[Paper Review] Structure Preserving Equivalent Martingale Measures for SCII Models

David Criens|arXiv (Cornell University)|Jun 8, 2016

Stochastic processes and financial applications51 references3 citations

TL;DR

This paper establishes a one-to-one correspondence between structure-preserving equivalent martingale measures (M^sp) and a set of measurable, integrable functions (Y) in semimartingale models with conditionally independent increments. It proves M^sp is non-empty if and only if Y is non-empty, and links both sets to the semimartingale characteristics of the driving process, providing explicit conditions for existence in Lévy and time-changed Lévy models.

ABSTRACT

In this article we relate the set of structure preserving equivalent martingale measures (\mathcal{M}^ extup{sp}) for financial models driven by semimartingales with conditionally independent increments to a set of measurable and integrable functions (\mathcal{Y}). More precisely, we prove that (\mathcal{M}^ extup{sp} ot = \emptyset) if and only if (\mathcal{Y} ot = \emptyset), and connect the sets (\mathcal{M}^ extup{sp}) and (\mathcal{Y}) to the semimartingale characteristics of the driving process. In addition we consider integrated L\'evy models with independent stochastic factors and time-changed L\'evy models and derive mild explicit conditions for (\mathcal{M}^ extup{sp} ot = \emptyset).

Motivation & Objective

• To characterize the set of structure-preserving equivalent martingale measures (M^sp) in financial models driven by semimartingales with conditionally independent increments.
• To establish a precise correspondence between M^sp and a set of measurable and integrable functions (Y).
• To connect the existence of M^sp to the semimartingale characteristics of the underlying process.
• To derive mild, explicit conditions ensuring M^sp is non-empty in integrated Lévy and time-changed Lévy models.

Proposed method

• The authors use the theory of semimartingale characteristics to relate the structure-preserving martingale measures to a set of measurable and integrable functions Y.
• They define M^sp as the set of equivalent martingale measures that preserve the semimartingale structure under change of measure.
• The key technical step involves expressing the Radon-Nikodým derivative of such measures in terms of the predictable characteristics of the semimartingale.
• The existence of M^sp is reduced to the existence of a non-empty set Y of integrable functions satisfying specific integrability and measurability conditions.
• The analysis is applied to integrated Lévy models and time-changed Lévy models, where the characteristics are explicitly known.
• Explicit conditions for non-emptiness of M^sp are derived using the structure of the predictable characteristics in these specific models.

Experimental results

Research questions

• RQ1 Under what conditions does a structure-preserving equivalent martingale measure exist in semimartingale models with conditionally independent increments?
• RQ2 How is the set of structure-preserving equivalent martingale measures related to the set of measurable and integrable functions Y?
• RQ3 What role do the semimartingale characteristics play in determining the existence of M^sp?
• RQ4 What explicit, mild conditions ensure M^sp is non-empty in integrated Lévy and time-changed Lévy models?

Key findings

• M^sp is non-empty if and only if the set Y of measurable and integrable functions is non-empty.
• The existence of structure-preserving equivalent martingale measures is fully characterized by the existence of solutions in the set Y.
• The semimartingale characteristics of the driving process directly determine the structure of both M^sp and Y.
• For integrated Lévy models, the condition for M^sp ≠ ∅ reduces to a condition on the characteristic exponent and integrability of the Lévy measure.
• In time-changed Lévy models, explicit mild conditions on the time change and the Lévy process ensure M^sp ≠ ∅.
• The framework provides a systematic way to verify the existence of M^sp without requiring full specification of the measure change.