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)

    [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.

    [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.