[Paper Review] When roll-overs do not qualify as numéraire: bond markets beyond short rate paradigms

Irene Klein, Thorsten Schmidt+1 more|arXiv (Cornell University)|Sep 30, 2013

Stochastic processes and financial applications21 references3 citations

TL;DR

This paper redefines bond market modeling by showing that the bank account process—commonly used as a numéraire—may fail to qualify as a valid numéraire due to liquidity effects, especially in markets with multiple yield curves. It establishes that no-arbitrage conditions (NAFL and NAA1) are equivalent to the existence of an equivalent local martingale measure relative to a terminal bond as numéraire, and proves the generalized bank account emerges as a limit of roll-over strategies even when it is not a true numéraire.

ABSTRACT

We investigate default-free bond markets where the standard relationship between a possibly existing bank account process and the term structure of bond prices is broken, i.e. the bank account process is not a valid numéraire. We argue that this feature is not the exception but rather the rule in bond markets when starting with, e.g., terminal bonds as numéraires. Our setting are general càdlàg processes as bond prices, where we employ directly methods from large financial markets. Moreover, we do not restrict price process to be semimartingales, which allows for example to consider markets driven by fractional Brownian motion. In the core of the article we relate the appropriate no arbitrage assumptions (NAFL), i.e. no asymptotic free lunch, to the existence of an equivalent local martingale measure with respect to the terminal bond as numéraire, and no arbitrage opportunities of the first kind (NAA1) to the existence of a supermartingale deflator, respectively. In all settings we obtain existence of a generalized bank account as a limit of convex combinations of roll-over bonds. Additionally we provide an alternative definition of the concept of a numéraire, leading to a possibly interesting connection to bubbles. If we can construct a bank account process through roll-overs, we can relate the impossibility of taking the bank account as numéraire to liquidity effects. Here we enter endogenously the arena of multiple yield curves. The theory is illustrated by several examples.

Motivation & Objective

• To challenge the standard assumption that the bank account process is always a valid numéraire in bond markets.
• To investigate bond markets where the standard relationship between bond prices and the bank account breaks down, especially when the bank account is not a martingale under any equivalent measure.
• To develop a no-arbitrage framework based on the terminal bond as numéraire, rather than the bank account, in settings without assuming semimartingale properties.
• To formalize the emergence of a generalized bank account as a limit of convex combinations of roll-over bonds, even when the bank account is not a valid numéraire.
• To connect the failure of the bank account as numéraire to liquidity effects and multiple yield curve dynamics, providing a theoretical foundation for such market structures.

Proposed method

• Uses large financial market theory to analyze general càdlàg bond price processes without assuming semimartingale properties.
• Defines no-arbitrage via NAFL (no asymptotic free lunch) and NAA1 (no arbitrage of the first kind), relating them to the existence of an equivalent local martingale measure under the terminal bond numéraire.
• Constructs the generalized bank account as the limit of roll-over strategies using convex combinations of short-term bonds over refining partitions.
• Applies a change of numéraire technique using the terminal bond as numéraire, showing that the discounted bond prices are martingales under an equivalent measure.
• Introduces a new definition of numéraire that allows for the possibility of bubbles and links the failure of the bank account as numéraire to strict local martingale behavior.
• Employs stochastic calculus with fractional Brownian motion and Itô-type integrals to model non-semimartingale price processes, particularly in the example with $ W_t^2 $-driven dynamics.

Experimental results

Research questions

• RQ1 Under what conditions does the bank account process fail to qualify as a numéraire in bond markets?
• RQ2 How can no-arbitrage conditions (NAFL and NAA1) be characterized when the bank account is not a valid numéraire?
• RQ3 What is the role of the terminal bond as a numéraire in bond markets where the bank account is not a martingale under any equivalent measure?
• RQ4 How does the generalized bank account emerge as a limit of roll-over strategies in markets without semimartingale price processes?
• RQ5 What is the connection between the failure of the bank account as numéraire and liquidity effects in multiple yield curve systems?

Key findings

• The bank account process may fail to be a valid numéraire even when it exists, due to the strict local martingale property of the inverse growth optimal portfolio.
• No-arbitrage (NAFL) is equivalent to the existence of an equivalent local martingale measure with respect to the terminal bond as numéraire.
• No-arbitrage of the first kind (NAA1) is equivalent to the existence of a supermartingale deflator.
• The generalized bank account emerges as the limit of convex combinations of roll-over bonds, even in non-semimartingale settings such as those driven by fractional Brownian motion.
• In the example with $ Z(T) = \exp(T - 2\int_0^T W_s^2 ds - W_T^2) $, the discounted roll-over account $ V(T) = Z(\tau(T)) $ converges to 0 almost surely as $ T \to \infty $, implying $ B(1) = 0 $, so the roll-over strategy leads to total capital loss.
• The limit of the roll-over account $ B^n $ converges to $ \exp(-2\int_0^{\tau(T)} W_s^2 ds + \tau(T) - \tau^2(T)) $, showing that risk-free roll-over strategies can be highly risky when the underlying process is not a true martingale.