# Curves and Term Structure Models: Definition, Calibration and Application of Rate Curves and Term Structure Models

22 Pages Posted: 1 Jan 2013 Last revised: 11 Apr 2013

See all articles by Christian P. Fries

## Christian P. Fries

Ludwig Maximilian University of Munich (LMU) - Faculty of Mathematics; DZ Bank AG

Date Written: December 31, 2012

### Abstract

In this note we discuss the definition, construction, interpolation and application of curves.

We will discuss discount curves, a tool for the valuation of deterministic cash-flows and forward curves, a tool for the valuation of linear cash-flows of (possibly) stochastic indices.

The aim of this note is to carefully derive the definition of discount and forward curves and work out their relation to market instruments: A curve is tool to value linear products, i.e., product which can be replicated by static hedges.

We will distinguish forward curves from discount curves. Since forward curves are associated with a discount curve (representing the collateralization of the forward), this motivates an alternative interpolation method, namely interpolation of the forward value (the product of the forward and the discount factor).

In addition, treating forward curves as native curves (instead of representing them by pseudo-discount curves) will avoid other problems, like that of overlapping instruments.

We discuss the calibration of the curves for which we give a generic object oriented implementation.

In the last section we will show how to define term-structure models (analog to a LIBOR market model) based on the definition of the performance index of an accrual account associated with a discount curve.

Source code is available at http://www.finmath.net/topics/curvecalibration/

Keywords: Curves, Discount Curve, Forward Curve, Calibration, Bootstrapping, Multi-Curve, Tenor-Basis, Cross-Currency-Basis, Collateralization, Funding, OIS Discounting, Funding Curve, Spread Curve

JEL Classification: G13

Suggested Citation

Fries, Christian P., Curves and Term Structure Models: Definition, Calibration and Application of Rate Curves and Term Structure Models (December 31, 2012). Available at SSRN: https://ssrn.com/abstract=2194907 or http://dx.doi.org/10.2139/ssrn.2194907