Abstract
In option pricing one typically starts from a “realistic” underlying (martingale) stochastic processes and hopes it produces easy-to-use equations for describing market values. But it is also possible to proceed the other way around: start from option values which admit no-arbitrage (“increasing in convex order” functionals) and then devise a process which admits marginals fitting those equations. A prime example is Dupire [3] equation, but more sophisticated techniques, not necessarily involving continuous processes, are by now known (as expounded by e.g. Madan and Yor [4]). However, the processes obtained are typically very general and do not make use of the properties of any specific distributions implicit in option prices. We have recently discovered in [1] that certain “natural” (i.e. arising from functions popular in applied sciences) expressions for vanilla option values yield to distributions of logistic type for the underlying (or its logarithm), which are known to be infinitely-divisible. When an appropriate term function (i.e. not allowing calendar arbitrage) is supplied to the valuation equation the corresponding family of time-dependent infinitelydivisible distributions determines an additive process for the underlying security price, which turns out to be a martingale.
Therefore parsimonious and simple martingale process exist supporting elementary option valuation equation, capturing returns skewness, kurtosis, self-similarity and other stylized facts, whose additive structure also allows for path-dependent derivative valuation along the lines of well-established methodologies. A first probabilistic question is then how general is the attainability of additive processes starting from no-arbitrage call functions.
In further developments, in an effort to increase the amount of implied volatility skew and convexity picked up by the models, we modified the underlying distributions of the additive processes involved with an additional parameter, without perturbing the martingale property.
Vanilla option formulae can be seen to overlap with those studied by [5], but with the added dimension entailed by a consistent non arbitrage term structure leading to a proper stochastic process and martingale dynamics. A further extension is represented by the randomization of a dispersion parameter (the “bewilderment”) which coincides with the time dependent scale of the Dagum distribution, much in the spirit of what is done for stochastic volatility models and Lévy subordinated models. The obtained processes are no longer additive and a related probabilistic question would be how to characterize them in terms of their generator.
Empirical and numerical tests of these two latter modelling approaches are ongoing.
References
P. Carr and L. Torricelli (2021). Additive logistic processes in option pricing. To appear in Finance and Stochastics.
P. Carr and L. Torricelli (2021). Beta option prices. Ongoing. B. Dupire (1994). Pricing with a smile. Risk, 7, 18–20.
D. B. Madan and M. Yor (2002). Making Markov marginals meet martingales: with explicit constructions. Bernoulli, 8, 509–536.
J. B. McDonald and R. M. Bookstaber (1991). Option pricing for generalized distributions.
Communications in Statistics – Theory and Methods, 20, 4053–4068
Organizzatore
Sabrina Mulinacci