In this volume of the ‘Mastering Mathematical Finance’ series we relax the assumption of constant interest rates adopted in the binomial or the Black–Scholes market models covered in earlier volumes, in particular [DMFM] and [BSM]. In general, interest rates are time dependent and random. Being closely linked to, and indeed determined by, fixed-income instruments traded in the market, the rates also depend on the maturity dates of the underlying instruments. This gives rise to the notion of term structure, i.e. the family of interest rates parameterised by the maturity date. We are going to study models describing the random evolution through time of the term structure, that is, of the entire family of interest rates for various maturities.

Because the rates for different maturities are related to one another and evolve simultaneously in time, their joint evolution is more intricate than that of a single quantity such as a stock price. There is not a single term structure model universally adopted as a benchmark to play a similar role as the Black–Scholes model does for stock prices. Instead, a range of alternative and to some extent complementary models are in use to capture various aspects of the evolution of the term structure. A selection of such models will be presented along with the associated interest rate derivative securities.

The prerequisites for this book are covered in some other volumes of the ‘Mastering Mathematical Finance’ series. These include probability theory [PF], stochastic calculus [SCF], and the Black–Scholes model [BSM]. Familiarity with Monte Carlo simulations [NMFC] will also be helpful.

We begin with various fundamental notions and properties associated with fixed-income instruments in Chapter 1 and the basic ‘vanilla’ interest rate derivatives in Chapter 2. Here we also cover the change of numeraire technique and introduce the notion of forward measure, a very useful alternative to the risk-neutral measure when pricing interest rate derivatives.