M5MF2 Numerical Methods in Finance, MSc Mathematics and Finance, Spring term 2017
In this course, we shall endeavour to cover the following topics:
Finite difference methods for parabolic PDEs;
Fourier transform and quadrature methods;
Numerical optimisation;
Linear programming.
Time: Tuesdays 11am-1pm (Huxley 140) and Wednesdays 11am-1pm (Huxley 658)
Course Material
Lecture Notes (this version: 21/2/2017)
Zanadu Platform
IPython / Jupyter platform
Problem Class: Finite Differences (01/02/2017)
Problem Class: Fourier Methods (22/02/2017)
Case study: Variations around Crank-Nicolson (27/01/2017)
Code
[IPynb, PDF] Generating non-uniform grids
[IPynb, PDF] Finite differences for the heat equation
[IPynb, PDF] Finite differences for the Black-Scholes Call price
[IPynb, PDF] Finite difference for first-order derivatives
[IPynb, PDF] Interpolation of option prices / implied volatility
[Matlab] Explicit scheme for the heat equation
[Matlab] American options in Black-Scholes using an implicit scheme
[Matlab] American options in Black-Scholes using a SOR scheme
SPX options data
Additional material
[WEB]
W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery.
Numerical Recipes.
[PDF]
Moler, Van Loan. Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later.
[WEB]
Interesting article about the Julia programming language.
The Heston model
[PDF]
S. Heston.
A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.
[PDF]
H. Albrecher, P. Mayer, W. Schoutens, J. Tistaert.
The Little Heston Trap.
Finite differences
[PDF]
G. Fusai, S. Sanfelici and A. Tagliani.
Practical Problems in the Numerical Solution of PDEs in finance.
[PDF]
J.F. Harper.
Reducing Parabolic Partial Differential Equations to canonical forms.
[PDF]
O. Osterby.
Five Ways of Reducing the Crank-Nicolson Oscillations.
[PDF]
D. Duffy.
A critique of the Crank-Nicolson scheme strengths and weaknesses for financial instrument pricing.
[PDF]
J. Tropp.
An elementary proof of the spectral radius formula for matrices.
[PDF]
J.M. Varah.
On the solution of block-tridiagonal systems arising from certain finite-difference equations.
[PDF]
E. Ekstrom, P. Lotstedt, L. von Sydow, J. Tysk.
Numerical option pricing in the presence of bubbles.
Fourier transforms
[PDF]
P. Carr and D. Madan.
Option valuation using the fast Fourier transform.
[PDF]
D. Bailey and P. Swarztrauber.
The fractional Fourier transform and applications.
[PDF]
S. Drapeau, M. Kupper and A. Papapantoleon.
A Fourier Approach to the Computation of CVaR and Optimized Certainty Equivalents.
[PDF]
E. Eberlein, K. Glau and A. Papapantoleon.
Analysis of Fourier transform valuation formulas and applications.
[PDF]
C. Rogers and O. Zane.
Saddlepoint approximations to option prices.
Linear programming and duality
[PDF]
G.B. Dantzig.
Application of the simplex method to a transportation problem
[PDF]
G.B. Dantzig.
Maximization of a linear function of variables subject to linear inequalities
[PDF]
D. Gale, H.W. Kuhn and A.W. Tucker.
Linear programming and the theory of games
[PDF]
T.Pennanen.
Introduction to convex optimization in financial markets.
[PDF]
V.Piterbarg.
Spread options, Farkas's lemma and linear programming.
Miscellaneous
[PDF]
F. Delbaen and W. Schachermayer.
What is a free lunch?
[PDF]
J.F. Grcar.
Mathematicians of Gaussian Elimination.
Project Papers (2015-2016)
[PDF]
and
[PDF]
Higher-order finite difference schemes for Heston.
[PDF]
ADI method for Heston.
[PDF]
Pricing and Hedging Asian option.
[PDF]
Pricing with jumps.
[PDF]
Spread options with illiquidity.
[PDF]
Spread options.
Past exams
[PDF]
Exam 2014.
[PDF]
Exam 2015.
[PDF]
Exam 2016.