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
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).
The book of Wakker (2010) provides a thorough treatment on prospect theory.
See Quiggin (1993), p 56.
In the same paper Prelec derives two other probability weighting functions: the conditionally-invariantexponential-power and the projection-invarianthyperbolic-logarithm function.
This is not the case for weighting function (12); when \(a \ne b\), both parameters controls for curvature and all parameters may influence elevation.
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).
Alternative dynamics could in principle be considered.
In other numerical trials we used alternative values of the parameters a, b and \(\lambda \); in particular \(20\,\%\) of the TK sentiment.
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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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DOI: https://doi.org/10.1007/s10287-018-0324-y