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Computational Management Science

, Volume 16, Issue 1–2, pp 249–274 | Cite as

European option pricing under cumulative prospect theory with constant relative sensitivity probability weighting functions

  • Martina NardonEmail author
  • Paolo Pianca
Original Paper
  • 64 Downloads

Abstract

In this contribution, we evaluate European financial options under continuous cumulative prospect theory. In prospect theory, risk attitude and loss aversion are shaped via a value function, while a probability weighting function models probabilistic risk perception. We focus on investors’ probability risk attitudes, as probability weighting may be one of the possible causes of the differences between empirically observed options prices and theoretical prices obtained with the Black and Scholes formula. We consider alternative probability weighting functions; in particular, we adopt the constant relative sensitivity weighting function, whose parameters have a direct interpretation in terms of curvature and elevation. Curvature models optimism and pessimism when one moves from extreme probabilities, whereas elevation can be interpreted as a measure of relative optimism. We performed a variety of numerical experiments and studied the effects of these features on options prices and implied volatilities.

Keywords

European option pricing Cumulative prospect theory Probability weighting function Curvature Elevation 

Mathematics Subject Classification

91G20 91G60 65C20 

References

  1. Allais M (1953) Le comportement de l’homme rationel devant le risque: critique des axioms et postulates de l’école Americaine. Econometrica 21(4):503–546Google Scholar
  2. Allais M (1988) The general theory of random choices in relation to the invariant cardinal utility function and the specific probability function. The \((U, \theta )-\) model: a general overview. In: Munier BR (ed) Risk, decision and rationality. D. Reidel Publishing Company, Dordrecht, pp 231–289Google Scholar
  3. Abbink K, Rockenbach B (2006) Option pricing by students and professional traders: a behavioral investigation. Manag Decis Econ 27(6):497–510Google Scholar
  4. Abdellaoui M, L’Haridon O, Zank H (2010) Separating curvature and elevation: a parametric probability weighting function. J Risk Uncertain 41:39–65Google Scholar
  5. Barberis N, Thaler RH (2003) A survey of behavioral finance. In: Constantinides GM, Harris M, Stulz R (eds) Handbook of the economics of finance. Elsevier Science, Amsterdam, pp 1051–1121Google Scholar
  6. Bell DE (1985) Disappointment in decision making under uncertainty. Oper Res 33:1–27Google Scholar
  7. Birnbaum MH, McIntosh WR (1996) Violations of branch independence in choices between gambles. Organ Behav Hum Decis Process 67:91–110Google Scholar
  8. Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81(3):637–654Google Scholar
  9. Breuer W, Perst A (2007) Retail banking and behavioral financial engineering: the case of structured products. J Bank Financ 31(3):827–844Google Scholar
  10. Currim IS, Sarin RK (1989) Prospect versus utility. Manag Sci 35(1):22–41Google Scholar
  11. Davies GB, Satchell SE (2007) The behavioural components of risk aversion. J Math Psychol 51(1):1–13Google Scholar
  12. Diecidue E, Schmidt U, Zank H (2009) Parametric weighting functions. J Econ Theory 144(3):1102–1118Google Scholar
  13. Goldstein WM, Einhorn HJ (1987) Expression theory and the preference reversal phenomena. Psychol Rev 94(2):236–254Google Scholar
  14. Gonzalez R, Wu G (1999) On the shape of the probability weighting function. Cogn Psychol 38:129–166Google Scholar
  15. Han B (2008) Investor sentiment and option prices. Rev Financ Stud 21(1):387–414Google Scholar
  16. Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6(2):327–343Google Scholar
  17. Hey JD, Orme C (1994) Investigating generalizations of expected utility theory using experimental data. Econom 62(6):1291–1326Google Scholar
  18. Hoffmann AOI, Fischer ET (2012) Behavioral aspects of covered call writing: an empirical investigation. J Behav Financ 13:66–79Google Scholar
  19. Kahneman D, Tversky A (1979) Prospect theory: an analysis of decision under risk. Econometrica 47(2):263–292Google Scholar
  20. Karmarkar US (1978) Subjectively weighted utility: a descriptive extension of the expected utility model. Organ Behav Hum Perform 21:61–72Google Scholar
  21. Karmarkar US (1979) Subjectively weighted utility and the Allais Paradox. Organ Behav Hum Perform 24:67–72Google Scholar
  22. Kilka M, Weber M (2001) What determines the shape of the probability weighting function under uncertainty? Manag Sci 47(12):1712–1726Google Scholar
  23. Kothiyal A, Spinu V, Wakker PP (2011) Prospect theory for continuous distributions: a preference foundation. J Risk Uncertain 42:195–210Google Scholar
  24. Lattimore PK, Baker JR, Witte AD (1992) The influence of probability on risky choice: a parametric examination. J Econ Behav Organ 17(3):377–400Google Scholar
  25. Loomes G, Moffatt PG, Sugden R (2002) A microeconometric test of alternative stochastic theories of risk choise. J Risk Uncertain 24:103–130Google Scholar
  26. Luce DR (2000) Utility of gains and losses: measurement-theoretical and experimental approaches. Lawrence Erlbaum Publishers, LondonGoogle Scholar
  27. Luce DR (2001) Reduction invariance and Prelec’s weighting functions. J Math Psychol 45:167–179Google Scholar
  28. Luce DR, Mellers BA, Chang SJ (1993) Is choice the correct primitive? On using certainty equivalents and reference levels to predict choices among gambles. J Risk Uncertain 6:115–143Google Scholar
  29. Merton RC (1973) Theory of rational option pricing. Bell J Econ Manag 4:141–183Google Scholar
  30. Nardon M, Pianca P (2014) A behavioural approach to the pricing of European options. In: Corazza M, Pizzi C (eds) Mathematical and statistical methods for actuarial sciences and finance. Springer, Berlin, pp 217–228Google Scholar
  31. Pfiffelmann M (2011) Solving the St. Petersburg paradox in cumulative prospect theory: the right amount of probability weighting. Theory Dec 75:325–341Google Scholar
  32. Polkovnichenko V, Zhao F (2013) Probability weighting functions implied in options prices. J Financ Econ 107:580–609Google Scholar
  33. Poteshman AM, Serbin V (2003) Clearly irrational financial market behavior: evidence from the early exercise of exchange traded stock options. J Financ 58(1):37–70Google Scholar
  34. Prelec D (1998) The probability weighting function. Econom 66(3):497–527Google Scholar
  35. Quiggin J (1982) A theory of anticipated utility. J Econ Behav Organ 3:323–343Google Scholar
  36. Quiggin J (1993) Generalized expected utility theory: the rank-dependent model. Springer, NetherlandsGoogle Scholar
  37. Rieger MO, Wang M (2006) Cumulative prospect theory and the St Petrsburg paradox. J Econ Theory 28:665–679Google Scholar
  38. Rieger MO, Wang M (2008) Prospect theory for continuous distributions. J Risk Uncertain 36:83–102Google Scholar
  39. Röell A (1987) Risk aversion in Quiggin and Yaari’s rank-order model of choice under uncertainty. Econ J 97:143–159Google Scholar
  40. Safra Z, Segal U (1998) Constant risk aversion. J Econ Theory 83:19–42Google Scholar
  41. Schmeidler D (1989) Subjective probability and expected utility without additivity. Econometrica 57:571–587Google Scholar
  42. Schoutens W (2003) Lévy processes in finance: pricing financial derivatives. Wiley, ChichesterGoogle Scholar
  43. Shefrin H, Statman M (1993) Behavioral aspects of the design and marketing of financial products. Financ Manag 22(2):123–134Google Scholar
  44. Shiller RJ (1999) Human behavior and the efficiency of the financial system. In: Taylor JB, Woodford M (eds) Handbook of macroeconomics. Elsevier, Amsterdam, pp 1305–1340Google Scholar
  45. Subrahmanyam A (2007) Behavioural finance: a review and synthesis. Eur Financ Manag 14(1):12–29Google Scholar
  46. Thaler RH (1985) Mental accounting and consumer choice. Mark Sci 4:199–214Google Scholar
  47. Thaler RH (1999) Mental accounting matters. J Behav Decis Mak 12:183–206Google Scholar
  48. Tversky A, Fox CR (1995) Weighting risk and uncertainty. Psychol Rev 102:269–283Google Scholar
  49. Tversky A, Kahneman D (1992) Advances in prospect theory: cumulative representation of the uncertainty. J Risk Uncertain 5:297–323Google Scholar
  50. Versluis C, Lehnert T, Wolff CCP (2010) A cumulative prospect theory approach to option pricing. Working paper, LSF Research Working Paper Series 09–03, Luxembourg School of FinanceGoogle Scholar
  51. Wakker PP (2010) Prospect theory: for risk and ambiguity. Cambridge University Press, CambridgeGoogle Scholar
  52. Walther H (2003) Normal randomness expected utility, time preferences and emotional distortions. J Econ Behav Organ 52:253–266Google Scholar
  53. Wu G, Gonzalez R (1996) Curvature of the probability weighting function. Manag Sci 42(12):1676–1690Google Scholar
  54. Wu G, Gonzalez R (1999) Nonlinear decision weights in choice under uncertainty. Manag Sci 45(1):74–85Google Scholar
  55. Yaari M (1987) The dual theory of choice under risk. Econometrica 55(1):95–115Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of EconomicsCa’ Foscari University of VeniceVeneziaItaly

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