講演会

過去の記録 ~02/05次回の予定今後の予定 02/06~


2010年01月28日(木)

13:00-14:10   数理科学研究科棟(駒場) 122号室
Olivier Alvarez 氏 (Head of quantitative research, IRFX options Asia, BNP Paribas)
Partial differential equations in Finance II
[ 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