Time-Changed Brownian Motions and Copulas: Estimation, Inference, and Applications

Zinn, Toke Christian

4. term

Mathematics-Economics, Master

2020

111

This thesis studies how to model temporal dependence in continuous‑time finance by combining time‑changed Brownian motions and copulas. We propose an estimator for the conditional copula in models representable as time‑changed Brownian motions, based on realized variance and nonparametric estimates of quadratic variation. Under suitable restrictions we show that the estimator is consistent and we derive a limit theorem in a highly restricted case. Numerical analysis suggests that the results extend to a broader class of models and that the asymptotic behavior is partly preserved. We then develop a semiparametric procedure to forecast a portfolio’s distribution by incorporating temporal copulas, and demonstrate its validity in a controlled simulation using data from the Heston model, evaluating performance with a conditional coverage test. Although the simulation design is idealized, we outline extensions that could increase flexibility and practical usefulness.

Denne afhandling undersøger, hvordan afhængighed over tid kan modelleres i kontinuertids finans ved at kombinere tidsændrede Brownske bevægelser og copulaer. Vi foreslår en estimator for den betingede copula i modeller, der kan repræsenteres som tidsændrede Brownske bevægelser, baseret på realiseret varians og ikke‑parametriske estimater af kvadratisk variation. Under passende antagelser viser vi, at estimatoren er konsistent, og vi etablerer en grænsesætning i et meget begrænset tilfælde. Numeriske analyser antyder, at resultaterne kan generaliseres til en bredere klasse af modeller, og at den asymptotiske adfærd delvist bevares. Vi udvikler dernæst en semiparametrisk metode til at fremskrive fordelingen af en portefølje ved at inkorporere tidslige copulaer og demonstrerer dens gyldighed i en kontrolleret simulationsundersøgelse med data fra Heston‑modellen, hvor validiteten vurderes med en betinget dæknings‑test. Selvom opsætningen er idealiseret, skitserer vi udvidelser, der kan øge fleksibiliteten og den praktiske anvendelighed.

[This apstract has been generated with the help of AI directly from the project full text]

Copula ; Brownian motion ; realised variance

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