On the Geometry of Interest Rate Models

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in Forthcoming in Springer Lecture Notes in Mathematics AbstractIn this paper, which is a substantial extension of an earlier essay [3], we give an overview of some recent work on the geometric properties of the evolution of the forward rate curve in an arbitrage free bond market. The main problems to be discussed are as follows.• When is a given forward rate model consistent with a given family of forward rate curves?• When can the inherently infinite dimensional forward rate process be realized by means of a Markovian finite dimensional state space model.We consider interest rate models of Heath-Jarrow-Morton type, where the forward rates are driven by a multidimensional Wiener process, and where he volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Within this framework we give necessary and sufficient conditions for consistency, as well as for the existence of a finite dimensional realization, in terms of the forward rate volatilities. We also study stochastic volatility HJM models, and we provide a systematic method for the construction of concrete realizations.JEL: E43, G13.

Arbitrage-Free Interpolation of the Swap Curve

“…A number of authors, starting from the work of Björk and Christenssen (see [1], Filipović [7] and Filipović and Teichmann [8]), have studied this problem within the HJM [10] framework in an infinite-dimensional space by looking at specific classes of functions and asking whether these functions are invariant under the HJM dynamics. They obtain some negative results but later extended the class of functions by using an infinite-dimensional version of the Frobenius theorem.…”

A Black-Scholes Approach to Satisfying the Demand in a Failure-Prone Manufacturing System

“…System theoretic concepts and tools developed by control theoreticians have been and continue to be applied in finance problems (see, e.g., [1]- [5] and the references therein). The main goal of this paper, however, is to apply a finance model and strategy to a systems application.…”

“…Random quantities in this paper will be in boldface. Let x(t) c R be the inventory of the part being produced at time t; c + and cdenote the surplus and shortage costs per unit of production per unit time, respectively; and x + (t) = max(x (t), 0) and x-(t) = max(-x(t), 0) the corresponding levels of inventory surplus and shortage at time t. An objective function to be minimized, for example, is mrin E{ liri ,c++(t) + c--(t)dt} (1) subject to d+x (t) = u(t) -d, x (0) = 0, where d is a constant demand. The inventory level is random due to the random production rates u(t) that depend on the random failure and repair times.…”

“…The optimality of the solution for a single machine and single part with a discounted version of the cost function in (1) appeared in [6]. In [7] the optimal solution was derived for the cost function with a constant demand d in (1), assuming that the times between successive machine failures and repair times follow an exponential distribution. The explicit optimal solution, when the machine is not in the failure state, is…”

Lie Theory: Applications to problems in Mathematical Finance and Economics

“…We also want to recall to Gaspar [8], who studied a general model for the structure of forward prices by using the methodology of Lie algebras given by Björk. Finally, Björk [4,5] studied some applications of Lie algebras to certain economic concepts like constant volatility and constant direction volatility (i.e. considering that the volatility follows a constant vector field as its "direction").…”