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Paris-Princeton Lectures on Mathematical Finance 2003 (Lecture Notes in Mathematics) Paperback – December 3, 2004
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In the article entitled "On the Geometry of Interest Rate Models" the author is interested in Heath-Jarrow-Morton (HJM) interest rate models, which were invented to go beyond short-rate models and which view the instantaneous forward rates as being the fundamental quantities. In an HJM model one can get an arbitrage-free framework for the stochastic evolution of the entire yield curve. The instantaneous volatility structures fully specify the forward-rates 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 forward-rate arbitrage-free, the drift must also depend on the drift rates of a collection of zero-coupon bond prices.
The author's goal is take a HJM model with forward rates following a multi-dimensional 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 infinite-dimensional object: the forward rate curve. The forward rate curve process then evolves in an infinite-dimensional 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 (infinite-dimensional) 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 curve-fitting 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 Nelson-Siegel family of forward rate curves and analyzes their consistency with the Ho-Lee and Hull-White interest rate models. The author finds that the Ho-Lee and Hull-White models are not consistent with the Nelson-Siegel 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 Nelson-Siegel manifold is consistent with the Ho-Lee 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) finite-dimensional realization. He states, but does not prove, that a forward rate process has a finite-dimensional realization if and only if there exists an invariant finite-dimensional 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 finite-dimensional. He then states the main result, which is an existence theorem for finite-dimensional 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 finite-dimensional 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 Ho-Lee and Hull-White 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 finite-dimensional hidden Markov process. The main problems still involve finding necessary and sufficient conditions for the existence of a finite-dimensional 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 finite-dimensionality of the Lie bracket of drift and volatility terms around the initial rate curve guarantees the existence of a (generic) finite-dimensional realization. This result is generalized to the case of finite-dimensional 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.