Journal of Economics and Financial Analysis, 3 (2), pp. 41-83, [2019]
URI: https://ojs.tripaledu.com/jefa/article/view/49/57

Convexity Adjustments Made Easy: An Overview of Convexity Adjustment Methodologies in Interest Rate Markets


University of Reading, United Kingdom


DOI: http://dx.doi.org/10.1991/jefa.v3i2.a28

Abstract

Interest rate instruments are typically priced by creating a non-arbitrage replicating portfolio in a risk-neutral framework. Bespoke instruments with timing, quanto1 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 multicurve 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; Radon-Nykodym Derivative; Shifted- Lognormal; Linear Swap Rate Method; Libor In-Arrears Swaps; Constant Maturity Swaps; CMS Caplets, Floorlets and Swaplets.

JEL Classification

G100, G120.

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References

Andersen, L.B.G., and Piterbarg, V.V. (2010). Interest Rate Modelling. Volume I: Foundations and Vanilla Models. London: Atlantic Financial Press.

Baxter, M., and Rennie, A. (1996). Financial Calculus: An Introduction to Derivative Pricing. Cambridge, UK: Cambridge University Press.

Brigo, D., and Mercurio, F. (2006). Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit. 2nd ed., Springer Finance.

Boenkost, W., and Schmidt, W.M. (2016). Notes on Convexity and Quanto Adjustments for Interest Rates and Related Options. Available at SSRN: https://ssrn.com/abstract=1375570 or http://dx.doi.org/10.2139/ssrn.1375570

Burgess, N. (2014). Martingale Measures & Change of Measure Explained. Available at SSRN: https://ssrn.com/abstract=2961006 or http://dx.doi.org/10.2139/ssrn.2961006

Burgess, N. (2017). How to Price Swaps in Your Head - An Interest Rate Swap & Asset Swap Pricer. Available at SSRN: https://ssrn.com/abstract=2815495 or http://dx.doi.org/10.2139/ssrn.2815495

Carr, P., and Madan, D. (2001). Optimal Partitioning in Derivative Securities. Quantitative Finance, 1(1), pp. 19-37.

Caspers, P. (2012). Normal Libor in Arrears. Available at SSRN: https://ssrn.com/abstract=2188619 or http://dx.doi.org/10.2139/ssrn.2188619

Caspers, P. (2015). Libor Timing Adjustments. Available at SSRN: https://ssrn.com/abstract=2170721 or http://dx.doi.org/10.2139/ssrn.2170721

Derman, E. (2008). Static Hedging and Implied Distributions. Columbia University Smile Course, Lecture 5. http://www.emanuelderman.com/writing/category/smile-course

Hagan, P. (2003). Convexity Conundrums: Pricing CMS swaps, caps and floors. Wilmott Magazine, pp. 38-44.

Hull, J.C. (2011). Options, Futures and Other Derivatives. 8th ed., Pearson Education.

Hunt, P., and Kennedy, J. (2000). Financial Derivatives in Theory and Practice. John Wiley & Sons.

Karouzakis, N., Hatgioannides, J., and Andriosopoulos, K. (2018). Convexity Adjustment for Constant Maturity Swaps in a Multi-Curve Framework. Annals of Operations Research, 266(1-2), pp. 159-181

Lesniewski, A. (2008). Convexity. Math NYU Lecture Series Spring 2007, www.math.nyu.edu

Pelsser, A. (2003). Mathematical Foundation of Convexity Correction. Quantitative Finance, 3(1), pp. 59-65.

Pelsser, A. (2004). Efficient Methods for Evaluating Interest Rate Derivatives, 2nd ed., Springer Finance

Schlenkrich, S. (2015). Multi-Curve Convexity. Available at SSRN: https://ssrn.com/abstract=2667405 or http://dx.doi.org/10.2139/ssrn.2667405




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