Multiple curve Lévy forward price model allowing for negative interest rates

Abstract

In this paper we develop a framework for discretely compounding interest rates which is based on the forward price process approach. This approach has a number of advantages, in particular in the current market environment. Compared to the classical as well as the Lévy Libor market model, it allows in a natural way for negative interest rates and has superb calibration properties even in the presence of extremely low rates. Moreover, the measure changes along the tenor structure are simplified significantly. These properties make it an excellent base for a post-crisis multiple curve setup. Two variants for multiple curve constructions are discussed. Timeinhomogeneous Lévy processes are used as driving processes. An explicit formula for the valuation of caps is derived using Fourier transform techniques. Based on the valuation formula, we calibrate the two model variants to market data.

Stochastic interest rate modelling using a single or multiple curves: an empirical performance analysis of the Lévy forward price model

“…For a formal derivation of the single curve model, see appendix 2. To compare the LFPM under the single and multiple curve approach, we provide a brief outline of the multiple curve construction of the LFPM, based on the derivations of Eberlein and Gerhart (2018) and Eberlein et al (2019).…”

“…However, in practice we observe quite substantial spreads between different interest rates, implying that it is actually inconsistent to discount and generate cash flows with the same curve. The multiple curve approach of Eberlein and Gerhart (2018) and Eberlein et al (2019) does take the credit and liquidity risk into account by explicitly incorporating the tenor dependence of each term structure. More specifically, it distinguishes between a basic, 'risk-free' discount curve denoted with index 0 and m ∈ N risky tenor-dependent curves.…”

“…From the construction of the forward price process, it has become apparent that interest rate derivatives can be priced consistently in the LFPM using closed-form solutions (appendix 2). In this paper, we adopt the explicit Fourier-based formula of Eberlein et al (2016) and Eberlein et al (2019) to value one of the most popular and liquid interest rate derivatives in the market nowadays, namely an interest rate cap.…”

“…One approach that enables the pricing of these individual caplets, is suggested by Eberlein et al (2016) and Eberlein et al (2019). Their approach incorporates Fourier-based methods as well as the cumulant and moment generating function to derive a closed-form, explicit formula for valuing standard interest rate derivatives in the LFPM.…”

“…The empirical performance of these eight model specifications is assessed under the risk-neutral measure by adopting three distinct performance measures including the goodnessof-fit, the parameter stability and the out-of-sample pricing. The model specifications are calibrated under the Normal Inverse Gaussian (NIG) distribution by approximating the analytical pricing formula of Eberlein et al (2016) or Eberlein et al (2019) in the single or multiple curve framework, respectively, for an interest rate caplet with the trapezoidal method. Finally, the relevant market data on March 30th, 2012 up until April 28th, 2017 are obtained by applying a stripping procedure and the bootstrap method under the assumption of complete markets and the Efficient Market Hypothesis (EMH).…”