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Lectures
Seminar information archive ～05/02｜Next seminar｜Future seminars 05/03～
2010/01/28
13:00-14:10 Room #122 (Graduate School of Math. Sci. Bldg.)
Olivier Alvarez (Head of quantitative research, IRFX options Asia, BNP Paribas)
Partial differential equations in Finance II
Olivier Alvarez (Head of quantitative research, IRFX options Asia, BNP Paribas)
Partial differential equations in Finance II
[ Abstract ]
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
