Trajectorial asset models with operational assumptions

Ferrando, Sebastian; Fleck, Andrew; Gonzalez, Alfredo L.; Rubtsov, Alexey

doi:10.3934/qfe.2019.4.661

Cited by 2 publications

(3 citation statements)

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“…In practice, the coordinates S i are multidimensional in order to allow for multiple sources of uncertainty ( [7]), for simplicity we restrict to S i ∈ R. One can also extend the theory to allow for several traded assets S k i . Notice S (S,0) = S and, if S ∈ S (S, j) , then S ( S, j) = S (S, j) .…”

Section: Trajectorial Settingmentioning

confidence: 99%

“…Item (7) of Proposition 3 shows that property (L j ) is crucial to have an appropriate conditional integral defined by σ j . In order to obtain conditions for its validity, some new concepts need to be introduced.…”

Section: Conditions For Validity Of Property (L J )mentioning

confidence: 99%

“…, from where, (6) holds. (7) Taking f m ≡ 0 for any m ≥ 0, it follows that σ j 0 ≤ 0. In case that σ j 0 = 0, it then follows that there exist at least some S ∈ S satisfying σ j 0(S) < 0.…”

Section: Proof Of Propositionmentioning

confidence: 99%

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Conditional Non-Lattice Integration, Pricing and Superhedging

Bender,

Ferrando,

Gonzalez

2021

Preprint

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Closely motivated by financial considerations, we develop an integration theory which is not classical i.e. it is not necessarily associated to a measure. The base space, denoted by S and called a trajectory space, substitutes the set Ω in probability theory and provides a fundamental structure via conditional subsets S (S, j) that allows the definition of conditional integrals. The setting is a natural by-product of no arbitrage assumptions that are used to model financial markets and games of chance (in a discrete infinite time framework). The constructed conditional integrals can be interpreted as required investments, at the conditioning node, for hedging an integrable function, the latter characterized a.e. and in the limit as we increase the number of portfolios used. The integral is not classical due to the fact that the original vector space of portfolio payoffs is not a vector lattice. In contrast to a classical stochastic setting, where price processes are associated to conditional expectations (with respect to risk neutral measures), we uncover a theory where prices are naturally given by conditional non-lattice integrals. One could then study analogues of classical probabilistic notions in such non-classical setting, the paper stops after defining trajectorial martingales the study of which is deferred to future work.

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Section: Trajectorial Settingmentioning

confidence: 99%

Section: Conditions For Validity Of Property (L J )mentioning

confidence: 99%