# Multiple Curves and Multiple Regimes: Libor Market Models on Switching (co-Jump) Diffusions

Author(s):
Rodney Hoskinson, PhD
Keywords:

## Abstract :

Multiple Curve Libor Market Models on Hybrid Switching Diffusions: This paper introduces a comprehensive approach to modelling the term structure of interest rates and volatility by setting double curve Libor market models (“LMMs”) on a hybrid switching diffusion. Modelling interest rate volatility over the whole term structure is an important part of building a model for pricing interest rates options. This is of practical interest due to the large size of the international overthe- counter (OTC) market for interest rate derivatives — which according to the Bank for International Settlement was US\$435,000 billion as of June 2015 — as well as in fixed income instruments with embedded options such as callable bonds and structured notes.

Multiple Curve Libor Market Models on Hybrid Switching Diffusions with Time-Varying Parameters: In this paper we assume all the definitions and results from mathematical finance theory and interest rate modelling with multiple curves from Sections A.1 and A.2 of Hoskinson 2016a. In the vanilla interest rate derivative markets, it can be difficult to accurately calibrate a model with constant parameters to the market. The market can display different skews and/or smiles for different cap expiries or swaption expiries and maturities. To address this, (Vladimir Piterbarg 2005) derived a method to extend displaced diffusion stochastic volatility models to time-varying skew and volatility parameters. The method is based on finding fast and accurate averaging formulae, that can determine, from the model parameters, the equivalent constant parameters over each interval given the time-varying parameters. In Andersen and V. Piterbarg 2010, Piterbarg includes parameter averaging results for volatility of variance. Application of parameter averaging methods permits use of time-varying skew models and facilitates an efficient calibration approach. In essence the parameter averaging theory is an application of the “mimicking” theorem (Gyöngy 1986) to stochastic volatility models. This papers introduces an extension of the class of multiple curve Libor Market Models set on Hybrid switching diffusions introduced in Hoskinson 2016a to the case of time-varying model parameters. The extended specification is introduced in Section 2.3. The purpose of allowing for time-varying parameters is to achieve a better calibration of the model to the market, for our models set on switching diffusions.

Multiple Curve Libor Market  Models on Hybrid Switching Co-Jump Diffusions: In Hoskinson 2016c and Hoskinson 2016d we introduced a class of multi-curve Libor Market Models with regime-switching stochastic volatility set on a hybrid switching diffusion. The switching volatility regimes were obtained by applying Markov switching to the mean reversion level of a square-root stochastic volatility process. Among other advantages, the switching models calibrated to the implied volatility of vanilla interest rate options implied a more realistic path-by-path dynamics of volatility than did the equivalent models without switching. We did not, however, consider any links between the volatility regimes and the level of forward rates. In this paper we propose further model extensions, with simultaneous jumps in forward rates on the discount curve and switches in the stochastic volatility regime. The primary motivation for the jump extension is not calibration accuracy, but rather increased realism of the implied joint path-by-path dynamics of volatility and the forward curve in the model calibrated to optionimplied volatility. Also, in the uncertain global economic and financial environment emerging since the 2007 financial crisis, interest rates, option prices and implied volatility can change quickly. Models with jumps are arguably more important in this environment.

Multiple Curve Libor Market Models on Hybrid Switching Co-Jump Diffusions with Time-Varying Parameters: In this paper we assume all the definitions and results from mathematical finance theory and interest rate modelling with multiple curves from Sections A.1 and A.2 of Hoskinson 2016b and Sections 3.3 and C.1 of Hoskinson 2016a . In the vanilla interest rate derivative markets, it can be difficult to accurately calibrate a model with constant parameters to the market. The market can display different skews and/or smiles for different cap expiries or swaption expiries and maturities. To address this, Piterbarg (Vladimir Piterbarg 2005) derived a method to extend displaced diffusion stochastic volatility models to time-varying skew and volatility parameters. The method is based on finding fast and accurate averaging formulae, that can determine, from the model parameters, the equivalent constant parameters over each interval given the time-varying parameters. In Andersen and V. Piterbarg 2010, Piterbarg includes parameter averaging results for volatility of variance. Application of parameter averaging methods permits use of time-varying skew models and facilitates an efficient calibration approach. In essence the parameter averaging theory is an application of the “mimicking” theorem (Gyöngy 1986) to stochastic volatility models (or more generally to semi-martingale models).

Publication date of the thesis
27-04-2016

Thesis committee

Supervisors: Frank Fabozzi, EDHEC Business School, Jakša Cvitanić, Caltech

External reviewer: Marcel Rindisbacher, Boston University

Other committee member: René Garcia, EDHEC Business School