Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues

2

The term structure or yield curve

2.1 INTRODUCTION TO THE YIELD CURVE

A term structure or yield curve can be defined as the graph of spot rates or zeroes1 in function of their maturity. Since most of the time interest rates are higher with longer maturities, one talks of a “normal” yield curve if it is going upwards, and of an “inverse” yield curve if and when longer rates are lower than shorter rates.

Alternatively, the term structure can be built on discount factors, as functions of the zero rates, but this way is less used in practice.

Yield curves can be built with mid rates – the most usual way – or with borrowing or lending rates. The two main uses of a yield curve are:

• to determine the corresponding interest rate for a given maturity, by interpolation on the yield curve;
• to serve as the “spinal column” for the pricing of any kind of financial instruments involving future cash flows, such as bonds, stocks, and all kinds of derivative products. Indeed, derivatives being basically forward products – their valuation is subject to the value of yields relative to the corresponding forward maturities involved.

Before moving on, it is worthwhile mentioning an unsolvable methodological problem: dealing with the yield curve implies using swaps and bonds data. But dealing with swaps and bonds implies using the yield curve. We have opted for starting with the yield curve – given it is a corner stone in financial mathematics of the markets – and refer the reader to the subsequent ...