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

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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.

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Notes

  1. One can introduce jumps, stochastic volatility (Heston 1993), or consider Lévy processes (see e.g. Schoutens 2003).

  2. Inspired by Allais’ paradox, the crucial idea underpinning rank dependent utility (RDEU) and cumulative prospect theory (CPT) models is that small probabilities (of a large gain or a large loss) do matter. According to Quiggin (1993) “the overweight of small probabilities should be applied only to low probability extreme outcomes, and not to low probability intermediate outcomes” (we are grateful to an anonymous referee for pointing out this).

  3. The book of Wakker (2010) provides a thorough treatment on prospect theory.

  4. See Quiggin (1993), p 56.

  5. See also Rieger and Wang (2008), Wakker (2010), and Kothiyal et al. (2011).

  6. In the same paper Prelec derives two other probability weighting functions: the conditionally-invariantexponential-power and the projection-invarianthyperbolic-logarithm function.

  7. This is not the case for weighting function (12); when \(a \ne b\), both parameters controls for curvature and all parameters may influence elevation.

  8. This approach could be applied also for the evaluation of over-the-counter structured financial products and mortgages, when some option is embedded in the contract, and for the evaluation of options implicit in investment projects (real options).

  9. Alternative dynamics could in principle be considered.

  10. In other numerical trials we used alternative values of the parameters a, b and \(\lambda \); in particular \(20\,\%\) of the TK sentiment.

References

  • Allais M (1953) Le comportement de l’homme rationel devant le risque: critique des axioms et postulates de l’école Americaine. Econometrica 21(4):503–546

    Article  Google Scholar 

  • 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 231–289

    Chapter  Google Scholar 

  • Abbink K, Rockenbach B (2006) Option pricing by students and professional traders: a behavioral investigation. Manag Decis Econ 27(6):497–510

    Article  Google Scholar 

  • Abdellaoui M, L’Haridon O, Zank H (2010) Separating curvature and elevation: a parametric probability weighting function. J Risk Uncertain 41:39–65

    Article  Google Scholar 

  • 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 1051–1121

    Google Scholar 

  • Bell DE (1985) Disappointment in decision making under uncertainty. Oper Res 33:1–27

    Article  Google Scholar 

  • Birnbaum MH, McIntosh WR (1996) Violations of branch independence in choices between gambles. Organ Behav Hum Decis Process 67:91–110

    Article  Google Scholar 

  • Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81(3):637–654

    Article  Google Scholar 

  • Breuer W, Perst A (2007) Retail banking and behavioral financial engineering: the case of structured products. J Bank Financ 31(3):827–844

    Article  Google Scholar 

  • Currim IS, Sarin RK (1989) Prospect versus utility. Manag Sci 35(1):22–41

    Article  Google Scholar 

  • Davies GB, Satchell SE (2007) The behavioural components of risk aversion. J Math Psychol 51(1):1–13

    Article  Google Scholar 

  • Diecidue E, Schmidt U, Zank H (2009) Parametric weighting functions. J Econ Theory 144(3):1102–1118

    Article  Google Scholar 

  • Goldstein WM, Einhorn HJ (1987) Expression theory and the preference reversal phenomena. Psychol Rev 94(2):236–254

    Article  Google Scholar 

  • Gonzalez R, Wu G (1999) On the shape of the probability weighting function. Cogn Psychol 38:129–166

    Article  Google Scholar 

  • Han B (2008) Investor sentiment and option prices. Rev Financ Stud 21(1):387–414

    Article  Google Scholar 

  • Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6(2):327–343

    Article  Google Scholar 

  • Hey JD, Orme C (1994) Investigating generalizations of expected utility theory using experimental data. Econom 62(6):1291–1326

    Article  Google Scholar 

  • Hoffmann AOI, Fischer ET (2012) Behavioral aspects of covered call writing: an empirical investigation. J Behav Financ 13:66–79

    Article  Google Scholar 

  • Kahneman D, Tversky A (1979) Prospect theory: an analysis of decision under risk. Econometrica 47(2):263–292

    Article  Google Scholar 

  • Karmarkar US (1978) Subjectively weighted utility: a descriptive extension of the expected utility model. Organ Behav Hum Perform 21:61–72

    Article  Google Scholar 

  • Karmarkar US (1979) Subjectively weighted utility and the Allais Paradox. Organ Behav Hum Perform 24:67–72

    Article  Google Scholar 

  • Kilka M, Weber M (2001) What determines the shape of the probability weighting function under uncertainty? Manag Sci 47(12):1712–1726

    Article  Google Scholar 

  • Kothiyal A, Spinu V, Wakker PP (2011) Prospect theory for continuous distributions: a preference foundation. J Risk Uncertain 42:195–210

    Article  Google Scholar 

  • Lattimore PK, Baker JR, Witte AD (1992) The influence of probability on risky choice: a parametric examination. J Econ Behav Organ 17(3):377–400

    Article  Google Scholar 

  • Loomes G, Moffatt PG, Sugden R (2002) A microeconometric test of alternative stochastic theories of risk choise. J Risk Uncertain 24:103–130

    Article  Google Scholar 

  • Luce DR (2000) Utility of gains and losses: measurement-theoretical and experimental approaches. Lawrence Erlbaum Publishers, London

    Google Scholar 

  • Luce DR (2001) Reduction invariance and Prelec’s weighting functions. J Math Psychol 45:167–179

    Article  Google Scholar 

  • 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 6:115–143

    Article  Google Scholar 

  • Merton RC (1973) Theory of rational option pricing. Bell J Econ Manag 4:141–183

    Article  Google Scholar 

  • 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 217–228

    Google Scholar 

  • Pfiffelmann M (2011) Solving the St. Petersburg paradox in cumulative prospect theory: the right amount of probability weighting. Theory Dec 75:325–341

    Article  Google Scholar 

  • Polkovnichenko V, Zhao F (2013) Probability weighting functions implied in options prices. J Financ Econ 107:580–609

    Article  Google Scholar 

  • Poteshman AM, Serbin V (2003) Clearly irrational financial market behavior: evidence from the early exercise of exchange traded stock options. J Financ 58(1):37–70

    Article  Google Scholar 

  • Prelec D (1998) The probability weighting function. Econom 66(3):497–527

    Article  Google Scholar 

  • Quiggin J (1982) A theory of anticipated utility. J Econ Behav Organ 3:323–343

    Article  Google Scholar 

  • Quiggin J (1993) Generalized expected utility theory: the rank-dependent model. Springer, Netherlands

    Book  Google Scholar 

  • Rieger MO, Wang M (2006) Cumulative prospect theory and the St Petrsburg paradox. J Econ Theory 28:665–679

    Article  Google Scholar 

  • Rieger MO, Wang M (2008) Prospect theory for continuous distributions. J Risk Uncertain 36:83–102

    Article  Google Scholar 

  • Röell A (1987) Risk aversion in Quiggin and Yaari’s rank-order model of choice under uncertainty. Econ J 97:143–159

    Article  Google Scholar 

  • Safra Z, Segal U (1998) Constant risk aversion. J Econ Theory 83:19–42

    Article  Google Scholar 

  • Schmeidler D (1989) Subjective probability and expected utility without additivity. Econometrica 57:571–587

    Article  Google Scholar 

  • Schoutens W (2003) Lévy processes in finance: pricing financial derivatives. Wiley, Chichester

    Book  Google Scholar 

  • Shefrin H, Statman M (1993) Behavioral aspects of the design and marketing of financial products. Financ Manag 22(2):123–134

    Article  Google Scholar 

  • Shiller RJ (1999) Human behavior and the efficiency of the financial system. In: Taylor JB, Woodford M (eds) Handbook of macroeconomics. Elsevier, Amsterdam, pp 1305–1340

    Chapter  Google Scholar 

  • Subrahmanyam A (2007) Behavioural finance: a review and synthesis. Eur Financ Manag 14(1):12–29

    Google Scholar 

  • Thaler RH (1985) Mental accounting and consumer choice. Mark Sci 4:199–214

    Article  Google Scholar 

  • Thaler RH (1999) Mental accounting matters. J Behav Decis Mak 12:183–206

    Article  Google Scholar 

  • Tversky A, Fox CR (1995) Weighting risk and uncertainty. Psychol Rev 102:269–283

    Article  Google Scholar 

  • Tversky A, Kahneman D (1992) Advances in prospect theory: cumulative representation of the uncertainty. J Risk Uncertain 5:297–323

    Article  Google Scholar 

  • 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 Finance

  • Wakker PP (2010) Prospect theory: for risk and ambiguity. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Walther H (2003) Normal randomness expected utility, time preferences and emotional distortions. J Econ Behav Organ 52:253–266

    Article  Google Scholar 

  • Wu G, Gonzalez R (1996) Curvature of the probability weighting function. Manag Sci 42(12):1676–1690

    Article  Google Scholar 

  • Wu G, Gonzalez R (1999) Nonlinear decision weights in choice under uncertainty. Manag Sci 45(1):74–85

    Article  Google Scholar 

  • Yaari M (1987) The dual theory of choice under risk. Econometrica 55(1):95–115

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Economics, Ca’ Foscari University of Venice, San Giobbe - Cannaregio 873, 30121, Venezia, Italy

    Martina Nardon & Paolo Pianca

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  1. Martina Nardon
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  2. Paolo Pianca
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Correspondence to Martina Nardon.

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Nardon, M., Pianca, P. European option pricing under cumulative prospect theory with constant relative sensitivity probability weighting functions. Comput Manag Sci 16, 249–274 (2019). https://doi.org/10.1007/s10287-018-0324-y

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