Advertisement

Mathematical Programming

, Volume 134, Issue 1, pp 157–186 | Cite as

Introduction to convex optimization in financial markets

  • Teemu PennanenEmail author
Full Length Paper Series B

Abstract

Convexity arises naturally in financial risk management. In risk preferences concerning random cash-flows, convexity corresponds to the fundamental diversification principle. Convexity is a basic property also of budget constraints both in classical linear models as well as in more realistic models with transaction costs and constraints. Moreover, modern securities markets are based on trading protocols that result in convex trading costs. The first part of this paper gives an introduction to certain basic concepts and principles of financial risk management in simple optimization terms. The second part reviews some convex optimization techniques used in mathematical and numerical analysis of financial optimization problems.

Mathematics Subject Classification

90C25 91Gxx 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Artzner P., Delbaen F., Eber J.M., Heath D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Artzner P., Delbaen F., Koch-Medona P.: Risk measures and efficient use of capital. Astin Bull. 39(1), 101–116 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ben-Tal A., Goryashko A., Guslitzer E., Nemirovski A.: Adjustable robust solutions of uncertain linear programs. Math. Program. 99(2, Ser. A), 351–376 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization. MPS/SIAM Series on Optimization [Analysis, algorithms, and engineering applications]. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001)Google Scholar
  5. 5.
    Ben-Tal A., Teboulle M.: Expected utility, penalty functions, and duality in stochastic nonlinear programming. Manag. Sci. 32(11), 1445–1466 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Ben-Tal A., Teboulle M.: An old-new concept of convex risk measures: the optimized certainty equivalent. Math. Finance 17(3), 449–476 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bertsekas D.P.: Necessary and sufficient conditions for existence of an optimal portfolio. J. Econ. Theory 8(2), 235–247 (1974)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Black F., Scholes M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)CrossRefGoogle Scholar
  9. 9.
    Bühlmann H.: Mathematical Methods in Risk Theory. Die Grundlehren der mathematischen Wissenschaften, Band 172. Springer, New York (1970)Google Scholar
  10. 10.
    Carmona, R. (ed.): Indifference Pricing: Theory and Applications. Princeton Series in Financial Engineering. Princeton University Press, Princeton (2009)Google Scholar
  11. 11.
    Çetin U., Rogers L.C.G.: Modelling liquidity effects in discrete time. Math. Finance 17(1), 15–29 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Cvitanić J., Karatzas I.: Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2(4), 767–818 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Cvitanić J., Karatzas I.: Hedging and portfolio optimization under transaction costs: a martingale approach. Math. Finance 6(2), 133–165 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Dalang R.C., Morton A., Willinger W.: Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch. Stoch. Rep. 29(2), 185–201 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Davis M.H.A., Norman A.R.: Portfolio selection with transaction costs. Math. Oper. Res. 15(4), 676–713 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Delbaen F., Schachermayer W.: The Mathematics of Arbitrage. Springer Finance. Springer, Berlin (2006)Google Scholar
  17. 17.
    Dempster M.A.H., Evstigneev I.V., Taksar M.I.: Asset pricing and hedging in financial markets with transaction costs: an approach based on the Von Neumann–Gale model. Ann. Finance 2(4), 327–355 (2006)zbMATHCrossRefGoogle Scholar
  18. 18.
    Dermody J.C., Prisman E.Z.: No arbitrage and valuation in markets with realistic transaction costs. J. Finan. Quant. Anal. 28(1), 65–80 (1993)CrossRefGoogle Scholar
  19. 19.
    Dermody J.C., Rockafellar R.T.: Cash stream valuation in the face of transaction costs and taxes. Math. Finance 1(1), 31–54 (1991)zbMATHCrossRefGoogle Scholar
  20. 20.
    Dermody J.C., Rockafellar R.T.: Tax basis and nonlinearity in cash stream valuation. Math. Finance 5(2), 97–119 (1995)zbMATHCrossRefGoogle Scholar
  21. 21.
    Drapeau, S., Kupper, M.: Risk preferences and their robust representation (2010, preprint)Google Scholar
  22. 22.
    ElKaroui N., Ravanelli C.: Cash subadditive risk measures and interest rate ambiguity. Math. Finance 19(4), 561–590 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Evstigneev I.V., Schürger K., Taksar M.I.: On the fundamental theorem of asset pricing: random constraints and bang-bang no-arbitrage criteria. Math. Finance 14(2), 201–221 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Föllmer H., Schied A.: Convex measures of risk and trading constraints. Finance Stoch 6(4), 429–447 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Föllmer, H., Schied, A.: Stochastic Finance [An introduction in discrete time]. Walter de Gruyter & Co., Berlin, extended edition (2011)Google Scholar
  26. 26.
    Harris L.: Trading and Exchanges, Market Microstructure for Practitioners. Oxford University Press, Oxford (2002)Google Scholar
  27. 27.
    Harrison J.M., Kreps D.M.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20(3), 381–408 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Harrison J.M., Pliska S.R.: Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11(3), 215–260 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Haugh M.B., Kogan L.: Pricing american options: a duality approach. Oper. Res. 52(2), 258–270 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Hilli P., Koivu M., Pennanen T.: Cash-flow based valuation of pension liabilities. Eur. Actuar. J. 1, 329–343 (2011)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Hilli P., Koivu M., Pennanen T.: Optimal construction of a fund of funds. Eur. Actuar. J. 1, 345–359 (2011)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Hodges S.D., Neuberger A.: Optimal replication of contingent claims under transaction costs. Rev. Futur. Mark. 8, 222–239 (1989)Google Scholar
  33. 33.
    Jouini E., Kallal H.: Arbitrage in securities markets with short-sales constraints. Math. Finance 5(3), 197–232 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Jouini E., Kallal H.: Martingales and arbitrage in securities markets with transaction costs. J. Econ. Theory 66(1), 178–197 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Juditsky, A., Nemirovski, A.: First order methods for nonsmooth convex large-scale optimization, i: general purpose methods. In: Optimization for Machine Learning, pp. 121–148. MIT Press (2011)Google Scholar
  36. 36.
    Juditsky, A., Nemirovski, A.: First order methods for nonsmooth convex large-scale optimization, ii: utilizing problems structure. In: Optimization for Machine Learning, pp. 149–183. MIT Press (2011)Google Scholar
  37. 37.
    Kabanov Y.M.: Hedging and liquidation under transaction costs in currency markets. Finance Stoch. 3(2), 237–248 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Kabanov Y.M., Safarian M.: Markets with Transaction Costs. Springer Finance (Mathematical theory). Springer, Berlin (2009)Google Scholar
  39. 39.
    King, A.J.: Duality and martingales: a stochastic programming perspective on contingent claims. Math. Program. 91(3, Ser. B), 543–562 (2002). ISMP 2000, Part 1 (Atlanta, GA)Google Scholar
  40. 40.
    Koivu M., Pennanen T.: Galerkin methods in dynamic stochastic programming. Optimization 59(3), 339–354 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Kramkov D., Schachermayer W.: The condition on the asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9(3), 904–950 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Kreps D.M.: Arbitrage and equilibrium in economies with infinitely many commodities. J. Math. Econ. 8(1), 15–35 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Lan, G., Nemirovski, A., Shapiro, A.: Validation analysis of mirror descent stochastic approximation method. Math. Program. (online first)Google Scholar
  44. 44.
    Malo, P., Pennanen, T.: Reduced form modeling of limit order markets. Quant. Finance (to appear)Google Scholar
  45. 45.
    Markowitz, H.M.: Portfolio Selection: Efficient diversification of Investments. Cowles Foundation for Research in Economics at Yale University, Monograph 16. Wiley, New York (1959)Google Scholar
  46. 46.
    Merton R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 3, 373–413 (1969)MathSciNetGoogle Scholar
  47. 47.
    Nemirovski A., Juditsky A., Lan G., Shapiro A.: Robust stochastic approximation approach to stochastic programming. SIAM J. Optim. 19(4), 1574–1609 (2008)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Nemirovsky, A.S., Yudin, D.B.: Problem Complexity and Method Efficiency in Optimization. A Wiley-Interscience Publication. Wiley, New York (1983). Translated from the Russian and with a preface by E.R. Dawson, Wiley-Interscience Series in Discrete MathematicsGoogle Scholar
  49. 49.
    Nesterov Y.: Introductory Lectures on Convex Optimization, vol. 87 of Applied Optimization (A Basic Course). Kluwer, Boston (2004)Google Scholar
  50. 50.
    Novak E., Woźniakowski H.: Tractability of multivariate problems. Vol. 1: Linear information, vol. 6 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich (2008)CrossRefGoogle Scholar
  51. 51.
    Novak E., Woźniakowski H.: Tractability of multivariate problems. Volume II: Standard information for functionals, vol. 12 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich (2010)CrossRefGoogle Scholar
  52. 52.
    Olsen, P.: Discretizations of multistage stochastic programming problems. Math. Program. Stud. 6, 111–124 (1976). Stochastic systems: modeling, identification and optimization, II (Proc. Sympos. Univ. Kentucky, Lexington, Ky.,1975)Google Scholar
  53. 53.
    Pennanen, T.: Indifference pricing in illiquid markets (submitted)Google Scholar
  54. 54.
    Pennanen T.: Epi-convergent discretizations of multistage stochastic programs. Math. Oper. Res. 30(1), 245–256 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Pennanen T.: Epi-convergent discretizations of multistage stochastic programs via integration quadratures. Math. Program. Ser. B 116(1), 461–479 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Pennanen T.: Arbitrage and deflators in illiquid markets. Finance Stoch. 15(1), 57–83 (2011)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Pennanen T.: Convex duality in stochastic optimization and mathematical finance. Math. Oper. Res. 36(2), 340–362 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Pennanen T.: Dual representation of superhedging costs in illiquid markets. Math. Financ. Econ. 5, 233–248 (2011)CrossRefGoogle Scholar
  59. 59.
    Pennanen T.: Superhedging in illiquid markets. Math. Finance 21(3), 519–540 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Pennanen T., Penner I.: Hedging of claims with physical delivery under convex transaction costs. SIAM J. Financ. Math. 1, 158–178 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Pennanen, T., Perkkiö, A.-P.: Stochastic programs without duality gaps. Math. Program. (to appear)Google Scholar
  62. 62.
    Pflug G.C., Römisch W.: Modeling, Measuring and Managing Risk. World Scientific, Singapore (2007)zbMATHCrossRefGoogle Scholar
  63. 63.
    Pliska S.R.: Introduction to Mathematical Finance: Discrete Time Models. Blackwell, Malden (1997)Google Scholar
  64. 64.
    Rockafellar R.T.: Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton (1970)Google Scholar
  65. 65.
    Rockafellar R.T.: Conjugate Duality and Optimization. Society for Industrial and Applied Mathematics, Philadelphia (1974)zbMATHCrossRefGoogle Scholar
  66. 66.
    Rockafellar R.T., Uryasev S.P.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)Google Scholar
  67. 67.
    Rockafellar, R.T., Uryasev, S.P.: The Fundamental Risk Quadrangle in Risk Management, Optimization and Statistical Estimation. Working paper (2011)Google Scholar
  68. 68.
    Rockafellar R.T., Wets R.J.-B.: Variational Analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1998)Google Scholar
  69. 69.
    Rogers L.C.G.: Monte Carlo valuation of American options. Math. Finance 12(3), 271–286 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Schachermayer W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Finance 14(1), 19–48 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Shapiro, A.: Monte Carlo sampling methods. In: Stochastic Programming, vol. 10 of Handbooks Oper. Res. Management Sci., pp. 353–425. Elsevier, Amsterdam (2003)Google Scholar
  72. 72.
    Shapiro, A., Nemirovski, A.: On complexity of stochastic programming problems. In: Continuous Optimization, vol. 99 of Appl. Optim, pp. 111–146. Springer, New York (2005)Google Scholar
  73. 73.
    Sloan I.H., Wozniakowski H.: When are quasi-monte carlo algorithms efficient for high dimensional integrals?. J. Complex. 14, 1–33 (1997)MathSciNetCrossRefGoogle Scholar
  74. 74.
    Traub J.F., Werschulz A.G.: Complexity and Information. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar

Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.Department of MathematicsKing’s College LondonLondonUK

Personalised recommendations