Putting the Modern Pricing Approach in Perspective

The set of techniques to price interest-rate derivatives that stemmed from the original work of Heath, Jarrow and Morton (HJM) in the late 1980s (HJM 1989) are referred to in this book as the ‘modern’ or the ‘LIBOR-marketmodel’ approach. At a superficial glance, the differences between the various ‘incarnations’ of the approach might appear greater than what they have in common. The state variables could be instantaneous or discretely compounded rates; they could be swap rates or forward rates; they might be normally or log-normally (or otherwise) distributed; the associated numeraire could be a zero-coupon bond, a swap annuity or the money-market account; and so on. Despite these non-trivial differences, these approaches share one essential common feature: the recognition that the no-arbitrage evolution of the state variables (however chosen) can be expressed purely as a function of the volatilities of, and of the correlations among, the state variables themselves. Different choices of numeraires will give rise to different combinations for these covariance elements, but this fundamental result, which goes back to the original insight of HJM, is shared by all the approaches that will be dealt with in this book. This result and its implications are sufficiently fundamental and far-reaching to justify a self-contained and unified treatment. Given the various ‘versions’, ‘implementations’ and choices of numeraires, no general agreement exists in the financial community on how to call this set of approaches: the terms ‘BGM (Brace, Gatarek and Musiela) model’ and ‘Jamshidian approach’ are often used, but ‘pricing in the forward measure’, the ‘LIBOR market model’ and other terms are also frequently encountered. Some purists insist on calling the approach simply the ‘HJM model’. The difficulty in establishing intellectual priority is compounded by the fact that