Economy – Quantitative Finance
Scientific paper
Jun 2004
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2004qufin...4c..34r&link_type=abstract
Quantitative Finance, Volume 4, Issue 3, pp. C34 (2004).
Economy
Quantitative Finance
Scientific paper
Having a mostly equity derivatives background, I was concerned that I would find this book too complex. I therefore started reading it with misgiving. However, the first thing Rebonato does is to give a concise yet thorough history of the evolution of interest-rate derivatives.
Starting with the famous Black model, one assumes that either the forward rate or the swap rate (fixed rate) is log-normal. Using the forward rate would allow us to deal with the `pull to par' phenomenon: the fact that the price of a bond must be par at its maturity. Note that, as Rebonato points out, some were even ignoring this phenomenon and using the Black - Scholes pricing formula. The justification was that in their cases, the bond maturity was far beyond the option maturity. This model has been, and still is, popular due to its simplicity and the fact that it allows us to price many vanilla instruments (caps and swaptions) exactly.
However, as Rebonato keeps telling us, this exact replication is deceiving. Indeed they take place under different probability measures and, therefore, the replications are inconsistent with one another. This means that if you were to price an exotic derivative with multiple call dates or exercise dates, you would not have one internally plausible framework to do so.
Then came the short-rate models such as the Vasicek model or the Cox, Ingersoll and Ross (CIR) model. They treated the instantaneous short interest rate as the driving force of the term structure. This short rate therefore followed a stochastic differential equation (SDE) of its own which was mean reverting. However, these models did not guarantee matching of the market.
An extension to these models, mainly by Hull and White on the one hand and Jamshidian on the other hand, was the following: make the drift term a time-dependent deterministic function that we would resolve in order to match various vanilla prices. But once again, the same issue raises its ugly head. If we are pricing non-path-dependent European securities these models are perfectly fine, but as soon as we are dealing with more complex derivatives, we need consistency between the joint distribution of the forward rates at various points in time.
Heath, Jarrow and Morton (HJM) as well as Brace, Gatarek and Musiela (BGM) then came along with the modern models that give this book its subtitle: the LIBOR market model. Starting with the zero-coupon bond SDE, HJM derive a relationship between the drift and the volatility of the instantaneous forward rate and then obtain the SDE of the instantaneous short rate. This provides us with a non-Markov equation where the history of the rate matters, and which would be difficult (but not impossible!) to implement in a tree but simple to implement via Monte Carlo. As Rebonato points out, some of the fortuitous reasons that the HJM model became popular were the advances made in the field of Monte Carlo simulations. These included the use of low discrepancy numbers.
It is interesting to note that the HJM model is the continuous limit of a discrete model with a rolling measure where the famous forward risk neutrality used in the Black model rolls from one cap date to the next. This is the essence of the modern LIBOR market models.
Many consider the Monte Carlo implementation inappropriate for transactions with American or Bermudan features, where early exercises could occur. However recent regression-based algorithms, and in particular the one by Longstaff and Schwartz, allow the use of Monte Carlo implementations for such contracts with surprising straightforwardness. This makes the usefulness of HJM and BGM models even more significant than before.
Elegantly educational
Rebonato's writing style is probably the most elegant I have ever seen in a quantitative finance book. His ideas are conveyed in a brief and clear manner yet remain pedagogical. The book starts with an introduction on the concepts of interest rates and the underlying mathematical models. The author then describes various popular vanilla instruments as well as their pricing. Following this, the above concept of LIBOR market models is introduced. Rebonato shows how this model could be used to price more complex path-dependent derivatives. The no-arbitrage condition on the drifts is also discussed at this time.
In the second and third parts of the book, Rebonato discusses the concepts of input volatilities as well as correlations. He points to the crucial issues regarding the calibration to the market. Many practical and useful tips are provided at this point. Finally in the fourth part of the book he discusses more advanced issues concerning the `smile' and the ideas of stochastic volatility in this framework. This explains the subtitle of the book, `The LIBOR market model and beyond'.
I thoroughly enjoyed this book since it allowed me to discover a whole new world in a fast and painless fashion. I would therefore recommend it to everyone who has any interest in the fascinating universe of fixed-income derivatives.
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