Approximate pricing of swaptions in affine and quadratic models

Abstract

This paper proposes new bounds on the prices of European-style swaptions for affine and quadratic interest rate models. These bounds are computable whenever the joint characteristic function of the state variables is known. In particular, our lower bound involves the computation of a one-dimensional Fourier transform independently of the swap length. In addition, we control the error of our method by providing a new upper bound on swaption price that is applicable to all considered models. We test our bounds on different affine models and on a quadratic Gaussian model. We also apply our procedure to the multiple curve framework. The bounds are found to be accurate and computationally efficient.

Keywords:

• Pricing
• Swaptions
• Affine quadratic models
• Fourier transform
• Bounds

AMS Subject Classifications:

• G12
• G13

Acknowledgements

The authors are indebted to Laura Ballotta and the participants of XVI Workshop on Quantitative Finance, (Università degli studi di Parma) and the conference Challenges in Derivatives Markets, (Technische Universität München) for useful conversations and constructive comments. We would like to thank Laura Ballotta for useful feedback. All remaining errors are ours. We acknowledge financial support by Universitá degli Studi del Piemonte Orientale, Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA).

Notes

No potential conflict of interest was reported by the authors.

1 The three approximations presented in Kim (2014) are lower bounds, as proved in section 2. Therefore, the most precise is the one that produces the highest price, which was not discussed in the Kim paper.

3 Schrager and Pelsser (2006) and Duffie and Singleton (1997) for the two-factor CIR model.

4 diag() means the diagonalization of the vector and chol() means the Cholesky decomposition of the correlation matrix , where and are the volatility vector and the correlation matrix, respectively, of the original paper.