Option-Implied Objective Measures of Market Risk with Leverage

  • Matthias Leiss
  • Heinrich H. NaxEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 214)


Leverage has been shown to be procyclical and indicative of financial market risk. Here, we present a novel, inherently forward-looking way to estimate market leverage ratios based on derivative prices, option hedging, and the ‘operational’ riskiness measure by Foster and Hart (J Polit Econ 117(5):785–814, 2009). Furthermore, we report option-implied ‘optimal’ leverage levels inferred via the (Kelly, IRE Trans. Inf. Theory 2(3):185–189, 1956) criterion. The resulting measure of leverage exhibits strong procyclicality prior to the Global Financial Crisis of 2008. Finally, we find it to successfully predict large stock market downturns.


Objective risk Foster-Hart Leverage Risk-neutral densities 



Leiss acknowledges support from the ETH Risk Center and through SNF grant The Anatomy of Systemic Financial Risk, Nax from the European Commission through the ERC Advanced Investigator Grant Momentum (Grant No. 324247).


  1. Adrian, T., Shin, H.S.: Liquidity and leverage. J. Financ. Intermed. 19(3), 418–437 (2010)CrossRefGoogle Scholar
  2. Adrian, T., Shin, H.S.: Procyclical leverage and value-at-risk. Rev. Financ. Stud. 27(2), 373–403 (2014)CrossRefGoogle Scholar
  3. Anand, A., Li, T., Kurosaki, T., Kim, Y.S.: Foster–Hart optimal portfolios. J. Bank. Financ. 68, 117–130 (2016)CrossRefGoogle Scholar
  4. Aumann, R.J., Serrano, R.: An economic index of riskiness. J. Polit. Econ. 116(5), 810–836 (2008)CrossRefzbMATHGoogle Scholar
  5. Basel Committee on Banking Supervision: Basel III: a global regulatory framework for more resilient banks and banking systems. Technical report, Bank for International Settlements (2010)Google Scholar
  6. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  7. Breeden, D.T., Litzenberger, R.H.: Prices of state-contingent claims implicit in option prices. J. Bus. 51(4), 621–651 (1978)CrossRefGoogle Scholar
  8. Chicago Board Options Exchange: The CBOE volatility index – VIX. Technical report, White Paper (2009)Google Scholar
  9. Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300(1), 463–520 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  10. Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events: For Insurance and Finance, vol. 33. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  11. Embrechts, P., Frey, R., McNeil, A.: Quantitative Risk Management, vol. 10. Princeton Series in Finance, Princeton (2005)zbMATHGoogle Scholar
  12. Figlewski, S.: Estimating the implied risk neutral density. In: Bollerslev, T., Russell, J., Watson, M. (eds.) Volatility and Time Series Econometrics. Oxford University Press, Oxford (2010)Google Scholar
  13. Foster, D.P., Hart, S.: An operational measure of riskiness. J. Polit. Econ. 117(5), 785–814 (2009)CrossRefGoogle Scholar
  14. Geanakoplos, J.: The leverage cycle. In: NBER Macroeconomics Annual 2009, vol. 24, pp. 1–65. University of Chicago Press, Chicago (2010)Google Scholar
  15. Gorton, G., Metrick, A.: Securitized banking and the run on repo. J. Financ. Econ. 104(3), 425–451 (2012)CrossRefGoogle Scholar
  16. Grossman, S.J., Vila, J.-L.: Optimal dynamic trading with leverage constraints. J. Financ. Quant. Anal. 27(02), 151–168 (1992)CrossRefGoogle Scholar
  17. Hadar, J., Russell, W.R.: Rules for ordering uncertain prospects. Am. Econ. Rev. 59(1), 25–34 (1969)Google Scholar
  18. Hanoch, G., Levy, H.: The efficiency analysis of choices involving risk. Rev. Econ. Stud. 36(3), 335–346 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  19. Hildebrand, P.M.: Is Basel II Enough? The Benefits of a Leverage Ratio. Philipp M. Hildebrand, Vice-Chairman of the Governing Board Swiss National Bank, in a Financial Markets Group Lecture at the London School of Economics on December 15, 2008. (2008)
  20. Jackwerth, J.C.: Option-Implied Risk-Neutral Distributions and Risk Aversion. Research Foundation of AIMR Charlotteville (2004)Google Scholar
  21. Kadan, O., Liu, F.: Performance evaluation with high moments and disaster risk. J. Financ. Econ. 113(1), 131–155 (2014)CrossRefGoogle Scholar
  22. Kelly, J.L.: A new interpretation of information rate. IRE Trans. Inf. Theory 2(3), 185–189 (1956)CrossRefGoogle Scholar
  23. Leiss, M., Nax, H.H.: Option-implied objective measures of market risk. Social Science Research Network Working Paper Series, 2690476, Quantitative Economics (2015, submitted)Google Scholar
  24. Leiss, M., Nax, H.H., Sornette, D.: Super-exponential growth expectations and the global financial crisis. J. Econ. Dyn. Control 55, 1–13 (2015)CrossRefMathSciNetGoogle Scholar
  25. Newey, W.K., West, K.D.: A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55(3), 703–708 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  26. Newey, W.K., West, K.D.: Automatic lag selection in covariance matrix estimation. Rev. Econ. Stud. 61(4), 631–653 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  27. Riedel, F., Hellmann, T.: The Foster-Hart measure of riskiness for general gambles. Theor. Econ. 10(1), 1–9 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  28. Rothschild, M., Stiglitz, J.E.: Increasing risk: I. A definition. J. Econ. Theory 2(3), 225–243 (1970)CrossRefMathSciNetGoogle Scholar
  29. Schularick, M., Taylor, A.M.: Credit booms gone bust: monetary policy, leverage cycles, and financial crises, 1870–2008. Am. Econ. Rev. 102(2), 1029–1061 (2012)CrossRefGoogle Scholar
  30. Sharpe, W.F.: The sharpe ratio. J. Portf. Manag. 21(1), 49–58 (1994)CrossRefGoogle Scholar
  31. Shimko, D.C., Tejima, N., Van Deventer, D.R.: The pricing of risky debt when interest rates are stochastic. J. Fixed Income 3(2), 58–65 (1993)CrossRefGoogle Scholar
  32. Sircar, R.K., Papanicolaou, G.: General Black-Scholes models accounting for increased market volatility from hedging strategies. Appl. Math. Finance 5(1), 45–82 (1998)CrossRefzbMATHGoogle Scholar
  33. Thurner, S., Farmer, J.D., Geanakoplos, J.: Leverage causes fat tails and clustered volatility. Quant. Finan. 12(5), 695–707 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  34. Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B Methodol. 58(1), 267–288 (1996)zbMATHMathSciNetGoogle Scholar
  35. Von der Becke, S., Sornette, D.: Toward a unified framework of credit creation. Technical Report 14-07, Swiss Finance Institute Research Paper (2014)Google Scholar
  36. Whitworth, W.: Choice and Chance. Deighton, Bell and Co, Cambridge (1870)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of HumanitiesSocial and Political Sciences, ETH ZurichZurichSwitzerland

Personalised recommendations