Annals of Finance

, Volume 15, Issue 4, pp 563–600 | Cite as

Semi-nonparametric approximation and index options

  • Julia Jiang
  • Weidong TianEmail author
Research Article


In an arbitrage-free securities market, all state-contingent claims and the stochastic discount factors can be approximated appropriately by index options with a semi-nonparametric method. These index options are constructed by efficient algorithms and uniform approximation error under these efficient algorithms are derived. This paper suggests a method to examine state-contingent claims and stochastic discount factors using index options in financial market regardless the market is complete or not.


Semi-nonparametric Index option Universal approximation error 

JEL Classification

G12 G13 


  1. Ait-Sahalia, Y., Lo, A.W.: Nonparametric estimation of state-price densities implicit in financial asset prices. J Finance 53(2), 499–547 (1998)Google Scholar
  2. Bakshi, G., Madan, D.: Spanning and derivative-security valuation. J Financ Econ 55, 205–238 (2000)Google Scholar
  3. Bansal, R., Viswanathan, S.: No arbitrage and arbitrage pricing: a new approach. J Finance 48(40), 1231–1262 (1993)Google Scholar
  4. Barron, A.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans Inf Theory 39, 930–945 (1993)Google Scholar
  5. Barron, A., Cohen, A., Dahmen, W., DeVore, E.: Approximation and learning by greedy algorithmis. Ann Stat 36(4), 64–94 (2008)Google Scholar
  6. Bertsimas, D., Kogan, L., Lo, A.: Hedging derivatives securities and incomplete markes: an \(\epsilon \)-arbitrage approach. Oper Res 49(3), 372–397 (2001)Google Scholar
  7. Bogle, J.C.: The index mutual fund: 40 years of growth, change, and challenge. Financ Anal J. 72, 9–13 (2016)Google Scholar
  8. Borovicka, J., Hansen, L.P., Scheinkman, J.: Misspecified recovery. J Finance 71, 2493–2544 (2016)Google Scholar
  9. Boyle, T., Tian, W.: Quadratic interest rate models as approximation to effective rate model. J Fixed Income 9, 69–80 (1999)Google Scholar
  10. Breeden, D., Litzenberger, R.: Prices of state-contingent claims implicit in option prices. J Bus 51(4), 621–651 (1978)Google Scholar
  11. Carr, P., Ellis, K., Gupta, V.: Static hedging of exotic options. J Finance 53(2), 1165–1190 (2002)Google Scholar
  12. Christensen, O.: An Introduction to Frames and Riesz Bases. Boston: Birkauser (2003)Google Scholar
  13. Christoffersen, P., Heston, S., Jacobs, K.: Capturing option anomalies with a variance-dependent pricing kernel. Rev Financ Stud 26, 1963–2006 (2013)Google Scholar
  14. Cochrane, J., Saa-Requejo, J.: Beyond arbitrage: good-deal asset price bounds in incomplete markets. J Polit Econ 108(1), 79–119 (2000)Google Scholar
  15. Cybenko, G.: Approximation by superposition of a sigmoidal function. Math Control Signals Syst 2, 303–314 (1989)Google Scholar
  16. Detemple, J., Kitapbayev, Y.: On American VIX options under the generalized 3/2 and 1/2 models. Math Finance 28, 550–581 (2018)Google Scholar
  17. Diaconis, P., Shahshahani, M.: On nonlinear functions of linear combinations. SIAM J Sci Stat Comput 5(1), 175–191 (1984)Google Scholar
  18. Dittmar, R.: Nonlinear pricing kernels, kurtosis preference, and evidence from the cross section on equity return. J Finance 57(1), 369–403 (2002)Google Scholar
  19. DeVore, R.: Optimal Computation. International Congress of Mathematicians, Plenary Lecture, Madrid (2006)Google Scholar
  20. Donahue, M., Gurvits, L., Darken, C., Sontag, E.: Rates of convex approximation in non-Hilbert spaces. Constr Approx 13(2), 187–220 (1997)Google Scholar
  21. Duan, J., Moreau, A., Sealey, C.W.: Spanning with index options. J Financ Quant Anal 27(2), 303–309 (1992)Google Scholar
  22. Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6), 1343–1376 (2000)Google Scholar
  23. Gagliardini, P., Gourieroux, C., Renault, E.: Efficient derivative pricing by the extended method of moments. Econometrica 79(4), 1181–1232 (2011)Google Scholar
  24. Galvani, V., Troitsky, V.G.: Options and efficiency in spaces of bounded claims. J Math Econ 46(4), 616–619 (2010)Google Scholar
  25. Hansen, L., Jagannathan, R.: Implications of security market data for models of dynamic economies. J Polit Econ 99, 225–262 (1991)Google Scholar
  26. Harvey, C.R., Siddique, A.: Conditional skewness in asset pricing tests. J Finance 55(3), 1263–1295 (2000)Google Scholar
  27. Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Netw 4, 251–257 (1991)Google Scholar
  28. Hornik, K., Stinchcombe, M., White, H.: Universal approximations of an unknown mapping and its derivatives using multilayer feedforward networks. Neural Netw 3, 551–560 (1990)Google Scholar
  29. Hutchinson, J., Lo, A., Poggio, T.: A nonparametric approach to the pricing and hedging of derivative securities via learning networks. J Finance 49, 851–889 (1994)Google Scholar
  30. Jarrow, R.A., Jin, X., Madan, D.B.: The second fundamental theorem of asset pricing. Math Finance 9(3), 255–273 (1999)Google Scholar
  31. Kirkby, J.L., Deng, S.: Static hedging and pricing of exotic options with payoff frames. Math Finance (2018). (forthcoming)
  32. Nachman, D.C.: Efficient funds for meager asset spaces. J Econ Theory 43(2), 335–347 (1987)Google Scholar
  33. Nachman, D.C.: Spanning and completeness with options. Rev Financ Stud 1, 311–328 (1988)Google Scholar
  34. Ross, S.A.: Options and efficiency. Q J Econ 90, 75–89 (1976)Google Scholar
  35. Ross, S.A.: Neoclassical Finance. Princeton University Press, Princeton (2005)Google Scholar
  36. Ross, S.A.: The recovery theorem. J Finance 70(2), 615–648 (2015)Google Scholar
  37. Stutzer, M.: A simple nonparametric approach to derivative security valuation. J Finance 51(5), 1633–1652 (1996)Google Scholar
  38. Tian, W.: Spanning with index. J Math Econ 53, 111–118 (2014)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Belk College of BusinessUniversity of North Carolina at CharlotteCharlotteUSA

Personalised recommendations