Modern Pricing of Interest-Rate Derivatives: The LIBOR Market Model and Beyond by Riccardo Rebonato

Modern Pricing of Interest-Rate Derivatives: The LIBOR Market Model and Beyond by Riccardo Rebonato

1

Putting the Modern Pricing
Approach in Perspective
1.1 Historical Developments
1.1.1 Introduction
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 frequendy encountered.
Some purists insist on calling the approach simply the 'HJM model'. The
difficulty in establishing intellectual priority is compounded by the fact that 

many of the key results were first obtained (and used) by practitioners but,
for obvious reasons, not published in the academic press. I have therefore
avoided any identification of the approach with names of academics or market
professionals, and used the more neutral terms 'LIBOR market model' or
'modern pricing approach' - much as I am sure the latter may read rather
quaint in a few years' time.
This introductory chapter is meant to provide a brief map of the development

of interest-rate derivative pricing from its earliest (modern) days to
the present. I have chosen to present such an introduction not only for its
intrinsic historical interest, but also because it illustrates rather clearly that
an uneven combination of market events, 'right choices made for the wrong
reasons', computational expediency and sound judgement have conspired to
produce the market standard that the later, more sophisticated, models have
striven to recover. In other words, the modern approach is, justifiably, so loved
by practitioners because of its ability to price exotic products while at the same
time recovering exactly the prices of the relevant plain-vanilla options (caplets
or European swaptions). I shall explain below how the market consensus has
crystallized around the Black framework, pardy for sound financial reasons,
but partly also by historical accident. If this analysis is correct, there is nothing
'inevitable' about the current market standard, and it is quite possible that the
target the modern approach has been trying to hit might in the near future
turn out to be a rather rapidly moving one.
Indeed, this phenomenon is already becoming apparent: as discussed in
Part IV of this book, in the last few years the prices of plain-vanilla options
have been able to be strait-jacketed into their log-normal-rate Black framework

only by increasingly complex ad hoc adjustments.1 As a consequence,
just when the pricing of exotic products had finally been successfully tuned
onto the log-normal-rate wavelength, the prices of the underlying vanilla

instruments have ceased to inhabit the same (Black) world. The brief account
of the developments that brought about this state of affairs is presented below,
and should give a clear indication of the fact that the 'modern' approach is
virtually certain to be anything but the last step in interest-rate derivative pricing.

The reader keen to delve into the quantitative aspects of the pricing can
safely skip these pages. She would miss, however, not only a good story, but
also some perspective useful in appreciating what aspects of today's market
consensus are more likely to be challenged and superseded tomorrow. 

Modern Pricing of Interest-Rate Derivatives: The LIBOR Market Model and Beyond by Riccardo Rebonato