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講演会
過去の記録 ～10/09｜次回の予定｜今後の予定 10/10～
2010年01月28日(木)
10:40-12:10 数理科学研究科棟(駒場) 123号室
Olivier Alvarez 氏 (Head of quantitative research, IRFX options Asia, BNP Paribas)
Partial differential equations in Finance I
Olivier Alvarez 氏 (Head of quantitative research, IRFX options Asia, BNP Paribas)
Partial differential equations in Finance I
[ 講演概要 ]
1. Markov processes and Partial differential equations (PDE)
- Markov processes, stochastic differential equations and infinitesimal generator
- The Feynman Kac formula and the backward Kolmogorov equation
- The maximum principle
- Exit time problems and Dirichlet boundary conditions
- Optimal time problems and obstacle problems
2. Application to the pricing of exotic options
- The model equation
- The Black-Scholes equation : absence of arbitrage and dynamical hedging
- Recovering the Black-Scholes formula
- Pricing exotic options : Knock-out / knock-in, american, Asian, lookback
- Overview of affine models and semi-closed formulae
- Heston model : valuing European options
- The Hull White model for IR exotics : valuing zero-coupons, caplets and swaptions.
3. Finite difference methods in Finance
- Basic concepts for numerical schemes : consistency, stability, accuracy and
convergence; the Lax equivalence theorem
- Finite difference methods in dimension 1 : Explicit, implicit, Crank-Nicholson methods for the heat equation : overview, accuracy and convergence
Incorporating first-order derivatives : upwind derivative, stability
- Finite difference methods in dimension 2 : presentation of various schemes :explicit, implicit, alternating direction implicit (ADI), Hopscotch method
- Solving high dimensional linear systems :
LU decomposition, iterative methods
- Finite difference and Monte Carlo methods
4. Optimal control in finance
- Introduction to optimal control
- The dynamic programming principle and the Hamilton-Jacobi-Bellman equation
- The verification theorem and the determination of the optimal control policy
- Utility maximization and Merton's problem
- Pricing with uncertain parameters
- Pricing with transaction costs
- Finite difference methods for optimal control
1. Markov processes and Partial differential equations (PDE)
- Markov processes, stochastic differential equations and infinitesimal generator
- The Feynman Kac formula and the backward Kolmogorov equation
- The maximum principle
- Exit time problems and Dirichlet boundary conditions
- Optimal time problems and obstacle problems
2. Application to the pricing of exotic options
- The model equation
- The Black-Scholes equation : absence of arbitrage and dynamical hedging
- Recovering the Black-Scholes formula
- Pricing exotic options : Knock-out / knock-in, american, Asian, lookback
- Overview of affine models and semi-closed formulae
- Heston model : valuing European options
- The Hull White model for IR exotics : valuing zero-coupons, caplets and swaptions.
3. Finite difference methods in Finance
- Basic concepts for numerical schemes : consistency, stability, accuracy and
convergence; the Lax equivalence theorem
- Finite difference methods in dimension 1 : Explicit, implicit, Crank-Nicholson methods for the heat equation : overview, accuracy and convergence
Incorporating first-order derivatives : upwind derivative, stability
- Finite difference methods in dimension 2 : presentation of various schemes :explicit, implicit, alternating direction implicit (ADI), Hopscotch method
- Solving high dimensional linear systems :
LU decomposition, iterative methods
- Finite difference and Monte Carlo methods
4. Optimal control in finance
- Introduction to optimal control
- The dynamic programming principle and the Hamilton-Jacobi-Bellman equation
- The verification theorem and the determination of the optimal control policy
- Utility maximization and Merton's problem
- Pricing with uncertain parameters
- Pricing with transaction costs
- Finite difference methods for optimal control
