Valuation of Interest Rate Swaps Under a Discrete-Time No-Arbitrage Framework: Theory, Derivations,and Numerical Applications
Abstract
This paper presents a rigorous treatment of interest rate swap valuation within a discrete-time no-arbitrage framework. We derive closed-form expressions for the par swap rate under both general (time-varying notional) and level-notional conditions, establishing the equivalence between the two approaches through the telescoping property of forward rates and discount factors. We extend the standard framework to deferred-start swaps and derive a corresponding simplified formula. All theoretical results are accompanied by detailed numerical examples that demonstrate practical implementation. A sensitivity analysis illustrates how the term structure of interest rates affects swap pricing, providing both pedagogical and practical insights for risk management. Our unified treatment bridges the gap between introductory textbook presentations and the measure-theoretic continuous-time models used in advanced quantitative finance.
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