Numerical methods for mathematical finance (seminar course) (2014/2015)

Obiettivi formativi

In this course we will study basic numerical methods for option pricing.

Binary Trees
- riskless portfolio
- risk neutral probability
- calibrating the tree
Continuous Time
- Geometric Brownian Motion
- the Black-Scholes PDE
- Feynman-Kac and risk-neutrality
Estimating the Volatility from historical data
Acceleration of the backfolding of the tree using the FFT
Other fast(er) methods for trees
Path Dependent Options using a Tree
- Lookback Puts and Calls
- the Cheuk-Vorst algorithm
- American Lookback options
Numerical Methods for Advection-Diffusion Equations
- Euler Explicit
- Euler Implicit
- Crank-Nicholson
- application to the Black-Scholes PDE
Asian Options
- an associated PDE
- eliminating the advection term
American Options
- linear complementary problems
- numerical solution
The Carr-Madan Fourier algorithm (for a call)
Jump Diffusion Models
- compound Poisson processes and their simulation
- the Merton Model
- the Merton formula
- the Carr-Madan algorithm for the Merton Model
- American options under the Merton Model
The fast Gauss Transform and its application to Option Pricing
Calibration using Hisorical Data
- the Schwartz Mean Reverting Model for Comodities
- its calibration using Forward Prices
Monte Carlo Methods
- accelerations
- application to Basket Options
Discretization of SDEs
- Euler discretization
- Milstein discretization

The final mark will be bsed on homework assignments as well as a project completed by the student.