Arbitrage-free approaches for pricing interest rate derivatives under the SABR model

Abstract

This thesis is about pricing swaptions under the SABR model or a variant thereof. In the interest market a stochastic local volatility is often used by practitioners to describe the volatility curve in the strike dimension of swaptions. It is a fast approach to inter- and extrapolate market quotes. It is however well-known that this approach is not arbitrage-free. This led to the investigation of approaches that are arbitrage-free. Several approaches have been proposed in the literature to resolve the arbitrage. Computationally rapid approaches that are arbitrage-free are desired as an alternative for Hagan's formulas. These approaches can be used to describe the volatility curve in the strike dimension. The focus will be on an approach that is analytically exact under the SABR model, an approach that reduces the dimensionality in the dynamics of the SABR process and the stochastic collocation method (SCM). The SCM will be used to remove the arbitrage in the stochastic local volatility approach. All approaches will be calibrated to swaption volatility curves and the impact on the inter- and extrapolation of the market quotes is investigated. This is done by investigating the extrapolation of the volatilities, the sensitivities and pricing constant maturity swap (CMS) derivatives using a convexity adjustment method dependent on the complete volatility curve.