Arbitrage with Fractional Brownian Motion

Rogers, L. C. G.

doi:10.1111/1467-9965.00025

Cited by 479 publications

(273 citation statements)

References 7 publications

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“…We refrain from drawing any conclusions regarding the potential implications of our findings for finance, since this is a complex issue outside the scope of this paper. One could, of course, investigate other models that exhibit long range dependence both with and without arbitrage, see for example, Rogers (1997), but our main aim here was to show how our method can be used to compare different models. …”

Section: Exchange Rates: Model Comparisonmentioning

confidence: 99%

Inference on fractal processes using multiresolution approximation

Falconer

Fernández

2007

Biometrika

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We consider Bayesian inference via Markov chain Monte Carlo (MCMC) for a variety of fractal Gaussian processes on the real line. These models have unknown parameters in the covariance matrix, requiring inversion of a new covariance matrix at each MCMC iteration.The processes have no suitable independence properties so this becomes computationally prohibitive. We surmount these difficulties by developing a computational algorithm for likelihood evaluation based on a 'multiresolution approximation' to the original process. The method is computationally very efficient and widely applicable, making likelihood-based inference feasible for large datasets. A simulation study indicates that this approach leads to accurate estimates for underlying parameters in fractal models, including functional parameters in the recently introduced multifractional Brownian motion. We apply the method to a variety of real data sets and illustrate its application to prediction and to model selection.

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Section: Exchange Rates: Model Comparisonmentioning

confidence: 99%

Inference on fractal processes using multiresolution approximation

Falconer

Fernández

2007

Biometrika

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“…( [1], [2], [3], [4], [5] & [6]). The crux of assumption for deriving the Black-Scholes Model (BSM) was that the market is complete, that is, investors do not incur transaction cost in the trading with a risk-free asset (bond with constant return) and a risky asset (stock).Moreover, the price is governed by a geometric Brownian motion with constant rate of return and constant volatility ( [7], [8] & [9]]).…”

Section: Introductionmentioning

confidence: 99%

with Transaction Cost for Pricing European Options and Hedging

Oyediran¹

2016

IJCTS

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Abstract:In this paper models with transaction costs for pricing of European options using mixed fractional Brownian motion (fbm) and Partial differential equation (PDE) model are considered. Investigation on price sensitivity to volatility and formulation of asymptotic strategy for replicating self financing assets are also made. Simulation experiments for the models are run to obtain the call and put prices using fbm with Hurst parameter 1 3H  and the Crank-Nicolson method for the numerical solution to the PDE model.It is found that stock prices increase with time to maturity dates, whereas, the call prices decrease steadily as time approaches maturity dates in conformity with theta hedging strategy.Keywords: Simulation, European option, fractional Brownian motion, stochastic volatility. Dedicated to evergreen memory of Late Professor M. S. Sasey Author's contributionsIn the recent times, development of financial instruments for pricing options are on increasing and the use of mixed fractional Brownian motion (fbm) for simulation of derivatives are not too common. In this paper, simulation experiments are designed and implemented using codes in MATLAB for the models considered. We obtained the call and put prices using fbm with Hurst parameter 1 3H  and the Crank-Nicolson method for the numerical solution to the PDE model using fbm realization. Investigation on price sensitivity to volatility and formulation of asymptotic strategy for replicating self financing assets are also made.

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“…However it was soon realized [12], [13], [14], [15] that this replacement implied the existence of arbitrage. These results might be avoided either by restricting the class of trading strategies [16], introducing transaction costs [17] or replacing pathwise integration by a different type of integration [18] [19].…”

Section: Introductionmentioning

confidence: 99%

No-arbitrage, leverage and completeness in a fractional volatility model

Mendes

Oliveira

Rodrigues

2015

Physica A: Statistical Mechanics and its Applications

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Based on a criterion of mathematical simplicity and consistency with empirical market data, a stochastic volatility model has been obtained with the volatility process driven by fractional noise. Depending on whether the stochasticity generators of log-price and volatility are independent or are the same, two versions of the model are obtained with different leverage behavior. Here, the no-arbitrage and completeness properties of the models are studied.

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Arbitrage with Fractional Brownian Motion

Cited by 479 publications

References 7 publications

Inference on fractal processes using multiresolution approximation

Inference on fractal processes using multiresolution approximation

with Transaction Cost for Pricing European Options and Hedging

No-arbitrage, leverage and completeness in a fractional volatility model

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