Arbitrage with fractional Brownian motion

By Christian Bender, Tommi Sottinen, Esko Valkeila. 2007

343 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/97dc29943a2998b7e3ac0d6df09d0b107bb5ba68

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Citations in the corpus

An Introduction to (Stochastic) Calculus with Respect to Fractional Brownian Motion (2007, background): “These significant properties make fractional Brownian motion a natural candidate as a model for noise in mathematical finance (see Comte and Renault [CR96], Rogers [ Rog97 ], Cheridito, [Che03], and Duncan, [Dun04]); in hydrology (see Hurst, [Hur51]), in communication networks (see, for instance Lel”
Fractional Processes as Models in Stochastic Finance (2010, methodology): “that Rogers [36] used a representation of the fBm starting from −∞ .”
Consistent price systems and face-lifting pricing under transaction costs (2008, background): “However, in a frictionless setting, it turns out that these models lead to arbitrage opportunities [6, 12, 14, 34, 35, 38] and therefore cannot be meaningfully employed for studying optimal investment and derivatives pricing.”
Long range dependence in financial markets (2005, unknown): “FBM is not a semimartingale implied the existence of φ ∈ A verifying (18): Rogers [63] offers examples of such strategies.”
Financial Markets with Memory I: Dynamic Models (2005, unknown): “However this approach is not entirely satisfactory since fractional Brownian motion is not a semimartingale (Liptser and Shiryaev [15], Lin [14], Rogers [17]), and as a result, the market is not arbitrage free (Cutland et al. [6], Rogers [17]).”
Variational solutions for partial differential equations driven by a fractional noise (2006, background): “…of fractional Brownian motion make this stochastic process quite relevant to the mathematical modelling of a host of applications in engineering, the natural sciences and mathematical finance, to name only a few (see, for instance, [10], [12], [13], [36], [37], [46] and their numerous references).”
Modeling stock prices by multifractional Brownian motion: an improved estimation of the pointwise regularity (2013, background): “the cornerstone of all modern financial theory (Rogers 1997).”
The fundamental theorem of asset pricing for continuous processes under small transaction costs (2010, background): “2] implies the existence of arbitrage opportunities in the absence of transaction costs (more precisely: a free lunch with vanishing risk with respect to simple integrands; compare also to [21], [2]).”
Chapter 2 Jump-Diffusion Models for Asset Pricing in Financial Engineering (2007, background): “However, as Rogers (1997) pointed out these models may lead to arbitrage opportunities.”
A decomposition of the bifractional Brownian motion and some applications (2008, background): “As a consequence of the Ergodic Theorem and the scaling property of the fractional Brownian motion, it is easy to show (see, for instance, [12]) that the fractional Brownian motion with Hurst parameter H ∈ (0, 1) has an 1 H -variation equals to CHt, where CH = E(|ξ|H) and ξ is a standard normal rand”
Estimation and pricing under long-memory stochastic volatility (2012, methodology): “In this way we introduce long-memory in our model, but not directly in the returns, as was suggested by Cheridito (2003, [5]), Rogers (1997, [23]) and Sottinen (2001, [25]).”
Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion (2008, methodology): “Rogers, L.C.G. 1997.”
NO ARBITRAGE UNDER TRANSACTION COSTS, WITH FRACTIONAL BROWNIAN MOTION AND BEYOND (2006, methodology): “We refer to Rogers & Williams (2000) for all definitions on Excursion Theory for Markov Processes.”
MULTIFRACTIONAL STOCHASTIC VOLATILITY MODELS (2011, unknown): “The white noise-based, or Wick-Itô, integral with respect to fBm was developed in [31, 9, 12], and applied to option pricing in a fractional Black and Scholes model in [31] (such a fractional Black and Scholes model raises both financial and economical issues, see [11, 14]).”
Volatility in Equilibrium: Asymmetries and Dynamic Dependencies (2009, unknown): “Some care is needed because of subtleties related to possible arbitrage opportunities under long-memory type dependencies (Rogers, 1997).”
Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H∈(0,12) (2005, background): “(1.1) If H = 12, then BH is a two-sided Brownian motion, but forH = 12, {BHt , t 0} is not a semimartingale (for proof in the caseH ∈ (12,1) see Example 4.9.2 in Liptser and Shiryaev [18], for a general proof see Mahes and Sims [19] or Rogers [31]).”
Fractional Brownian motion (2006, background): “Rogers has proved in [40] that the fBmBH has finite 1/H -variation equals to cpT , where cp = E(|BH1 |p).”
Bipower Variation for Gaussian Processes with Stationary Increments (2009, unknown): “An interesting discussion on this point can be found in [11].”