Pricing without martingale measure

Julien Baptiste (1) , Laurence Carassus (2) , Emmanuel Lépinette (3)

1 PSL - Université Paris Sciences et Lettres
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

For several decades, the no-arbitrage (NA) condition and the martingale measures have played a major role in the financial asset's pricing theory. Here, we propose a new approach based on convex duality instead of martingale measures duality : our prices will be expressed using Fenchel conjugate and bi-conjugate. This is lead naturally to a weak condition of absence of arbitrage opportunity, called Absence of Immediate Profit (AIP), which asserts that the price of the zero claim should be zero. We study the link between (AIP), (NA) and the no-free lunch condition. We show in a one step model that, under (AIP), the super-hedging cost is just the payoff's concave envelop and that (AIP) is equivalent to the non-negativity of the super-hedging prices of some call option. In the multiple-period case, for a particular, but still general setup, we propose a recursive scheme for the computation of a the super-hedging cost of a convex option. We also give some numerical illustrations.

Mots clés: Financial market models; Super-hedging prices; No-arbitrage condition; Conditional support; Essential supremum

Domaines: Probabilités [math.PR]

Julien Baptiste, Laurence Carassus, Emmanuel Lépinette. Pricing without martingale measure. 2021. ⟨hal-01774150v3⟩