Stochastic Hyperbolic Dynamics for Infinite‐dimensional Forward Rates and Option Pricing

Abstract

We model the term-structure modeling of interest rates by considering the forward rate as the solution of a stochastic hyperbolic partial differential equation. First, we study the arbitrage-free model of the term structure and explore the completeness of the market. We then derive results for the pricing of general contingent claims. Finally we obtain an explicit formula for a forward rate cap in the Gaussian framework from the general results.

A note on arbitrage in term structure

“…The result obtained bears similarities to those of Aihara and Bagchi (2005) (in particular to Theorem 3.12), and gives a discrete-time counterpart of the continuous-time Brownian models treated there. See also the section "Diffusion market models" of the recent Rokhlin (2007).…”

Statistical inference for stochastic parabolic equations: a spectral approach

“…Stochastic parabolic equations are used in various economical and physical models, such as the term structure of interest rates for bonds with different maturities (Aihara and Bagchi [8], [9], Cont [16]), the temperature of the top layer of the ocean (Frankignoul [20], Piterbarg and Rozovskiȋ [57]), evolution of the population in time and space (Dawson [17], De [18]), spread of pollutants (Serrano and Adomian [70], Serrano and Unny [71]), etc. Equations of the type (1.1) provide a useful toy model for understanding the possible effect of the infinite number of dimensions and for deriving the bench-mark results about the corresponding estimators.…”

Adaptive Identification for Factor Model with Jump Processes

“…From Appendix, we finally obtain the factor process f (t, x) satisfies the following 1st order hyperbolic equation (see also [9,1]):…”