Lorenzo Torricelli, Università di Bologna
Date: 12 NOVEMBER 2021 from 15:00 to 17:00
Event location: Online event
Type: Statistics Seminars
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.
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.
L’Organizzatore Il Direttore
Prof. Sabrina Mulinacci Prof. Carlo Trivisano