Annals of Operations Research

, Volume 193, Issue 1, pp 129–158

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 (e.g., 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.


Optimisation heuristics Financial optimisation Portfolio optimisation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Acker, D., & Duck, N. W. (2007). Reference-day risk and the use of monthly returns data. Journal of Accounting, Auditing and Finance, 22(4), 527–557. Google Scholar
  2. Althöfer, I., & Koschnick, K.-U. (1991). On the convergence of “Threshold Accepting”. Applied Mathematics & Optimization, 24(1), 183–195. CrossRefGoogle Scholar
  3. Bakshi, G., Cao, C., Chen, Z. (1997). Empirical performance of alternative option pricing models. Journal of Finance, 52(5), 2003–2049. CrossRefGoogle Scholar
  4. Barr, R. S., Golden, B. L., Kelly, J. P., Resende, M. G. C., & Stewart, W. R. (1995). Designing and reporting on computational experiments with heuristic methods. Journal of Heuristics, 1(1), 9–32. CrossRefGoogle Scholar
  5. Bates, D. S. (2003). Empirical option pricing: a retrospection. Journal of Econometrics, 116(1–2), 387–404. CrossRefGoogle Scholar
  6. Blume, M. E. (1971). On the assessment of risk. Journal of Finance, 26(1), 1–10. CrossRefGoogle Scholar
  7. Chan, L. K. C., & Lakonishok, J. (1992). Robust measurement of beta risk. Journal of Financial and Quantitative Analysis, 27(2), 265–282. CrossRefGoogle Scholar
  8. Chan, L. K. C., Karceski, J., & Lakonishok, J. (1999). On portfolio optimization: forecasting covariances and choosing the risk model. The Review of Financial Studies, 12(5), 937–974. CrossRefGoogle Scholar
  9. Chekhlov, A., Uryasev, S., & Zabarankin, M. (2005). Drawdown measure in portfolio optimization. International Journal of Theoretical and Applied Finance, 8(1), 13–58. CrossRefGoogle Scholar
  10. Constantinides, G. M., & Malliaris, A. G. (1995). Portfolio theory. In R. A. Jarrow, V. Maksimovic, & W. T. Ziemba (Eds.), Handbooks in operations research and management science: Vol. 9: Finance (pp. 1–30). Amsterdam: North-Holland. Google Scholar
  11. Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1, 223–236. CrossRefGoogle Scholar
  12. Cont, R., & da Fonseca, J. (2002). Dynamics of implied volatility surfaces. Quantitative Finance, 2, 45–60. CrossRefGoogle Scholar
  13. Dacorogna, M. M., Gençay, R., Müller, U. A., Olsen, R. B., & Pictet, O. V. (2001). An introduction to high-frequency finance. San Diego: Academic Press. Google Scholar
  14. Dembo, R. S. (1991). Scenario optimization. Annals of Operation Research, 30(1), 63–80. CrossRefGoogle Scholar
  15. Diebold, F. X., & Li, C. (2006). Forecasting the term structure of government bond yields. Journal of Econometrics, 130(2), 337–364. CrossRefGoogle Scholar
  16. Dixit, A. K. (1990). Optimization in economic theory (2nd ed.). London: Oxford University Press. Google Scholar
  17. Dueck, G., & Scheuer, T. (1990). Threshold Accepting. A general purpose optimization algorithm superior to Simulated Annealing. Journal of Computational Physics, 90(1), 161–175. CrossRefGoogle Scholar
  18. Eberhart, R. C., & Kennedy, J. (1995). A new optimizer using Particle Swarm theory. In Proceedings of the sixth international symposium on micromachine and human science, Nagoya, Japan (pp. 39–43). CrossRefGoogle Scholar
  19. Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33, 3–56. CrossRefGoogle Scholar
  20. Fama, E. F., & French, K. R. (2004). The capital asset pricing model: theory and evidence. The Journal of Economic Perspectives, 18(3), 25–46. CrossRefGoogle Scholar
  21. Gaivoronski, A. A., & Pflug, G. (2005). Value-at-risk in portfolio optimization: properties and computational approach. The Journal of Risk, 7(2), 1–31. Google Scholar
  22. Gendreau, M., & Potvin, J.-Y. (Eds.) (2010). Handbook of metaheuristics (2nd ed.). Berlin: Springer. Google Scholar
  23. Genton, M. G., & Ronchetti, E. (2008). Robust prediction of beta. In: E. J. Kontoghiorghes, B. Rustem, & P. Winker (Eds.), Computational methods in financial engineering—essays in honour of Manfred Gilli. Berlin: Springer. Google Scholar
  24. Gill, P. E., Murray, W., & Wright, M. H. (1986). Practical optimization. Amsterdam: Elsevier. Google Scholar
  25. Gilli, M., & Këllezi, E. (2002a). A global optimization heuristic for portfolio choice with VaR and expected shortfall. In: E. J. Kontoghiorghes, B. Rustem, & S. Siokos (Eds.), Applied optimization series: Computational methods in decision-making, economics and finance (pp. 167–183). Dordrecht: Kluwer Academic. Google Scholar
  26. Gilli, M., & Këllezi, E. (2002b). The Threshold Accepting heuristic for index tracking. In P. Pardalos & V. K. Tsitsiringos (Eds.), Applied optimization series: Financial engineering, e-commerce and supply chain (pp. 1–18). Boston: Kluwer Academic. Google Scholar
  27. Gilli, M., & Schumann, E. (2010a). Distributed optimisation of a portfolio’s omega. Parallel Computing, 36(7), 381–389. CrossRefGoogle Scholar
  28. Gilli, M., & Schumann, E. (2010b). Portfolio optimization with “Threshold Accepting”: a practical guide. In S. E. Satchell (Ed.), Optimizing optimization: the next generation of optimization applications and theory. Amsterdam: Elsevier. Google Scholar
  29. Gilli, M., & Schumann, E. (2010c). Optimization in financial engineering—an essay on ‘good’ solutions and misplaced exactitude. Journal of Financial Transformation, 28, 117–122. Google Scholar
  30. Gilli, M., & Schumann, E. (2010d). Optimal enough? Journal of Heuristics. doi:10.1007/s10732-010-9138-y.
  31. Gilli, M., & Schumann, E. (2011). Risk-reward optimisation for long-run investors: an empirical analysis. European Actuarial Journal. Available from
  32. Gilli, M., & Winker, P. (2003). A global optimization heuristic for estimating agent based models. Computational Statistics & Data Analysis, 42(3), 299–312. CrossRefGoogle Scholar
  33. Gilli, M., & Winker, P. (2008). A review of heuristic optimization methods in econometrics. Swiss finance institute research paper No. 08-12. Google Scholar
  34. Gilli, M., & Winker, P. (2009). Heuristic optimization methods in econometrics. In D. A. Belsley & E. Kontoghiorghes (Eds.), Handbook of computational econometrics. New York: Wiley. Google Scholar
  35. Gilli, M., Këllezi, E., & Hysi, H. (2006). A data-driven optimization heuristic for downside risk minimization. The Journal of Risk, 8(3), 1–18. Google Scholar
  36. Gilli, M., Maringer, D., & Winker, P. (2008). Applications of heuristics in finance. In D. Seese, C. Weinhardt, & F. Schlottmann (Eds.), Handbook on information technology in finance. Berlin: Springer. Google Scholar
  37. Gilli, M., Schumann, E., di Tollo, G., Cabej G. (2010). Constructing 130/30-portfolios with the omega ratio. Journal of Asset Management. doi:10.1057/jam.2010.25. Google Scholar
  38. Gilli, M., Maringer, D., & Schumann, E. (2011). Numerical methods and optimization in finance. Amsterdam: Elsevier. Google Scholar
  39. Glover, F. (1986). Future paths for integer programming and links to artificial intelligence. Computers & Operations Research, 13(5), 533–549. CrossRefGoogle Scholar
  40. Glover, F. (2007). Tabu Search—uncharted domains. Annals of Operation Research, 149(1), 89–98. CrossRefGoogle Scholar
  41. Glover, F., & Laguna, M. (1997). Tabu Search. Dordrecht: Kluwer Academic. CrossRefGoogle Scholar
  42. Hamida, S. B., & Cont, R. (2005). Recovering volatility from option prices by evolutionary optimization. Journal of Computational Finance, 8(4), 1–2. Google Scholar
  43. Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bonds and currency options. The Review of Financial Studies, 6(2), 327–343. CrossRefGoogle Scholar
  44. Hillier, F. S. (1983). Heuristics: a Gambler’s roll. Interfaces, 13(3), 9–12. CrossRefGoogle Scholar
  45. Hochreiter, R. (2008). Evolutionary stochastic portfolio optimization. In A. Brabazon & M. O’Neill (Eds.), Natural computing in computational finance. Berlin: Springer. Google Scholar
  46. Holland, J. H. (1992). Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control and artificial intelligence. Cambridge: MIT Press. Google Scholar
  47. Hoos, H. H., & Stützle, T. (2004). Stochastic Local Search: foundations and applications. San Mateo: Morgan Kaufmann. Google Scholar
  48. Ince, O. S., & Porter, R. B. (2006). Individual equity return data from Thomson datastream: handle with care! Journal of Financial Research, 29(4), 463–479. CrossRefGoogle Scholar
  49. Keating, C., & Shadwick, B. (2002). An introduction to omega. AIMA Newsletter (April 2002). Google Scholar
  50. Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. P. (1983). Optimization by Simulated Annealing. Science, 220(4598), 671–680. CrossRefGoogle Scholar
  51. Kirman, A. P. (1992). Whom or what does the representative individual represent? The Journal of Economic Perspectives, 6(2), 117–136. Google Scholar
  52. Kirman, A. P. (1993). Ants, rationality, and recruitment. The Quarterly Journal of Economics, 108(1), 137–156. CrossRefGoogle Scholar
  53. Knez, P. J., & Ready, M. J. (1997). On the robustness of size and book-to-market in cross-sectional regressions. Journal of Finance, 52(4), 1355–1382. CrossRefGoogle Scholar
  54. LeBaron, B. (2000). Agent-based computational finance: suggested readings and early research. Journal of Economic Dynamics & Control, 24, 679–702. CrossRefGoogle Scholar
  55. LeBaron, B. (2006). Agent-based computational finance. In L. H. Tesfatsion & K. H. L. Judd (Eds.), Handbook of computational economics (Vol. II, pp. 1187–1233). Amsterdam: Elsevier. Google Scholar
  56. Lo, A. W. (2008). Hedge funds—an analytic perspective. Princeton: Princeton University Press. Google Scholar
  57. Luenberger, D. G. (1998). Investment science. Oxford: Oxford University Press. Google Scholar
  58. Madan, D. B. (2001). On the modelling of option prices. Quantitative Finance, 1, 481. CrossRefGoogle Scholar
  59. Manaster, S., & Koehler, G. (1982). The calculation of implied variances from the Black–Scholes model: a note. Journal of Finance, 37(1), 227–230. CrossRefGoogle Scholar
  60. Maniezzo, V., Stützle, T., & Voß, S. (Eds.) (2009). Matheuristics: hybridizing metaheuristics and mathematical programming. Berlin: Springer. Google Scholar
  61. Maringer, D. (2004). Finding the relevant risk factors for asset pricing. Computational Statistics & Data Analysis, 47, 339–352. CrossRefGoogle Scholar
  62. Maringer, D. (2005). Portfolio management with heuristic optimization. Berlin: Springer. Google Scholar
  63. Maringer, D. (2008). Risk preferences and loss aversion in portfolio optimization. In E. J. Kontoghiorghes, B. Rustem, & P. Winker (Eds.), Computational methods in financial engineering—essays in honour of Manfred Gilli. Berlin: Springer. Google Scholar
  64. Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 7(1), 77–91. CrossRefGoogle Scholar
  65. Markowitz, H. M. (1959). Portfolio selection. New York: Wiley. Google Scholar
  66. Martin, R. D., & Simin, T. T. (2003). Outlier-resistant estimates of beta. Financial Analysts Journal, 59(5), 56–69. CrossRefGoogle Scholar
  67. Michalewicz, Z., & Fogel, D. B. (2004). How to solve it: modern heuristics. Berlin: Springer. Google Scholar
  68. Mikhailov, S., & Nögel, U. (2003). Heston’s stochastic volatility model implementation, calibration and some extensions. Wilmott, pp. 74–79. Google Scholar
  69. Mladenović, N., & Hansen, P. (1997). Variable neighborhood search. Computers & Operations Research, 24(11), 1097–1100. CrossRefGoogle Scholar
  70. Moscato, P. (1989). On evolution, search, optimization, Genetic Algorithms and martial arts—towards memetic algorithms. Technical Report 790, CalTech California Institute of Technology. Google Scholar
  71. Moscato, P., & Fontanari, J. F. (1990). Stochastic versus deterministic update in Simulated Annealing. Physics Letters A, 146(4), 204–208. CrossRefGoogle Scholar
  72. Nelson, C. R., & Siegel, A. F. (1987). Parsimonious modeling of yield curves. Journal of Business, 60(4), 473–489. CrossRefGoogle Scholar
  73. Pisinger, D., & Ropke, S. (2010). Large neighborhood search. In M. Gendreau & J.-Y. Potvin (Eds.), Handbook of metaheuristics (2nd ed.). Berlin: Springer. Google Scholar
  74. Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. The Journal of Risk, 2(3), 21–41. Google Scholar
  75. Rousseeuw, P. J. (1984). Least median of squares regression. Journal of the American Statistical Association, 79(388), 871–880. CrossRefGoogle Scholar
  76. Schlottmann, F., & Seese, D. (2004). Modern heuristics for finance problems: a survey of selected methods and applications. In S. T. Rachev (Ed.), Handbook of computational and numerical methods in finance. Basel: Birkhäuser. Google Scholar
  77. Schrimpf, G., Schneider, J., Stamm-Wilbrandt, H., & Dueck, G. (2000). Record breaking optimization results using the ruin and recreate principle. Journal of Computational Physics, 159(2), 139–171. CrossRefGoogle Scholar
  78. Sortino, F., van der Meer, R., & Plantinga, A. (1999). The Dutch triangle. Journal of Portfolio Management, 26(1), 50–58. CrossRefGoogle Scholar
  79. Storn, R. M., & Price, K. V. (1997). Differential Evolution—a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 11(4), 341–359. CrossRefGoogle Scholar
  80. Svensson, L. E. O. (1994). Estimating and interpreting forward interest rates: Sweden 1992–1994. IMF Working Paper 94/114. Google Scholar
  81. Talbi, E.-G. (2002). A taxonomy of hybrid metaheuristics. Journal of Heuristics, 8(5), 541–564. CrossRefGoogle Scholar
  82. Vasicek, O. A. (1973). A note on the cross-sectional information in Bayesian estimation of security betas. Journal of Finance, 28(5), 1233–1239. CrossRefGoogle Scholar
  83. Winker, P. (2001). Optimization heuristics in econometrics: applications of Threshold Accepting. New York: Wiley. Google Scholar
  84. Winker, P., & Fang, K.-T. (1997). Application of Threshold-Accepting to the evaluation of the discrepancy of a set of points. SIAM Journal on Numerical Analysis, 34(5), 2028–2042. CrossRefGoogle Scholar
  85. Winker, P., & Gilli, M. (2004). Applications of optimization heuristics to estimation and modelling problems. Computational Statistics and Data Analysis, 47(2), 211–223. CrossRefGoogle Scholar
  86. Winker, P., & Maringer, D. (2007). The Threshold Accepting optimisation algorithm in economics and statistics. In E. J. Kontoghiorghes & C. Gatu (Eds.), Advances in computational management science: Vol. 9. Optimisation, econometric and financial analysis (pp. 107–125). Berlin: Springer. CrossRefGoogle Scholar
  87. Winker, P., Gilli, M., & Jeleskovic, V. (2007). An objective function for simulation based inference on exchange rate data. Journal of Economic Interaction and Coordination, 2, 125–145. CrossRefGoogle Scholar
  88. Zanakis, S. H., & Evans, J. R. (1981). Heuristic “optimization”: why, when, and how to use it. Interfaces, 11(5), 84–91. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of EconometricsUniversity of Geneva and Swiss Finance InstituteGeneva 4Switzerland
  2. 2.VIP Value Investment Professionals AGZugSwitzerland

Personalised recommendations