Affine Multiple Yield Curve Models

Abstract

We provide a general and tractable framework under which all multiple yield curve modeling approaches based on affine processes, be it short rate, Libor market, or HJM modeling, can be consolidated. We model a numéraire process and multiplicative spreads between Libor rates and simply compounded OIS rates as functions of an underlying affine process. Besides allowing for ordered spreads and an exact fit to the initially observed term structures, this general framework leads to tractable valuation formulas for caplets and swaptions and embeds all existing multi-curve affine models. The proposed approach also gives rise to new developments, such as a short rate type model driven by a Wishart process, for which we derive a closed-form pricing formula for caplets. The empirical performance of two specifications of our framework is illustrated by calibration to market data.2010 Mathematics Subject Classification. 91G30, 91B24, 91B70. JEL Classification E43, G12. Key words and phrases. Multiple yield curves, Libor rate, forward rate agreement, multiplicative spread, affine processes. Acknowledgements. The authors are grateful to two anonymous referees for their valuable comments that helped to significantly improve the paper. 1 Similarly as in [5,28], we denote by Xibor a generic interbank offered rate for unsecured term lending, such as the Libor rate in the London interbank market and the Euribor rate in the Eurozone. While the theoretical framework developed in the present paper applies to generic Xibor rates, the empirical results reported in Section 6 refer to Euribor rates. 1 arXiv:1603.00527v2 [q-fin.MF]

Multiple Yield Curve Modelling With CBI Processes

“…As illustrated by Figures 4, the model achieves a good fit to market data, across different strikes and maturities. We remark that, in terms of number of parameters, the model under consideration is even more parsimonious than the simple specifications calibrated in [CFG19b]. Motivated by the presence of forward rates, we also calibrated a version of the model where the OIS short rate is affected by an auxiliary Ornstein-Uhlenbeck process, in line with Remark 3.6.…”

“…In this section, we develop a general modelling framework based on CBI processes for financial markets with multiple curves. To this effect, we adapt the affine short rate multi-curve approach of [CFG19b], to which we refer for additional details on the general features of the post-crisis interest rate market. In this section, we focus on the construction and properties of the framework.…”

“…By relying on the theory of continuous-state branching processes with immigration (CBI processes), we develop a modelling framework that can capture all the empirical properties mentioned above and, at the same time, allows for an efficient valuation of interest rate derivatives written on Ibor rates. Exploiting the affine property of CBI processes, we design our modelling framework in the spirit of the affine multi-curve models recently studied in [CFG19b], taking multiplicative spreads and the OIS short rate as fundamental modelling objects. By construction, the model achieves a perfect fit to the initially observed term structures and can generate spreads greater than one and increasing with respect to the tenor (see Section 3).…”

Cross Currency Valuation and Hedging in the Multiple Curve Framework

“…One possibility is to extend the market by introducing instruments which are by definition fully collateralized, i.e. natively collateralized assets such as OIS bonds and (textbook) FRAs as in Cuchiero et al (2016) and Cuchiero et al (2019). The resulting model would allow for the joint evolution of interbank spreads, overnight rates, foreign exchange and risky assets.…”

“…Several spreads have emerged (more precisely widened) between certain interest rates (notably between overnight and unsecured Ibor rates) and these rates in turn differ from interest rates agreed in the context of repurchase agreements (repo rates). From a modeling perspective this resulted in the development of multi curve interest rate models as in Henrard (2007), Bianchetti (2010), Moreni and Pallavicini (2014), Mercurio (2010), Henrard (2014), Grbac et al (2015), Crépey et al (2015) Cuchiero et al (2016) and Cuchiero et al (2019) among others.…”