fukasawa, H. Hino, S. Iacus, K. Kamatani, Y. Koike, H. Masuda, R. Nomura, T. Ogihara, Y. Shimuzu, M. Uchida and N. Yoshida
  • Journal Title: Journal of Statistical Software Volume: 57 Pages: 4-4
  • Peer Reviewed
• [Journal Article] Limit theorems for random walks under irregular conductance (2013)
  • Author(s): Masaaki Fukasawa
  • Journal Title: Proceedings of the Japan Academy, Ser. A, Mathematical Sciences Volume: 89 Issue: 8 Pages: 87-91
  • DOI: 10.3792/pjaa.89.87
  • NAID: 40019906997
  • Peer Reviewed
• [Journal Article] Convex risk measure for good deal bounds (2013)
  • Author(s): T. Arai and M. Fukasawa
  • Journal Title: Mathematical Finance Volume: (to appear) Issue: 3 Pages: 464-484
  • DOI: 10.1111/mafi.12020
  • Peer Reviewed
• [Presentation] At-the-money short-term asymptotics under stochastic volatility models (2017)
  • Author(s): Masaaki Fukasawa
  • Organizer: Advances in Financial Mathematics
  • Place of Presentation: パリ第5大学（フランス，パリ）
  • Year and Date: 2017-01-10
  • Int'l Joint Research / Invited
• [Presentation] Asymptotic replication with modified volatility under small transaction costs (2016)
  • Author(s): Masaaki Fukasawa
  • Organizer: Winter Workshop on Operations Research, Finance and Mathematics 2016
  • Place of Presentation: サホロリゾート（北海道上川郡）
  • Year and Date: 2016-02-17
  • Int'l Joint Research / Invited
• [Presentation] Phase transitions in a control problem (2015)
  • Author(s): Masaaki Fukasawa
  • Organizer: 確率解析シンポジウム
  • Place of Presentation: 大阪大学基礎工学部
  • Year and Date: 2015-10-21
  • Int'l Joint Research / Invited
• [Presentation] Homogenization in hedging under transaction costs (2015)
  • Author(s): Masaaki Fukasawa
  • Organizer: RIMS合宿型セミナー「確率解析の新展開」
  • Place of Presentation: 国際高等研究所（京都府木津川市）
  • Year and Date: 2015-07-10
  • Int'l Joint Research / Invited
• [Presentation] Efficient discretization of stochastic integrals (2013)
  • Author(s): 深潭正彰
  • Organizer: 10_<th>JAFEE-Columbia Conference
  • Place of Presentation: 統計数理研究所(東京)(招待講演)
  • Year and Date: 2013-03-19
• [Presentation] Effective discretization of stochastic differential equations (2013)
  • Author(s): 深潭正彰
  • Organizer: Statistique Asymptotique des Processus Stochastiques IX
  • Place of Presentation: Universite du Maineルマン(フランス)(招待講演)
  • Year and Date: 2013-03-13
• [Presentation] Linear regression of drift in continuous semimartingale models (2013)
  • Author(s): Masaaki Fukasawa
  • Organizer: The 59th ISI World Statistics Congress
  • Place of Presentation: Hong Kong
  • Invited
• [Presentation] Efficient discretization of stochastic integrals (2012)
  • Author(s): 深澤正彰
  • Organizer: 8_<th> World Congress in Probability and Statistics
  • Place of Presentation: Grand Cevahir Hotel イスタンブール(トルコ)(招待講演)
  • Year and Date: 2012-07-12
• [Presentation] 高頻度データに対する Whittle 尤度
  • Author(s): 深澤正彰
  • Organizer: 日本数学会
  • Place of Presentation: 学習院大学
  • Invited
• [Remarks] M. FUKASAWA WEB
  • URL: http://www.math.sci.osaka-u.ac.jp/~fukasawa/
• [Remarks] M. FUKASAWA WEB
  • URL: http://www.math.sci.osaka-u.ac.jp/~fukasawa/
• [Remarks]
  • URL: http://www.math.sci.osaka-u.ac.jp/~fukasawa/
• [Funded Workshop] The fourth Asian quantitative finance conference (2016)
  • Place of Presentation: 大阪大学中之島センター
  • Year and Date: 2016-02-21
• [Funded Workshop] Seminar on Probability (2016)
  • Place of Presentation: 大阪大学理学部
  • Year and Date: 2016-01-12
• [Funded Workshop] Seminar on Probability (2015)
  • Place of Presentation: 大阪大学理学部
  • Year and Date: 2015-10-06
• [Funded Workshop] Seminar on Probability (2015)
  • Place of Presentation: 大阪大学理学部
  • Year and Date: 2015-07-14