The Carma Interest Rate Model

Abstract

In this paper, we present a multi-factor continuous-time autoregressive moving-average (CARMA) model for the short and forward interest rates. This model is able to present an adequate statistical description of the short and forward rate dynamics. We show that this is a tractable term structure model and provides closed-form solutions to bond prices, yields, bond option prices, and the term structure of forward rate volatility. We demonstrate the capabilities of our model by calibrating it to a panel of spot rates and the empirical volatility of forward rates simultaneously, making the model consistent with both the spot rate dynamics and forward rate volatility structure.

Pricing of Forwards and Options in a Multivariate Non-Gaussian Stochastic Volatility Model for Energy Markets

“…where dL (1)] under P. Thus, by appropriately choosing i we can obtain a higher or lower mean-reversion level, implying a higher or lower risk loading on the spike processes Y i under Q. Similar considerations hold for the volatility processes Z j .…”

“…In this section we derive the forward price dynamics based on the multivariate spot price model (1).…”

The exact discretisation of CARMA models with applications in finance

“…Andresen, Benth, Koekebakker and Zakamulin (2014), has suggested that CARMA models may be suitable representations for short-term interest rates. These authors propose a number of reasons why CARMA models may be preferable to the more commonly used first-order Vasicek-type models, not least the fact that they can provide a better empirical fit to the observed term structure dynamics.…”

“…Interest rate equations in the form of second-order stochastic differential equations are not without precedent. A CARMA(2, 0) specification (in effect) was used in the continuous time macroeconometric model of the United Kingdom by Bergstrom and Nowman (2007) while Andresen, Benth, Koekebakker and Zakamulin (2014) have more recently developed more general CARMA specifications.…”

“…This class of continuous time autoregressive moving average (CARMA) models has been discussed by Brockwell (2001) and Brockwell and Marquardt (2005) and has been applied to interest rates by Andresen, Benth, Koekebakker and Zakamulin (2014) and to electricity futures by Benth andŠaltytė Benth (2009), while Brockwell (2004) discusses their application to GARCH and stochastic volatility models. We explore three benchmark models in finance using recently developed techniques of Thornton (2012, 2016) to formulate the exact discrete representation of the CARMA model, that is to say the discrete time process which matches perfectly the first and second moments of the discretely observed process generated by the CARMA model and which lends itself to estimation using the Gaussian quasi-maximum likelihood method proposed by Bergstrom (1983).…”