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Unformatted text preview: Term-Structure Models: a Review Riccardo Rebonato QUARC ( QU Antitative R esearch C entre) - Royal Bank of Scotland Oxford University - OCIAM February 25, 2003 1 Introduction 1.1 Justification for Another Review of Interest-Rate Mod- els The topic of term-structure modelling for derivatives pricing has been covered in recent years, at least in book form (see, eg, [Rebonato (1998)], [James and Webber (2000)], [Hughston (2000)], [Brigo and Mercurio (2001)]). One obvious justification for updating these works is that the modelling of interest rates is still rapidly evolv- ing. There are, however, deeper reasons why a fresh examination of the current state of research in this area deserves attention. The first is that recently a qualitatively new dimension has been added to the modelling complexity, because of the appearance of pronounced and complex smiles in the implied volatility surfaces of caplets and swaptions. Sure enough, smiles of sorts were produced by some of the early modelling approaches (eg, by the [Hull and White (1993)] model), but these were at the time obtained as an ’afterthought’, and by and large regarded as an unpleasant feature to be ignored or excused away. Currently, the modelling of smiles is possibly the most active area of research in interest-rate derivatives pricing, and a sufficiently substantial body of work has accumulated to warrant a review of its achievements and of the problems still to be tackled. The second reason why a review of term structure modelling is timely is connected with two related market developments, namely the compression in profit margins for the well-established complex products and the simultaneous increase in complexity of many of the new-generation interest-rate derivatives. These joint market developments are putting under severe strain some of the basic underpinning theoretical concepts of the classic pricing approach, which ultimately derives from the [Black and Scholes (1973)] (BS in the following) paradigm (although, often, via the [Harrison and Pliska (1981)] route). In par- ticular, given the robustness of the BS model, market completeness and exact payoff replicability (both reasonable but inaccurate descriptions of financial re- ality) can constitute a valid framework when one is pricing, say, a simple cap 1 (and the profit margin is generous).However, the wisdom of adopting the same pricing approach (which implies irrelevance of attitudes towards risk) should at least be questioned if the trader is pricing a product with payoff depending on, say, the joint realizations of two currencies and of a non-particularly liquid currency pair over a 30-year period 1 . More generally, for most problems in asset pricing a mixture of the relative and absolute approaches (see, eg, [Cochrane (2000)]) tends to be more prof- itable. Traditionally, derivatives pricing has been the area of asset pricing where the relative approach has been most successful, and its conceptual corollaries (such as irrelevance of attitudes towards risk) have therefore been extended to...
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- Fall '08
- Volatility, Mathematical finance