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Fractional term structure models: No-arbitrage and consistency. (English) Zbl 1188.91229

Summary: We introduce Heath-Jarrow-Morton (HJM) interest rate models driven by fractional Brownian motions. By using support arguments we prove that the resulting model is arbitrage free under proportional transaction costs in the same spirit of P. Guasoni [Math. Finance 16, No. 3, 569–582 (2006; Zbl 1133.91421)]. In particular, we obtain a drift condition which is similar in nature to the classical HJM no-arbitrage drift restriction. The second part of this paper deals with consistency problems related to the fractional HJM dynamics. We give a fairly complete characterization of finite-dimensional invariant manifolds for HJM models with fractional Brownian motion by means of Nagumo-type conditions. As an application, we investigate consistency of Nelson-Siegel family with respect to Ho-Lee and Hull-White models. It turns out that similar to the Brownian case such a family does not go well with the fractional HJM dynamics with deterministic volatility. In fact, there is no nontrivial fractional interest rate model consistent with the Nelson-Siegel family.

MSC:

91G30 Interest rates, asset pricing, etc. (stochastic models)
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60G18 Self-similar stochastic processes

Citations:

Zbl 1133.91421
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References:

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