Introduction

Abstract

A term-structure is a function that relates a certain financial variable or parameter to its maturity. Prototypical examples are the term-structure of interest rates or zero-coupon bond prices. But there are also term-structures of option implied volatilities, credit spreads, variance swaps, etc. Term-structures are high-dimensional objects, which often are not directly observable. On the empirical side this requires estimation methods that are flexible enough to capture the entire market information. But flexibility often comes at the cost of irregular term-structure shapes and a great number of factors. Principal component analysis and parametric estimation methods can put things right. On the modeling side we find several challenging tasks. Bonds and other forward contracts expire at maturity where they have to satisfy a formally predetermined terminal condition. For example, a zero-coupon bond has value one at maturity, a European-style option has a predetermined payoff contingent on some underlying instrument, etc. Under the absence of arbitrage this has non-trivial implications for any dynamic term-structure model. As a consequence, various approaches to modeling the term-structure of interest rates have been proposed in the last decades, starting with the seminal work of Vasiček (J. Financ. Econ. 5:177–188, 1977). By arbitrage we mean an investment strategy that yields no negative cash flow in any future state of the world and a positive cash flow in at least one state; in simple terms, a risk-free profit. The assumption of no arbitrage is justified by market efficiency as a consequence of which prices tend to converge to arbitrage-free prices due to demand and supply effects.