Time Dependent Relative Risk Aversion

  • Enzo Giacomini
  • Michael Handel
  • Wolfgang K. Härdle
Part of the Contributions to Economics book series (CE)

Risk management has developed in the recent decades to be one of the most fundamental issues in quantitative finance. Various models are being developed and applied by researchers as well as financial institutions. By modeling price fluctuations of assets in a portfolio, the loss can be estimated using statistical methods. Different measures of risk, such as standard deviation of returns or confidence interval Value at Risk, have been suggested. These measures are based on the probability distributions of assets' returns extracted from the data-generating process of the asset.

However, an actual one dollar loss is not always valued in practice as a one dollar loss. Purely statistical estimation of loss has the disadvantage of ignoring the circumstances of the loss. Hence the notion of an investor's utility has been introduced. Arrow [2] and [10] were the first to introduce elementary securities to formalize economics of uncertainty. The so-called Arrow-Debreu securities are the starting point of all modern financial asset pricing theories. Arrow—Debreu securities entitle their holder to a payoff of 1$ in one specific state of the world, and 0 in all other states of the world. The price of such a security is determined by the market, on which it is tradable, and is subsequent to a supply and demand equilibrium. Moreover, these prices contain information about investors' preferences due to their dependence on the conditional probabilities of the state of the world at maturity and due to the imposition of market-clearing and general equilibrium conditions. The prices reflect investors' beliefs about the future, and the fact that they are priced differently in different states of the world implies, that a one-dollar gain is not always worth the same, in fact its value is exactly the price of the security.


Utility Function Risk Aversion Asset Price Option Price Implied Volatility 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ait Sahalia, Y. and Lo, A. W. [2000], ‘Nonparametric risk management and implied risk aversion’, Journal of Econometrics 94, 9–51CrossRefGoogle Scholar
  2. 2.
    Arrow, K. J. [1964], ‘The role of securities in the optimal allocation of risk bearing’, Review of Economic Studies 31, 91–96CrossRefGoogle Scholar
  3. 3.
    Arrow, K. J. [1965], ‘Aspects of the theory of risk-bearing’, Yr j ö Hahnsson Foundation, Helsinki Google Scholar
  4. 4.
    Black, F. and Scholes, M. [1973], ‘The pricing of options and corporate liabililties’, Journal of Political Economy 81, 637–654CrossRefGoogle Scholar
  5. 5.
    Breeden, D. and Litzenberger, R. [1978], ‘Prices of state-contingent claims implicit in option prices’, Journal of Business 51, 621–651CrossRefGoogle Scholar
  6. 6.
    Brown, D. P. and Jackwerth, J. C. [2004], ‘The pricing kernel puzzle: Reconciling index option data and economic theory’, Working Paper, University of Konstanz / University of Wisconsin Google Scholar
  7. 7.
    Campbell, J. and Cochrane, J. H. [1999], ‘A consumption-based explanation of aggregate stock market behavior’, Journal of Political Economy 107 Google Scholar
  8. 8.
    Cochrane, J. H. [2001], ‘Asset pricing’, Princeton University Press,Princeton Google Scholar
  9. 9.
    Constantinides, G. [1982], ‘Intertemporal asset pricing with heterogeneous consumers and without demand aggregation’, Journal of Business 55, 253–268CrossRefGoogle Scholar
  10. 10.
    Debreu, G. [1959], ‘The theory of value’, Wiley, New York Google Scholar
  11. 11.
    Derman, E. and Kani, I. [1994], ‘The volatility smile and its implied tree’, Quantitative strategies research notes, Goldman Sachs Google Scholar
  12. 12.
    Fengler, M. R. [2005], ‘Semiparametric modelling of implied volatility’, Springer, Berlin Google Scholar
  13. 13.
    Franke, J., Härdle, W. and Hafner, C. [2004], ‘Statistics of financial markets’, Springer, Heidelberg Google Scholar
  14. 14.
    Härdle, W. [1990], ‘Applied nonparametric regression’, Cambridge University Press, Cambridge Google Scholar
  15. 15.
    Härdle, W. and Hlávka, Z. [2005], ‘Dynamics of state price densities’, SFB 649 Discussion paper 2005-021, CASE, Humboldt University, Berlin Google Scholar
  16. 16.
    Härdle, W. and Zheng, J. [2002], ‘How precise are distributions predicted by implied binomial trees?’, in: W. Hardle, T. Kleinow and G. Stahl (eds.), Applied Quantitative Finance, Springer, Berlin, Ch. 7 Google Scholar
  17. 17.
    Huynh, K., Kervella, P. and Zheng, J. [2002], ‘Estimating state-price densities with nonparametric regression’, in: W. Hardle, T. Kleinow and G. Stahl (eds.), Applied Quantitative Finance, Springer, Berlin, Ch. 8 Google Scholar
  18. 18.
    Jackwerth, J. C. [2000], ‘Recovering risk aversion from option prices and realized returns’, Review of Financial Studies 13, 433–451CrossRefGoogle Scholar
  19. 19.
    Kwiatkowski, D., Phillips, P., Schmidt, P. and Shin, Y. [1992], ‘Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic series have a unit root’, Journal of Econometrics 54, 159–178CrossRefGoogle Scholar
  20. 20.
    Lucas, R. E. [1978], ‘Asset prices in an exchange economy’, Econometrica 46, 1429–1446CrossRefGoogle Scholar
  21. 21.
    Mas-Colell, A., Whinston, M. and Green, J. [1995], ‘Microeconomic theory’, Oxford University Press Google Scholar
  22. 22.
    McGrattan, E. and Prescott, E. [2003], ‘Taxes, regulations and the value of us and uk corporations’, Federal Reserve Bank of Minneapolis, Research Department Staff Report 309 Google Scholar
  23. 23.
    Mehra, R. and Prescott, E. [1985], ‘The equity premium — a puzzle’, Journal of Monetary Economics 15 Google Scholar
  24. 24.
    Merton, R. [1973], ‘Rational theory of option pricing’, Journal of Economics and Management Science 4, 141–183Google Scholar
  25. 25.
    Prais, S. J. and Winsten, C. B. [1954], ‘Trend estimators and serial correlation’, Cowles Commission Discussion Paper 383, Chicago 383 Google Scholar
  26. 26.
    Pratt, J. [1964], ‘Risk aversion in the small and in the large’, Econometrica 32 Google Scholar
  27. 27.
    Rookley, C. [1997], ‘Fully exploiting the information content of intra day option quotes: Applications in option pricing and risk management’, University of Arizona Google Scholar
  28. 28.
    Rosenberg, J. V. and Engle, R. F. [2002], ‘Empirical pricing kernels’, Journal of Financial Economics 64, 341–372CrossRefGoogle Scholar
  29. 29.
    Rubinstein, M. [1976], ‘The valuation of uncertain income streams and the pricing of options’, Bell Journal of Economics 7, 407–425CrossRefGoogle Scholar
  30. 30.
    Rubinstein, M. [1994], ‘Implied binomial trees’, Journal of Finance 49, 771–818CrossRefGoogle Scholar
  31. 31.
    Sen, A. and Srivastava, M. [1990], ‘Regression analysis: Theory, methods and applications’, Springer, New York Google Scholar
  32. 32.
    Veldkamp, L. [2005], ‘Media frenzies in markets for financial information’, American Economic Review, 96(3), 577–601CrossRefGoogle Scholar
  33. 33.
    Weil, P. [1989], ‘The equity premium puzzle and the risk-free rate puzzle’, Journal of Monetary Economics 24, 401–421CrossRefGoogle Scholar

Copyright information

© Physica-Verlag Heidelberg 2009

Authors and Affiliations

  1. 1.CASE — Center for Applied Statistics and EconomicsHumboldt-University of BerlinGermany
  2. 2.Dr. Nagler & Company GmbHMunichGermany

Personalised recommendations