On the Representation of General Interest Rate Models as Square Integrable Wiener Functionals

Details On the Representation of General Interest Rate Models as Square Integrable Wiener Functionals On the Representation of General Interest Rate Models as Square Integrable Wiener Functionals

2011-07-17

arxiv.org/abs/1107.3293v1

Economy

Quantitative Finance

General Finance

17 pages

Scientific paper

In the setting proposed by Hughston & Rafailidis (2005) we consider general interest rate models in the case of a Brownian market information filtration $(\mathcal{F}_t)_{t\geq0}$. Let $X$ be a square-integrable $\mathcal{F}_\infty$-measurable random variable, and assume the non-degeneracy condition that for all $t<\infty$ the random variable $X$ is not $\mathcal{F}_t$-measurable. Let ${\sigma_t}$ denote the integrand appearing in the representation of $X$ as a stochastic integral, write $\pi_t$ for the conditional variance of $X$ at time $t$, and set $r_t = \sigma^2_t / \pi_t$. Then $\pi_t$ is a potential, and as such can act as a model for a pricing kernel (or state price density), where $r_t$ is the associated interest rate. Under the stated assumptions, we prove the following: (a) that the money market account process defined by $B_t = \exp (\int_0^t r_s \,ds)$ is finite almost surely at all finite times; and (b) that the product of the money-market account and the pricing kernel is a local martingale, and is a martingale provided a certain integrability condition is satisfied. The fact that a martingale is thus obtained shows that from any non-degenerate element of Wiener space satisfying the integrability condition we can construct an associated interest-rate model. The model thereby constructed is valid over an infinite time horizon, with strictly positive interest, and satisfies the relevant intertemporal relations associated with the absence of arbitrage. The results thus stated pave the way for the use of Wiener chaos methods in interest rate modelling, since any such square-integrable Wiener functional admits a chaos expansion, the individual terms of which can be regarded as parametric degrees of freedom in the associated interest rate model to be fixed by calibration to appropriately liquid sectors of the interest rate derivatives markets.

Affiliated with

Also associated with

No associations

Rating

On the Representation of General Interest Rate Models as Square Integrable Wiener Functionals does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the Representation of General Interest Rate Models as Square Integrable Wiener Functionals, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the Representation of General Interest Rate Models as Square Integrable Wiener Functionals will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-479038