Fixed-Income Pricing and the Replication of Liabilities

This paper develops a model-free framework for static fixed-income pricing and the replication of liability cash flows. We show that the absence of static arbitrage across a universe of fixed-income instruments is equivalent to the existence of a strictly positive discount curve that reproduces all observed market prices. We then study the replication and super-replication of liabilities and establish conditions ensuring the existence of least-cost super-replicating portfolios, including a rigorous interpretation of swap–repo replication within this static framework. The results provide a unified foundation for discount-curve construction and liability-driven investment, with direct relevance for economic capital assessment and regulatory practice.

💡 Research Summary

This paper develops a completely model‑free, static framework for pricing fixed‑income securities and for replicating liability cash flows. The authors consider a universe of M fixed‑income instruments, each with a known price Pᵢ and a deterministic cash‑flow vector Cᵢ across a common set of N payment dates {x₁,…,x_N}. A portfolio q∈ℝᴹ has price qᵀP and generates cash flows qᵀC.

The first contribution is a fundamental theorem of fixed‑income pricing that links the absence of static arbitrage to the existence of a discount curve. Two notions of arbitrage are defined: (i) strict arbitrage (negative price, non‑negative cash flows) and (ii) arbitrage (non‑positive price, non‑negative cash flows with at least one strictly positive component). Theorem 2.3 shows that the absence of strict arbitrage is equivalent to the existence of a non‑negative discount function g with g(0)=1 such that P = C g(x). Theorem 2.4 strengthens this result: the absence of arbitrage (the usual no‑arbitrage condition) is equivalent to the existence of a strictly positive discount curve g > 0 satisfying the same pricing relation. Thus, the whole market can be represented by a single discount curve if and only if static arbitrage is ruled out. This result is a direct, cash‑flow‑matrix analogue of the classic state‑price vector theorems of Harrison‑Kreps and Dalang‑Mortensen‑Wang, but it is expressed entirely in terms of observable bond, swap and repo cash flows.

The second part of the paper addresses liability replication. Let Z be the vector of expected liability cash flows at the same dates. Exact replication requires Zᵀ to lie in the image of Cᵀ; in practice this condition is rarely met because asset and liability cash‑flow dates rarely align. The authors therefore introduce super‑replication: the set Q = { q | qᵀC ≥ Z } is a closed convex polyhedron containing all portfolios that dominate the liability cash flows. Theorem 3.1 characterises feasibility of super‑replication: Q is non‑empty if and only if there is no non‑negative vector v with C v = 0 and Z v > 0. This condition can be interpreted as the non‑existence of a “discount vector” that assigns zero value to every attainable cash flow while assigning positive value to the liability, i.e., a violation of the pricing rule.

When super‑replication is feasible, the authors consider the linear program
  min q∈Q qᵀP,
which seeks the least‑cost portfolio that dominates the liability. Theorem 3.3 provides sufficient conditions for the existence of an optimal solution: (i) no arbitrage (strictly positive discount curve), (ii) ker(Cᵀ) = {0} (full rank of the cash‑flow matrix), and (iii) super‑replication feasibility. Under these mild regularity assumptions, a least‑cost super‑replicating portfolio always exists.

A particularly insightful contribution is the rigorous treatment of swap‑repo replication within the static framework. The authors show that a receiver interest‑rate swap combined with a rolling repo position reproduces exactly the cash‑flow pattern of a coupon bond. Because the swap has zero net cost and the repo requires one unit of funding, the synthetic bond must have price 1 under the no‑dynamic‑arbitrage assumption. This provides a clean static justification for the common industry practice of constructing synthetic bonds from swaps and repos, and it explains why, in an arbitrage‑free world, government bond yields and swap rates should be priced off the same discount curve. The paper also discusses the empirical “swap‑spread puzzle” – the persistent non‑zero spread between government bonds and swaps – attributing it to market frictions, balance‑sheet constraints, and a safe‑asset premium, i.e., violations of the static no‑arbitrage condition.

The authors illustrate the theory with two concrete instrument families: (1) coupon bonds, whose cash‑flow matrix is directly constructed from coupon and principal payments; and (2) swap‑repo strategies, where matrices S (fixed‑rate payments), Ξ (floating‑rate payments), and F (notional repayments) are combined to form the overall cash‑flow matrix C = S + F. They then solve the linear program for a sample liability profile, obtaining the minimal super‑replication cost as qᵀ1, and describe the operational steps needed to implement the optimal swap‑repo hedge (initial repo funding, entering zero‑cost swaps, rolling repo positions, and offsetting floating legs).

In the conclusion, the paper emphasizes that its static, model‑free approach provides a solid theoretical foundation for regulatory discount‑curve construction (e.g., Solvency II, Swiss Solvency Test), economic capital assessment, and liability‑driven investment (LDI) strategies. Open research directions include uniqueness of the least‑cost super‑replicating portfolio, efficient numerical algorithms for large‑scale cash‑flow matrices, extensions to multi‑currency or credit‑risk‑adjusted settings, and integration of dynamic arbitrage considerations. Overall, the work bridges a gap between abstract arbitrage theory and practical fixed‑income liability management, offering both rigorous proofs and actionable guidance for practitioners.