A multiple-curve Lévy forward rate model in a two-price economy

Abstract

Abstract. In this thesis, we combine and merge the multiple-curve approach and the two-price theory based on acceptability indices in a Lévy interest rate model.A multiple-curve Heath-Jarrow-Morton (HJM) forward rate model driven by time-inhomogeneous Lévy processes (a multiple-curve Lévy term structure model) is presented. We nd deterministic conditions which ensure the monotonicity of the curves. Explicit valuation formulas for some interest rate derivatives are established, namely forward rate agreements, swaps, caps, oors and digital options. These formulas can numerically be evaluated very fast by using the Fourier based valuation method. Furthermore, we apply the two-price theory to this multiple-curve setting. Ask and bid model prices of caplets, oorlets and digital options are derived.A general procedure how to calibrate this two-price multiple-curve interest rate model to market data is described. As a practical application, the model is calibrated to market prices of caps for dates before and after the global nancial crisis. CONTENTS

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

“…As a result, the class of multi-curve stochastic interest rate models was introduced which is more consistent with this phenomenon observed in current financial markets but requires considerably more computational efforts to implement in practice (see, e.g. Grbac and Runggaldier 2015, Eberlein and Gerhart 2018, Sabelli et al 2018, Cuchiero et al 2019.…”

“…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.…”

Arbitrage-free Nelson–Siegel model for multiple yield curves

“…To further confirm the practical relevance of the model for derivative pricing we propose a simple calibration exercise. We build a surface of caplet prices on 15 Sep 2016 using discount bond values and cap implied volatilities, following the procedure outlined in [10]. We end up with market caplet prices for n T = 15 maturities T = {2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9, 9.5} years and n K = 5 strikes K = {−0.005, −0.0013, 0.0025, 0.01, 0.02}.…”

Multiple curve Lévy forward price model allowing for negative interest rates

“…Furthermore, we mention Crépey, Grbac, and Nguyen (2012) where a single risky rate is considered in addition to the basic one in a HJM-framework. In Eberlein and Gerhart (2018) a fully fledged model with an arbitrary number of tenor-dependent curves is developed. In addition this model considers multiple curves in the context of a two-price economy and therefore allows to exploit bid and ask quotes.…”