Red: no coding-exam
Section 1 Introduction:
Programming languages, programming basics, data types, operators, expressions, control structures (sequential, conditional and iterated execution), vector/array operations, input/output, plots, style, floating-point representation of real numbers, numerical errors.
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· Slide 1.1 and 1.2
· Numerical Recipes, Chapter 1
· Tools from Stochastic Analysis, Chapter 1 – 3.7
Section 2 Probability distributions:
Normal, exponential, log-normal, chi square, plot of their PDF and CDF, sampling with pseudo-random numbers, histograms, transformation from uniform to other distributions using the quantile function i.e. the inverse cumulative distribution function.
· The slides Truncation, discretisation and rounding errors; grids and histograms
· Tools from Stochastic Analysis, Chapters 10 and 11.1-2. Plotting and sampling distributions are good simple exercises to learn a programming language.
Section 3 Random numbers:
Linear congruential generators, requirements and statistical tests, pathologic cases, more advanced generators; inversion and transformation in one and more dimensions, acceptance-rejection method, Box-Muller method for normal deviates, polar method by Marsaglia, Ziggurat algorithm by Marsaglia and Tsang, correlated normal random variates, quasi-random numbers.
· Course Notes on Computational Finance, Chapter 2.
Section 4 Monte Carlo :
Diffusion equation, parabolic partial differential equations, stochastic differential equations, Feynman-Kac theorem, Black-Scholes-Merton equation, risk-neutral valuation of options, Euler-Maruyama algorithm for the numerical solution of a stochastic differential equation, approximation error, strong and weak solution, Milstein algorithm.
· Course Notes on Computational Finance, Chapter 3, first part.
Section 5 Drift-diffusion process:
Arithmetic and geometric Brownian motion, Ornstein-Uhlenbeck process and Vasicek model, Feller square-root process and Cox-Ingersoll-Ross model, constant elasticity of variance processes, Brownian bridge, Heston stochastic volatility model.
· Tools from Stochastic Analysis, Sections 3.8-9 (for arithmetic Brownian motion) and Chapter 5 (for all other processes).
Section 6 Jump-diffusion process:
Poisson and normal compound Poisson process, finite-activity jump-diffusion processes (Merton and Kou), time-changed Brownian motion (variance gamma, normal inverse Gaussian, CGMY), Lévy processes.
· Tools from Stochastic Analysis, Chapter 6.
Section 7 European options:
A simple program that prices European calls and puts with the analytical solution of the Black-Scholes-Merton equation, the Fourier transform, and Monte Carlo.
Section 8 Fourier Transform:
Definitions, inverse transform, properties, notable transform pairs (double-sided exponential/Lorentzian, Dirac delta/1, rectangular pulse/sinc, Gaussian/Gaussian), discrete and fast Fourier transform, characteristic function, moment- and cumulant-generating functions, correlation/convolution theorem, auto/cross-covariance and correlation, Parseval/Plancherel theorem, shift theorem, pricing in Fourier space.
· lecture notes
· Tools from Stochastic Analysis, Chapter 12
· Numerical Recipes, Chapters 12 and 13
Section 9 Model Calibration:
Implied volatility, Newton-Raphson method, Jäckel’s equivalent form, Jäckel’s modification, complex initial guess, attraction basin, fractals.
· For the Newton-Raphson method see Numerical Recipes, Section 9.4.
· Slides for section 9
Section 10 Exotic options:
Fourier transform methods for the numerical pricing of discretely and continuously monitored path-dependent options (barrier, hindsight, etc.).
Numerical pricing techniques for discretely monitored path-dependent options are shown as an example use of Lévy processes and Fourier transform methods in finance, but this advanced material, like all further reading, is not expected to be learned for the exam. The Fourier part of the code in Section 9 is a stripped-down version of the code used for the publication below, with only one monitoring date and thus no need for a Wiener-Hopf factorisation and an inverse z-transform.
Section 11 Partial differential equations:
Classification of second-order PDEs (elliptic, parabolic and hyperbolic) with examples, finite difference schemes.
· PDE Tools, Chapter 7.1-3
· Numerical Recipes, Chapter 20
Test format:
You have reported the structure of the final exam correctly as I announced it in class, except that the two “Matlab” questions should be more accurately called programming questions because you may supply your answers in almost any programming language you like, e.g. Matlab (including its free variants GNU Octave and Scilab), Julia, Python, C++, C, R, Fortran, Rust, C#, etc.
With respect to the answers to the past papers, I am not sure I will find the time to upload them, but you may disclose yours here on the forum and start a discussion with your peers to which I will contribute my feedback. I am also available during my weekly office hour.
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