Convexity Adjustments Made Easy - A Review of Convexity Adjustment Methodologies and Formulae in Interest Rate Markets

Abstract

Interest rate instruments are typically priced by creating a non-arbitrage replicating portfolio in a risk-neutral framework. Bespoke instruments with timing, quanto and other adjustments often present arbitrage opportunities, particularly in complete markets where the difference can be monetized. To eliminate arbitrage opportunities we are required to adjust bespoke instrument prices appropriately, such adjustments are typically non-linear and described as convexity adjustments.

We review convexity adjustments firstly using a linear rate model and then consider a more advanced static replication approach. We outline and derive the analytical formulae for Libor and Swap Rate adjustments in a single and multi-curve environment, providing examples and case studies for Libor In-Arrears, CMS Caplet, Floorlet and Swaplet adjustments in particular. In this paper we aim to review convexity adjustments with extensive reference to popular market literature to make what is traditionally an opaque subject more transparent and heuristic.

Keywords: Convexity Adjustments, Convexity Formulae, Convexity Conundrums, Change of Measure, Radon-Nykodym Derivative, Lognormal, Shifted-Lognormal, Normal, Hull's Method, Linear Swap Rate Method, Replication, Libor In-Arrears Swaps, Constant Maturity Swaps, CMS Caplets, Floorlets and Swaplets

JEL Classification: C00, C02, D40, D46, E40, E43, E44, F30, G10, G12, G15