Arbitrage-Free Interpolation of the Swap Curve

Abstract

We suggest an arbitrage free interpolation method for pricing zero-coupon bonds of arbitrary maturities from a model of the market data that typically underlies the swap curve; that is short term, future and swap rates. This is done first within the context of the Libor or the swap market model. We do so by introducing an independent stochastic process which plays the role of a short term yield, in which case we obtain an approximate closed-form solution to the term structure while preserving a stochastic implied short rate. This will be discontinuous but it can be turned into a continuous process (however at the expense of closed-form solutions to bond prices). We then relax the assumption of a complete set of initial swap rates and look at the more realistic case where the initial data consists of fewer swap rates than tenor dates and show that a particular interpolation of the missing swaps in the tenor structure will determine the volatility of the resulting interpolated swaps. We give conditions under which the problem can be solved in closed-form therefore providing a consistent arbitrage-free method for yield curve generation.

On Cross-Currency Models with Stochastic Volatility and Correlated Interest Rates

“…To define continuous time dynamics for a zero-coupon bond, interpolation techniques are available (see, for example, [Schlögl(a), 2002;Piterbarg, 2004;Davis and Mataix-Pastor, 2009;Beveridge and Joshi, 2009] We consider here the linear interpolation scheme, proposed in [Schlögl(a), 2002], which reads:…”

On Cross-Currency Models with Stochastic Volatility and Correlated Interest Rates

“…To define continuous time dynamics for a ZCB, interpolation techniques are available (see, e.g. Beveridge and Joshi, 2009;Davis and Mataix-Pastor, 2009;Piterbarg, 2004;Schlögl, 2002a). We consider here the linear interpolation scheme, proposed in Schlögl (2002a), which reads…”

In memoriam: Mark H. A. Davis and his contributions to mathematical finance

“…His research group quickly became one of the focal points for mathematical finance worldwide and Mark supervised a string of PhD students who went to make careers in both industry and academia. Among others, Mark worked on interest rates models with V. Mataix‐Pastor (2007, 2009), on sensitivities and Malliavin calculus with M. Johansson (2006) and on risk‐sensitive investment with L. Lleo (2008, 2010, 2011a, 2011b). From the late 2000s, Mark became increasingly interested in model uncertainty and model‐independent methods which led to a string of works, from the paper with D. Hobson (2007), through the work with J. Obłój and V. Raval (2014), to his collaboration with S. Badikov and A. Jacquier (2021), that is included in this volume.…”