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In 1992,Heath, Jarrow, and Morton(1992) (HJM) developed a general framework
to model the dynamics of the entire forward rate curve in an interest rate market.
The associated valuation approach is based on two main assumptions: the first one
postulates that it is not possible to gain riskless profit (no-arbitrage condition), and
the second one assumes the completeness of the financial market. The HJM model,
or strictly speaking the HJM framework, is a general model environment and incorporates many previously developed models like the model ofHo and Lee(1986),
Vasicek(1977)andHull and White(1990). The general setting mainly suffers from
two disadvantages: first of all the difficulty to apply the model in market practice
and second the extensive computational complexity caused by the high-dimensional
stochastic process of the underlying. The first disadvantage was improved by
the development of the LIBOR market model introduced by Brace, Gatarek, and
Musiela(1997), Jamshidian (1997)andMiltersen and Sandmann(1997), which
combines the general risk-neutral yield curve model with market standards. The
second disadvantage can be improved by restricting the general HJM model to a subset of models with a specific parametrization of the volatility function. The resulting
system of stochastic differential equations (SDE) describing the yield curve dynamics breaks down from a high-dimensional process into a low-dimensional structure
of Markovian processes. This approach was developed byCheyette(1994).
In practice, the Cheyette models usually incorporate several factors to achieve
sufficient flexibility to represent the market state. The structure of the models
supports a canonical construction of multifactor models as extensions of the onefactor model. The model dynamics consider all factors and might become a
high-dimensional SDE as each factorcaptures at least one dimension.
The purpose of this work is to show the application of the class of Cheyette models in practice. Therefore, we focus on thenecessary topics, namely, calibration, valuation and sensitivity analysis. Since we work in a Gaussian HJM framework, some
analytical formulas can be derived and various numerical methods are applicable. In
this book, we present the methods in detail and highlight the improvements due to
the special structure of Cheyette models. Thereby the main focus is not on the latest
implementation details, but we show the impact of the chosen interest rate model.
ix
x Preface
The class of Cheyette models is part of the general Gaussian HJM framework,
which is characterized by a specific parametrization of the volatility incorporating a
wide range of functions. Thus, the corresponding interest rate models become very
flexible and could capture almost all changes to the term structure of interest. This
positive effect is even intensified by including several stochastic factors, which can
be done easily in this setup. Hence, we deal with Cheyette models throughout this
book, which are Gaussian HJM models with time-dependent volatility.
The price of interest rate derivatives can be expressed as the expected value of the
terminal payoff under given model dynamics. Therefore, the computation comes up
as a multidimensional integral, which might be difficult to solve. Consequently, academics and practitioners have become interested in methods to handle these model
dynamics and compute accurate prices of interest rate derivatives in reasonable time.
Thereby, one can exploit the special structure of the class of Cheyette models and
achieve efficient valuation techniques as we see in this work.
In addition to pricing derivatives, practitioners always highlight the hedging of
financial positions and, thus, focus on risk sensitivities, the so-called Greeks. Since
we can derive analytical pricing formulas for bonds and caplets, we are able to
derive analytical Greek formulas for Model- and Market-Greeks in these cases as
well. Furthermore, we demonstrate how to compute Model- and Market-Greeks for
exotic interest rate derivatives numerically.
The book is divided into ten chapters and is organized as follows:
After a review of existing literature in Chap.1, Chap.2introduces to the class
of Cheyette models. First, we present the general HJM framework and imposing a
structure on the volatility function directly leads to the class of Cheyette models.
The setup covers well-known models like the Ho-Lee, the Hull-White or Vasicek
model. We explain the representation of these models in the Cheyette framework,
and finally, we distinguish the approach from alternative specifications of the HJM
framework.
本帖隐藏的内容
Statistical Inference for Financial Engineering.pdf
(2.53 MB)
Interest Rate Derivatives.pdf
(3.29 MB, 需要: 5 个论坛币)
In Chap.3, we present analytical pricing formulas for bonds and caplets/floorlets,
which are valid in the whole class of Cheyette models. The bond pricing formula
has been published byCheyette(1994) in the case of one-factor models only.
Furthermore, we derive explicit pricing formulas for caplets and floorlets based on
some general results ofMusiela and Rutkowski(2005). Applying their results to the
class of Cheyette models leads to these pricing formulas.
One of the most important and challenging parts of pricing interest rate derivatives is the model calibration to a given market state. In Chap.4, we formulate
the calibration problem and present some methodologies to solve it in the class
of Cheyette models. The theoretical aspects are illustrated by the calibration of a
multifactor model, which serves as a reference throughout the book.
Chapter5 treats the valuation of interest rate derivatives via Monte Carlo
simulation. Since we derive the distribution of the state variables, we obtain very fast
and accurate algorithms to value plain-vanilla and exotic derivatives. Implementing
a quasi-Monte Carlo simulation accelerates the evaluation further. Monte Carlo
simulation is a very robust method, and we apply it to value all kinds of plain
vanilla and exotic derivatives. Thus, we use Monte Carlo simulation as a stand-alone
Preface xi
method as well as a reference method todifferent valuation techniques based on
characteristic functions and PDEs. Furthermore, we pick up Monte Carlo simulation
again in Chap.9for the computation of risk sensitivities.
In Chap.6, we develop characteristic functions of Cheyette models as the solution
to a system of certain complex-valued ordinary differential equations. Due to the
structure of Cheyette models, these Ricatti equations can be solved explicitly, and
we end up with a fast and accurate pricing methodology for bonds and caplets.
In Chap.7, we derive the PDE to value interest rate derivatives. In general, the
computation via PDE is difficult, because the valuation PDE might become high
dimensional. So far, the application was limited to two-dimensional problems. We
use the sparse grid technique based on Finite Differences to solve the terminal
value problem numerically. The methodology was applied successfully to highdimensional problems byReisinger(2004)andBungartz and Griebel(2004)inthe
case of stock options. We transfer the technique to interest rate derivatives and make
use of some modifications developed byChiarella and Kang(2012).
In Chap.8, we summarize the applicability, numerical tractability and accuracy
of the previously presented pricing methodologies. Therefore, we analyze and
compare the approaches for plain-vanilla and exotic interest rate derivatives.
Chapter9deals with risk sensitivities, and we derive analytical formulas for
bonds and caplets. In the case where no analytical formulas are applicable, we show
how to compute the required Greeks numerically using Monte Carlo simulation.
Please note that the book is a revised version of my dissertation at Frankfurt
School of Finance & Management in the Centre for Practical Quantitative Finance
with the title ‘Valuation, Calibration and Sensitivity Analysis of Interest Rate
Derivatives in a Multifactor HJM Model with Time Dependent Volatility’.
Frankfurt, Germany Ingo Beyna这是Springer Lecture Notes in Economics and Mathematical Systems
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