Mathematics and Financial Economics

, Volume 13, Issue 4, pp 637–659 | Cite as

Golden options in financial mathematics

  • Alejandro BalbásEmail author
  • Beatriz Balbás
  • Raquel Balbás


This paper deals with the construction of “smooth good deals” (SGD), i.e., sequences of self-financing strategies whose global risk diverges to minus infinity and such that every security in every strategy of the sequence is a “smooth” derivative with a bounded delta. Since delta is bounded, digital options are excluded. In fact, the pay-off of every option in the sequence is continuos (and therefore jump-free) with respect to the underlying asset price. If the selected risk measure is the value at risk, then these sequences exist under quite weak conditions, since one can involve risks with both bounded and unbounded expectation, as well as non-friction-free pricing rules. Moreover, every strategy in the sequence is composed of a short European option plus a position in a riskless asset. If the chosen risk measure is a coherent one, then the general setting is more limited. Indeed, though frictions are still accepted, expectations and variances must remain finite. The existence of SGDs will be characterized, and computational issues will be properly addressed. It will be shown that SGDs often exist, and for the conditional value at risk, they are composed of the riskless asset plus easily replicable short European puts. The ideas presented may also apply in some actuarial problems such as the selection of an optimal reinsurance contract.


Golden option Risk measure Smooth good deal Dual approach 

Mathematics Subject Classification

91G10 91G20 91G80 91B06 

JEL Classification

G11 G13 C61 C65 



This research was partially supported by the University Carlos III of Madrid (Project 2009 / 00445 / 002). The authors sincerely thank the journal editor and the anonymous reviewer, whose comments led to significant improvements in this paper. The usual caveat applies.


  1. 1.
    Anderson, E.J., Nash, P.: Linear Programming in Infinite-dimensional Spaces. Wiley, New York (1987)zbMATHGoogle Scholar
  2. 2.
    Assa, H.: Natural risk measures. Math. Financ. Econ. 10, 441–456 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balbás, A., Balbás, B., Balbás, R.: Outperforming benchmarks with their derivatives: theory and empirical evidence. J. Risk 18(4), 25–52 (2016)CrossRefzbMATHGoogle Scholar
  5. 5.
    Balbás, A., Balbás, B., Balbás, R.: Differential equations connecting \(VaR\) and \(CVaR\). J. Comput. Appl. Math. 326, 247–267 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Balbás, A., Balbás, B., Balbás, R., Heras, A.: Optimal reinsurance under risk and uncertainty. Insur. Math. Econ. 60, 61–74 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Balbás, A., Balbás, R., Guerra, P.J.: Vector risk functions. Mediterr. J. Math. 9, 563–574 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bernardo, A.E., Ledoit, O.: Gain, loss, and asset pricing. J. Polit. Econ. 108, 144–172 (2000)CrossRefGoogle Scholar
  9. 9.
    Biagini, S., Pinar, M.C.: The best gain-loss ratio is a poor performance measure. SIAM J. Financ. Math. 4, 228–242 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bondarenko, O.: Why are put options so expensive? Q. J. Finance 4(3), 1450015 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cheung, K.C., Chong, W.F., Yam, S.: Convex ordering for insurance preferences. Insur. Math. Econ. 64, 409–416 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cochrane, J.H., Saa-Requejo, J.: Beyond arbitrage: good deal asset price bounds in incomplete markets. J. Polit. Econ. 108, 79–119 (2000)CrossRefGoogle Scholar
  13. 13.
    Duffie, D.: Security Markets: Stochastic Models. Academic Press, Cambridge (1988)zbMATHGoogle Scholar
  14. 14.
    Filipović, D., Kupper, M., Vogelpoth, N.: Approaches to conditional risk. SIAM J. Financ. Math. 3, 402–432 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jarrow, R., Larsson, M.: The meaning of market efficiency. Math. Finance 22, 1–30 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jouini, E., Meddeb, M., Touzi, N.: Vector-valued coherent risk measures. Finance Stoch. 8, 531–552 (2004)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Konstantinides, D.G., Zachos, G.C.: Exhibiting abnormal returns under a risk averse strategy. Methodol. Comput. Appl. Probab. (2018). (forthcoming) Google Scholar
  18. 18.
    Kopp, P.E.: Martingales and Stochastic Integrals. Cambridge University Press, Cambridge (2008)zbMATHGoogle Scholar
  19. 19.
    Kupper, M., Schachermayer, W.: Representation results for law invariant time consistent functions. Math. Financ. Econ. 2, 189–210 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Luenberger, D.G.: Optimization by Vector Spaces Methods. Wiley, New York (1969)zbMATHGoogle Scholar
  21. 21.
    Mausser, H., Saunders, D., Seco, L.: Optimizing omega. Risk Mag. 11, 88–92 (2006)Google Scholar
  22. 22.
    Pichler, A.: Insurance pricing under ambiguity. Eur. Actuar. J. 4, 335–364 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Generalized deviations in risk analysis. Finance Stoch. 10, 51–74 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rubinstein, M.: Implied binomial trees. J. Finance 49(3), 771–818 (1994)CrossRefGoogle Scholar
  25. 25.
    Salas-Molina, F.: Selecting the best risk measure in multiobjective cash management. Int. Trans. Oper. Res. (2018). (forthcoming) MathSciNetGoogle Scholar
  26. 26.
    Schaeffer, H.H.: Topological Vector Spaces. Springer, Berlin (1970)Google Scholar
  27. 27.
    Schaeffer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)CrossRefGoogle Scholar
  28. 28.
    Tamiz, M., Jones, D., Romero, C.: Goal programming for decision making: an overview of the current state-of-the-art. Eur. J. Oper. Res. 111, 569–581 (1998)CrossRefzbMATHGoogle Scholar
  29. 29.
    Zhao, P., Xiao, Q.: Portfolio selection problem with value-at-risk constraints under non-extensive statistical mechanics. J. Comput. Appl. Math. 298, 74–91 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Alejandro Balbás
    • 1
    Email author
  • Beatriz Balbás
    • 2
  • Raquel Balbás
    • 3
  1. 1.University Carlos III of MadridGetafe, MadridSpain
  2. 2.University of AlcaláAlcalá de Henares, MadridSpain
  3. 3.University Complutense of Madrid. SomosaguasPozuelo de Alarcón, MadridSpain

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