Introduction to convex optimization in financial markets

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.

This is a preview of subscription content, access via your institution.

References

  1. 1

    Artzner P., Delbaen F., Eber J.M., Heath D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2

    Artzner P., Delbaen F., Koch-Medona P.: Risk measures and efficient use of capital. Astin Bull. 39(1), 101–116 (2009)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

  5. 5

    Ben-Tal A., Teboulle M.: Expected utility, penalty functions, and duality in stochastic nonlinear programming. Manag. Sci. 32(11), 1445–1466 (1986)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7

    Bertsekas D.P.: Necessary and sufficient conditions for existence of an optimal portfolio. J. Econ. Theory 8(2), 235–247 (1974)

    MathSciNet  Article  Google Scholar 

  8. 8

    Black F., Scholes M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)

    Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12

    Cvitanić J., Karatzas I.: Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2(4), 767–818 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13

    Cvitanić J., Karatzas I.: Hedging and portfolio optimization under transaction costs: a martingale approach. Math. Finance 6(2), 133–165 (1996)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Google Scholar 

  15. 15

    Davis M.H.A., Norman A.R.: Portfolio selection with transaction costs. Math. Oper. Res. 15(4), 676–713 (1990)

    MathSciNet  MATH  Article  Google 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)

    MATH  Article  Google 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)

    Article  Google 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)

    MATH  Article  Google Scholar 

  20. 20

    Dermody J.C., Rockafellar R.T.: Tax basis and nonlinearity in cash stream valuation. Math. Finance 5(2), 97–119 (1995)

    MATH  Article  Google Scholar 

  21. 21

    Drapeau, S., Kupper, M.: Risk preferences and their robust representation (2010, preprint)

  22. 22

    ElKaroui N., Ravanelli C.: Cash subadditive risk measures and interest rate ambiguity. Math. Finance 19(4), 561–590 (2009)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24

    Föllmer H., Schied A.: Convex measures of risk and trading constraints. Finance Stoch 6(4), 429–447 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25

    Föllmer, H., Schied, A.: Stochastic Finance [An introduction in discrete time]. Walter de Gruyter & Co., Berlin, extended edition (2011)

  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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29

    Haugh M.B., Kogan L.: Pricing american options: a duality approach. Oper. Res. 52(2), 258–270 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30

    Hilli P., Koivu M., Pennanen T.: Cash-flow based valuation of pension liabilities. Eur. Actuar. J. 1, 329–343 (2011)

    MathSciNet  Article  Google Scholar 

  31. 31

    Hilli P., Koivu M., Pennanen T.: Optimal construction of a fund of funds. Eur. Actuar. J. 1, 345–359 (2011)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34

    Jouini E., Kallal H.: Martingales and arbitrage in securities markets with transaction costs. J. Econ. Theory 66(1), 178–197 (1995)

    MathSciNet  MATH  Article  Google 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)

  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)

  37. 37

    Kabanov Y.M.: Hedging and liquidation under transaction costs in currency markets. Finance Stoch. 3(2), 237–248 (1999)

    MathSciNet  MATH  Article  Google 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)

  40. 40

    Koivu M., Pennanen T.: Galerkin methods in dynamic stochastic programming. Optimization 59(3), 339–354 (2010)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42

    Kreps D.M.: Arbitrage and equilibrium in economies with infinitely many commodities. J. Math. Econ. 8(1), 15–35 (1981)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43

    Lan, G., Nemirovski, A., Shapiro, A.: Validation analysis of mirror descent stochastic approximation method. Math. Program. (online first)

  44. 44

    Malo, P., Pennanen, T.: Reduced form modeling of limit order markets. Quant. Finance (to appear)

  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)

  46. 46

    Merton R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 3, 373–413 (1969)

    MathSciNet  Google 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)

    MathSciNet  Article  Google 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 Mathematics

  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)

    Google 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)

    Google 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)

  53. 53

    Pennanen, T.: Indifference pricing in illiquid markets (submitted)

  54. 54

    Pennanen T.: Epi-convergent discretizations of multistage stochastic programs. Math. Oper. Res. 30(1), 245–256 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55

    Pennanen T.: Epi-convergent discretizations of multistage stochastic programs via integration quadratures. Math. Program. Ser. B 116(1), 461–479 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56

    Pennanen T.: Arbitrage and deflators in illiquid markets. Finance Stoch. 15(1), 57–83 (2011)

    MathSciNet  Article  Google Scholar 

  57. 57

    Pennanen T.: Convex duality in stochastic optimization and mathematical finance. Math. Oper. Res. 36(2), 340–362 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  58. 58

    Pennanen T.: Dual representation of superhedging costs in illiquid markets. Math. Financ. Econ. 5, 233–248 (2011)

    Article  Google Scholar 

  59. 59

    Pennanen T.: Superhedging in illiquid markets. Math. Finance 21(3), 519–540 (2011)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  61. 61

    Pennanen, T., Perkkiö, A.-P.: Stochastic programs without duality gaps. Math. Program. (to appear)

  62. 62

    Pflug G.C., Römisch W.: Modeling, Measuring and Managing Risk. World Scientific, Singapore (2007)

    Google 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)

    Google 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)

  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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

  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)

  73. 73

    Sloan I.H., Wozniakowski H.: When are quasi-monte carlo algorithms efficient for high dimensional integrals?. J. Complex. 14, 1–33 (1997)

    MathSciNet  Article  Google Scholar 

  74. 74

    Traub J.F., Werschulz A.G.: Complexity and Information. Cambridge University Press, Cambridge (1998)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Teemu Pennanen.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Pennanen, T. Introduction to convex optimization in financial markets. Math. Program. 134, 157–186 (2012). https://doi.org/10.1007/s10107-012-0573-4

Download citation

Mathematics Subject Classification

  • 90C25
  • 91Gxx