




ParisPrinceton Lectures on Mathematical Finance 2003 (Lecture Notes in Mathematics) [Paperback]
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Item description for ParisPrinceton Lectures on Mathematical Finance 2003 (Lecture Notes in Mathematics) by José A. Scheinkman Tomasz R. Bielecki...
The ParisPrinceton Lectures in Financial Mathematics, of which this is the second volume, will, on an annual basis, publish cuttingedge research in selfcontained, expository articles from outstanding  established or upcoming!  specialists. The aim is to produce a series of articles that can serve as an introductory reference for research in the field. It arises as a result of frequent exchanges between the finance and financial mathematics groups in Paris and Princeton. This volume presents the following articles: "Hedging of Defaultable Claims" by T. Bielecki, M. Jeanblanc, and M. Rutkowski; "On the Geometry of Interest Rate Models" by T. Bjrk;"Heterogeneous Beliefs, Speculationand Trading in Financial Markets" by J.A. Scheinkman, and W. Xiong. Promise Angels is dedicated to bringing you great books at great prices. Whether you read for entertainment, to learn, or for literacy  you will find what you want at promiseangels.com!
Item Specifications...
Pages 250
Est. Packaging Dimensions: Length: 9.3" Width: 6.1" Height: 0.6" Weight: 0.95 lbs.
Binding Softcover
Release Date Dec 3, 2004
Publisher Springer
ISBN 3540222669 ISBN13 9783540222668

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Reviews  What do customers think about ParisPrinceton Lectures on Mathematical Finance 2003 (Lecture Notes in Mathematics)?
 Suits its purpose May 16, 2008 
Mathematical finance has come a long way since in the last fifty years, as a perusal of the research literature will readily indicate. Since it is a branch of pure mathematics, its emphasis is on rigor and not practical results, and has to be approached with this in mind. It is doubtful that a mastery of some of the results in mathematical finance will make one a better trader, but they definitely will produce better modelers, due to its reliance on economy of thought and conciseness of concepts. This short book contains a few articles that should be of interest to those readers, such as this reviewer, who are interested in the level of abstraction brought about by mathematical finance.
In the article entitled "On the Geometry of Interest Rate Models" the author is interested in HeathJarrowMorton (HJM) interest rate models, which were invented to go beyond shortrate models and which view the instantaneous forward rates as being the fundamental quantities. In an HJM model one can get an arbitragefree framework for the stochastic evolution of the entire yield curve. The instantaneous volatility structures fully specify the forwardrates dynamics. The study of interest rate dynamics then becomes a study of the dynamics of a derived quantity (the instantaneous forward rate). The forward rate is governed by a diffusive stochastic process with the restriction that the drift cannot be freely specified, as it is dependent on the diffusion coefficient. Further, in order to have the dynamics of the forwardrate arbitragefree, the drift must also depend on the drift rates of a collection of zerocoupon bond prices.
The author's goal is take a HJM model with forward rates following a multidimensional Wiener process, and with volatility an arbitrary smooth functional of the present forward rate curve, and find out under what conditions this model can be realized in finite dimensions. The forward rate equation can be viewed as an infinite system of stochastic differential equations, but the author wants to view it as a single equation for an infinitedimensional object: the forward rate curve. The forward rate curve process then evolves in an infinitedimensional Hilbert space of forward rate curves. This Hilbert space (H) is a space of real, analytic functions that have holomorphic extensions to the entire complex plane. The author uses this Hilbert space to ensure that the differential operator (d/dx) with respect to the time to maturity (x) is bounded.
To formulate his main problem in terms of geometry, the author then considers a mapping (G) from a real, finite parameter space (Z) to H, giving a family of forward rate curves, called the forward curve manifold (Gc). An (infinitedimensional) interest rate volatility is then chosen so as to have components belonging to H, these components then forming a vector field on H. Given such a volatility function and the mapping G specifying Gc, the author's "main problem" is find out when Gc is invariant under the action of the forward rate dynamics. Using the Stratonovich calculus and assuming smoothness of the volatility, forward rate, and the mapping z>G(z) he states, but does not prove, a `(local) invariance theorem' for the manifold Gc with respect to the forward rate process. The notion of invariance that the author uses should be very familiar to readers who are acquainted with the notion of invariant manifold coming from the theory of dynamical systems. His notion of invariance is connected with the "consistency" problem for a given submanifold of forward rate curves.
Interestingly, this consistency problem has a practical interpretation in terms of the routine calibration of interest rate models using available market data. This market data is used daily to calibrate the model to observed bond prices with the goal of computing prices of interest rate derivatives. In actual practice of course one can only use a finite number of bonds in order to find the forward rate curve, which is accomplished using some curvefitting procedure. This calibration must be repeated again the next day using fresh data. Studying this recalibration in the framework of mathematical finance requires a rigorous notion of consistency between interest rate models and the family of forward curves. The notion of consistency that the author proposes is that an interest rate model (M) and a family of forward curves Gc is `consistent' if all forward curves that can be produced by M are members of Gc. If not, M and Gc are said to be `inconsistent.' Inconsistency represents the situation where the financial analyst is forced to repeatedly alter the model parameters. The author therefore wants to find necessary and sufficient conditions for the consistency between M and Gc, and this is done using the invariance conditions. He gives an explicit example using the NelsonSiegel family of forward rate curves and analyzes their consistency with the HoLee and HullWhite interest rate models. The author finds that the HoLee and HullWhite models are not consistent with the NelsonSiegel family, and this leads him to search for an interest rate model that is. This is done using the Filipovic state space formalism, and he shows that a simplified version of the NelsonSiegel manifold is consistent with the HoLee model.
Along these same lines, and the author devotes a major portion of his article to this, he wants to characterize those situations where a collection Gc that is finitely parametrized can be found that is consistent with a given model M. More rigorously, given a volatility mapping and an initial forward rate curve, he wants to find under what conditions the corresponding forward rate process has a (generic) finitedimensional realization. He states, but does not prove, that a forward rate process has a finitedimensional realization if and only if there exists an invariant finitedimensional submanifold Gc that contains the initial forward rate curve (r0). Some standard differential geometry is brought in to bear on this question for the case when the underlying space is finitedimensional. He then states the main result, which is an existence theorem for finitedimensional realizations, and involves the finite dimensionality of the Lie bracket of the drift and diffusion terms in a neighborhood of r0. But finding a concrete finitedimensional realization takes more work, and the author gives a brief outline of a procedure for doing so. Several examples of this procedure are given, including the case of deterministic volatility, with the HoLee and HullWhite models being special cases.
These constructions are then generalized to the case where the volatility is stochastic, i.e. where the volatility can depend on a finitedimensional hidden Markov process. The main problems still involve finding necessary and sufficient conditions for the existence of a finitedimensional realization of a given stochastic volatility model, and then constructing it explicitly. The theory of Lie algebras is again used for this purpose, and the condition of finitedimensionality of the Lie bracket of drift and volatility terms around the initial rate curve guarantees the existence of a (generic) finitedimensional realization. This result is generalized to the case of finitedimensional realizations of `orthogonal noise models', i.e. models where the rate and scalar evolutions are driven by independent Wiener processes. The explicit construction of realizations outlined before is then adapted to the stochastic volatility framework.
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