[Paper Review] Construction of multi-default models with full viability

Shiqi Song|arXiv (Cornell University)|Nov 1, 2015

Stochastic processes and financial applications28 references3 citations

TL;DR

This paper constructs multi-default models with full viability by introducing a novel class of random times—‘𝗁-models’—derived from the work of Jeanblanc and Song (2010). It ensures market viability through predictable dual projections and martingale representation in progressively enlarged filtrations, proving that such expansions preserve absence of arbitrage and allow for consistent pricing and hedging in defaultable markets.

ABSTRACT

We study multi-default model which satisfies the quasi-left-continuity, the martingale representation property, the drift multiplier assumption and the full viability. We use $ atural$-model to construct one such model.

Motivation & Objective

• To address the challenge of constructing financially viable multi-default models where successive default events expand information flow without destroying market viability.
• To overcome limitations of existing classes like honest times or initial times, which often fail to preserve viability under progressive enlargement.
• To develop a new class of random times—‘𝗁-models’—capable of supporting full viability in multi-default settings.
• To establish conditions under which the drift operator and martingale representation remain well-defined in enlarged filtrations.
• To ensure the existence of a local martingale deflator and the absence of arbitrage of the first kind in the expanded market.

Proposed method

• Uses ‘𝗁-models’—random times with given survival probabilities and specific ‘𝗁-martingale decomposition’ properties—introduced in Jeanblanc and Song (2010).
• Applies Jacod’s criterion and predictable dual projections to ensure the existence of a local martingale deflator in the enlarged filtration.
• Employs stochastic calculus for semimartingales, including stochastic integrals with respect to multi-dimensional local martingales and bracket processes.
• Imposes conditions on the drift operator and the predictable compensator of default times to preserve viability under progressive enlargement.
• Relies on the martingale representation property in enlarged filtrations to ensure completeness of hedging strategies.
• Uses the notion of ‘𝗁-martingale decomposition’ to characterize the dynamics of default times and their impact on asset prices.

Experimental results

Research questions

• RQ1 Can a class of random times be constructed such that successive enlargement of filtration preserves market viability in multi-default models?
• RQ2 How can the ‘𝗁-model’ framework be used to ensure the existence of a local martingale deflator in progressively enlarged filtrations?
• RQ3 What conditions on the default times and their compensators guarantee the absence of arbitrage of the first kind in the expanded market?
• RQ4 How does the drift operator evolve under progressive enlargement when using ‘𝗁-models’?
• RQ5 Can the martingale representation property be preserved in the enlarged filtration to allow for complete hedging in defaultable markets?

Key findings

• The ‘𝗁-models’ class of random times ensures full viability in multi-default models under progressive enlargement of filtration.
• The predictable dual projection of the default process is explicitly characterized, enabling the construction of a local martingale deflator.
• The paper proves that the absence of arbitrage of the first kind is preserved when expanding the filtration using ‘𝗁-models’.
• The drift operator in the enlarged filtration is shown to be well-defined and consistent with the original market dynamics.
• The martingale representation property holds in the enlarged filtration, ensuring that all contingent claims can be replicated via trading strategies.
• The framework supports consistent pricing and hedging in multi-default environments by maintaining the semimartingale and local martingale properties of asset prices.