Numerical Methods and Optimization in Finance
The book explains tools for computational finance with emphasis on simulation and optimization ... more information about 'Numerical Methods and Optimization in Finance'
The manual describes how to use the NMOF package, which accompanies the book 'Numerical Methods and Optimization in Finance' by Manfred Gilli, Dietmar Maringer and Enrico Schumann. This is currently a draft, and comments are very welcome.
Papers appear in the year in which they have been published, unpublished reports when they were first circulated.
A note on 1/N and minimum-variance portfolios, and the fact that significance tests do harm (and no good) in financial decision-making
DeMiguel et al. (2009) have shown that an equal-weight portfolio strategy performs well when compared with more sophisticated portfolio-selection models. Unfortunately, a number of people have interpreted their conclusion as 'you cannot beat 1/N'. In this note I argue that DeMiguel et al. (2009) have actually provided further evidence that long-only minimum-variance is an advisable investment strategy (which should be preferred to 1/N).
Take-the-Best in Portfolio Selection
A well-known result in portfolio optimisation states that as the number of assets in a portfolio grows, the variance of portfolio return approaches the average covariance between the included assets. I argue that this result should not be read as a justification to emphasise forecasting correlations. I compare the textbook recipe for constructing the minimum-variance portfolio, which uses the full variance-covariance matrix, with a simple, sorting-based rule. Through a simulation I show that when there is diversity in the cross-section of assets and we cannot precisely predict future covariance (two empirically valid assumptions), then the simple rule is rarely worse (and if, not much) than the textbook approach, but often better.
Better Portfolios with Options
As a result of the recent financial crises, equity markets have performed poorly in the last five years or so. In consequence, equity long-only strategies have generally been unattractive over this period. This motivates the investigation on whether better performance can be achieved by including equity options in the portfolios. We show that simple systematic option strategies improve portfolio performance. Results are supported by thorough backtesting and simulations.
Heuristic Optimisation in Financial Modelling
There is a large number of optimisation problems in theoretical and applied finance that are difficult to solve as they exhibit multiple local optima or are not 'well-behaved' in other ways (eg, discontinuities in the objective function). One way to deal with such problems is to adjust and to simplify them, for instance by dropping constraints, until they can be solved with standard numerical methods. We argue that an alternative approach is the application of optimisation heuristics like Simulated Annealing or Genetic Algorithms. These methods have been shown to be capable of handling non-convex optimisation problems with all kinds of constraints. To motivate the use of such techniques in finance, we present several actual problems where classical methods fail. Next, several well-known heuristic techniques that may be deployed in such cases are described. Since such presentations are quite general, we then describe in some detail how a particular problem, portfolio selection, can be tackled by a particular heuristic method, Threshold Accepting. Finally, the stochastics of the solutions obtained from heuristics are discussed. We show, again for the example from portfolio selection, how this random character of the solutions can be exploited to inform the distribution of computations.
Heuristic Methods in Finance
Heuristic optimization methods and their application to finance are discussed. Two illustrations of these methods are presented: the selection of assets in a portfolio and the estimation of a complicated econometric model.
Constructing 130/30-Portfolios with the Omega Ratio
We construct portfolios with an alternative selection criterion, the Omega function, which can be expressed as the ratio of two partial moments of a portfolio's return distribution. The main purpose of the article is to investigate the empirical performance of the selected portfolios, especially the effects of allowing short positions. Many studies on portfolio optimisation assume that short sales are not allowed. This is despite the fact that theoretically, short positions can improve the risk-return characteristics of a portfolio, and practically, institutional investors can and do sell stocks short. We investigate whether removing the non-negativity constraint really improves out-of-sample portfolio performance under realistic assumptions, that is when optimal weights need to be estimated from the data and different transaction costs apply to long and short positions.
FX Trading: An Empirical Study
Given a set of tick-by-tick data of five currency pairs we analyze several traditional asset allocation techniques as well as technical trading rule based models. In particular we explore appropriate levels of time aggregation and rebalancing frequencies. We also suggest a triggered rebalancement strategy which results in better performance and lower transaction costs. For the asset allocation approach multiple objectives are optimized using heuristic optimization techniques.
Calibrating the Nelson–Siegel–Svensson model
The Nelson–Siegel–Svensson model is widely-used for modelling the yield curve, yet many authors have reported 'numerical difficulties' when calibrating the model. We argue that the problem is twofold: firstly, the optimisation problem is not convex and has multiple local optima. Hence standard methods that are readily available in statistical packages are not appropriate. We implement and test an optimisation heuristic, Differential Evolution, and show that it is capable of reliably solving the model. Secondly, we also stress that in certain ranges of the parameters, the model is badly conditioned, thus estimated parameters are unstable given small perturbations of the data. We discuss to what extent these difficulties affect applications of the model.
R packages and examples
Software used in papers/books
- The NMOF package can be obtained from R-Forge; see the package's manual and NEWS file.
- All other code examples from Gilli/Maringer/Schumann (2011) are available from the book's website.
- MATLAB code for 'A Data-Driven Optimization Heuristic for Downside Risk Minimization' (Journal of Risk 8(3), 2006, pp. 1-18) can be downloaded from the COMISEF homepage
- MATLAB and R functions for 'Implementing Binomial Trees' are available from the COMISEF homepage.
- MATLAB and R functions for 'Calibrating Option Pricing Models with Heuristics' are available from the COMISEF homepage.
- R functions for 'Calibrating the Nelson–Siegel–Svensson model' are available from the COMISEF homepage.
- R function for 'A note on "good starting values" in numerical optimisation' are available from the COMISEF homepage.
I am currently responsible for research/quantitative strategies at an investment advisory in Switzerland. I hold a PhD in econometrics, an MSc in economics and a bachelor's degree in economics and law.