2004 Fiscal Year Final Research Report Summary
RESEARCH on PRICING DERIVATIVES BASED ON RISK MEASURES
| Project/Area Number |
13440029
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | THE UNIVERSITY OF TOKYO |
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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)
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| 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)
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| Project Period (FY) |
2001 – 2004
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| Keywords | Mathematical Finance / Derivatives / Risk measures / Numerical Analysis / Malliavin Calculus / Free Lie Algebra / Stochastic Differential Equations / Runge-KUtta Method |
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| 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.
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