When Is the Short Rate Markovian?

Carverhill, Andrew

doi:10.1111/j.1467-9965.1994.tb00060.x

Cited by 137 publications

(53 citation statements)

References 10 publications

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“…Now we formulate and prove the criterion for the short rate to be Markovian. Actually, we extend the results of [3], [8], [15] and [11] to the case where the forward rate is driven by a fractional Brownian motion. …”

Section: Markovian Short Ratesmentioning

confidence: 99%

“…We continue with formulating two assertions being extensions of the corresponding results in [3], [8] and [15] (see also [11]). Lemma 4.1.…”

Section: Markovian Short Ratesmentioning

confidence: 99%

“…This question for the bond market model driven by a Wiener process was studied by Carverhill [3]. It was proved that the short rate process is Markovian within the Heath-Jarrow-Morton framework with a deterministic volatility function if and only if this volatility factorizes into a product of two functions only depending on the current time and maturity time, respectively.…”

Section: Introductionmentioning

confidence: 99%

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On arbitrage and Markovian short rates in fractional bond markets

Gapeev

2004

Statistics & Probability Letters

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Statistics and Probability Letters (2004) 70(3) (211-222)We study a bond market model and related term structure of interest rates driven by a fractional Brownian motion with self-similarity parameter H ∈ (1/2, 1). We present a criterion on the deterministic forward rate volatility under which the short rate process is Markovian and construct an admissible self-financing portfolio realizing an arbitrage opportunity.

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Section: Markovian Short Ratesmentioning

confidence: 99%

“…We continue with formulating two assertions being extensions of the corresponding results in [3], [8] and [15] (see also [11]). Lemma 4.1.…”

Section: Markovian Short Ratesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On arbitrage and Markovian short rates in fractional bond markets

Gapeev

2004

Statistics & Probability Letters

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“…(5.3) vanishes, which is the case iff the η i (t, T ) are constant in t for all i and T . All in all, this shows that a Gaussian HJM model allows a description as an M dimensional Markov functional model if and only if the volatility vectors of bonds can be written in the form This generalizes the result of [Ca94]. Given these relations, one has the following explicit form for Eq.…”

Section: Gaussian Markov Functional Modelsmentioning

confidence: 92%

On the structure of Gaussian pricing models and Gaussian Markov functional models

Neumann¹

2007

Quantitative Finance

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C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a Software ENgineeringOn the structure of Gaussian pricing models and Gaussian Markov functional models ABSTRACT This article investigates the structure of Gaussian pricing models (that is, models in which future returns are normally distributed). Although much is already known about such models, this article differs in that it is based on a formulation of the theory of derivative pricing in which numeraire invariance is manifest, extending earlier work on this subject. The focus on symmetry properties leads to a deeper insight in the structure of these models. The central idea is the construction of the most general class of derived Gaussian tradables given a set of underlying tradables which are themselves Gaussian. These derived tradables are called "generalized power tradables" and they correspond to portfolios in which the fraction of total value invested in each asset is a deterministic function of time. Applying this theory to Gaussian HJM models, the new tradables give an explicit description of the interdependence of bonds implicit in such models. Given this structure, a simple condition is derived under which these models allow a description in terms of an M -factor Markov functional model, as introduced by Hunt, Kennedy and Pelsser. Finally, conditions are derived under which these Gaussian Markov functional models are time homogeneous (bond volatilities depending only on the time to maturity). This result is linked to recent results by Björk and Gombani.2000 Mathematics Subject Classification: 58J35, 58J70, 60H10, 91B24

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“…The application of such a drift approximation leads to gains in efficiency if we assume the instantaneous volatility structure of the market model is of a separable form, since this allows the market model to be approximated by a model driven by a low-dimensional Markov process (following ; see Section 2.3). For one of the first references on separability see Carverhill [1994]. For a one-factor LIBOR market model we say the model is separable if the instantaneous volatility function of each LIBOR at any time t is proportional to a common instantaneous volatility function σ t .…”

Section: Introductionmentioning

confidence: 99%

A Comparison of Markov–Functional and Market Models

Bennett¹,

Kennedy²

2005

JOD

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The LIBOR Markov-functional model is an efficient arbitrage-free pricing model suitable for callable interest rate derivatives. We demonstrate that the one-dimensional LIBOR Markov-functional model and the separable onefactor LIBOR market model are very similar. Consequently, the intuition behind the familiar SDE formulation of the LIBOR market model may be applied to the LIBOR Markov-functional model.The application of a drift approximation to a separable one-factor LIBOR market model results in an approximating model driven by a one-dimensional Markov process, permitting efficient implementation. For a given parameterisation of the driving process, we find the distributional structure of this model and the corresponding Markov-functional model are numerically virtually indistinguishable for short maturity tenor structures over a wide variety of market conditions, and both are very similar to the market model. A theoretical uniqueness result shows that any accurate approximation to a separable market model that reduces to a function of the driving process is effectively an approximation to the analogous Markov-functional model. Therefore, our conclusions are not restricted to our particular choice of driving process. Minor differences are observed for longer maturities, nevertheless the models remain qualitatively similar. These differences do not have a large impact on Bermudan swaption prices.Under stress-testing, the LIBOR Markov-functional and separable LI-BOR market models continue to exhibit similar behaviour and Bermudan prices under these models remain comparable. However, the drift approximation model now appears to admit arbitrage that is practically significant. In this situation, we argue the Markov-functional model is a more appropriate choice for pricing.

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When Is the Short Rate Markovian?

Cited by 137 publications

References 10 publications

On arbitrage and Markovian short rates in fractional bond markets

On arbitrage and Markovian short rates in fractional bond markets

On the structure of Gaussian pricing models and Gaussian Markov functional models

A Comparison of Markov–Functional and Market Models

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