Asymptotic proportion of arbitrage points in fractional binary markets

Cordero, Fernando; Klein, Irene; Perez-Ostafe, Lavinia

doi:10.1016/j.spa.2015.09.002

Cited by 3 publications

(10 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We briefly recall some estimations obtained in [4] for the quantities involved in the definition of the fractional binary markets, i.e., a…”

Section: Some Useful Estimationsmentioning

confidence: 99%

“…Sottinen showed in [16] that the fractional binary markets without friction admit arbitrage and such an opportunity is explicitly constructed using the path information starting from time zero. Moreover, in the recent work [4], it was proved that the asymptotic proportion of arbitrage points in the fractional binary markets is strictly positive and a characterization of that quantity, in terms of the Hurst parameter, is provided.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the concept of asymptotic arbitrage is recalled. We end this part with a brief presentation of fractional binary markets and recall some estimates obtained in [4] for the involved quantities. In Section 3, Section 4 and Section 5 are concentrated the main results.…”

Section: Introductionmentioning

confidence: 99%

“…For the sake of simplicity, we make use of the results obtained in [4] to provide here an alternative, but equivalent definition of the fractional binary market. First, we consider a sequence of i.i.d.…”

Section: Fractional Binary Markets Sottinen Introduces Inmentioning

confidence: 99%

See 3 more Smart Citations

Critical Transaction Costs and 1-Step Asymptotic Arbitrage in Fractional Binary Markets

Cordero

Perez-Ostafe

2015

Int. J. Theor. Appl. Finan.

Self Cite

View full text Add to dashboard Cite

Abstract. We study the arbitrage opportunities in the presence of transaction costs in a sequence of binary markets approximating the fractional BlackScholes model. This approximating sequence was constructed by Sottinen and named fractional binary markets. Since, in the frictionless case, these markets admit arbitrage, we aim to determine the size of the transaction costs needed to eliminate the arbitrage from these models. To gain more insight, we first consider only 1-step trading strategies and we prove that arbitrage opportunities appear when the transaction costs are of order o(1/ √ N ). Next, we characterize the asymptotic behavior of the smallest transaction costs λ, called "critical" transaction costs, starting from which the arbitrage disappears. Since the fractional Black-Scholes model is arbitrage-free under arbitrarily small transaction costs, one could expect that λ converges to one. We explain this apparent contradiction and conclude that it is appropriate to see the fractional binary markets as a large financial market and to study its asymptotic arbitrage opportunities. Finally, we construct a 1-step asymptotic arbitrage in this large market when the transaction costs are of order o(1/N H ), whereas for constant transaction costs, we prove that no such opportunity exists.

show abstract

“…We briefly recall some estimations obtained in [4] for the quantities involved in the definition of the fractional binary markets, i.e., a…”

Section: Some Useful Estimationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Fractional Binary Markets Sottinen Introduces Inmentioning

confidence: 99%

See 2 more Smart Citations

Critical Transaction Costs and 1-Step Asymptotic Arbitrage in Fractional Binary Markets

Cordero

Perez-Ostafe

2015

Int. J. Theor. Appl. Finan.

Self Cite

View full text Add to dashboard Cite

show abstract

“…From this point on, he constructs a discrete model, called "fractional binary market", approximating (2.1). Based on the results in [8], we provide here a simplified, but equivalent, presentation of these binary models.…”

Section: Fractional Binary Marketsmentioning

confidence: 99%

Strong asymptotic arbitrage in the large fractional binary market

Cordero

Perez-Ostafe

2015

Math Finan Econ

Self Cite

View full text Add to dashboard Cite

We study, from the perspective of large financial markets, the asymptotic arbitrage (AA) opportunities in a sequence of binary markets approximating the fractional Black-Scholes model. This approximating sequence was introduced by Sottinen and named fractional binary market. The large financial market under consideration does not satisfy the standard assumptions of the theory of AA. For this reason, we follow a constructive approach to show first that a strong AA (SAA) exists in the frictionless case. Indeed, with the help of an appropriate version of the law of large numbers and a stopping time procedure, we construct a sequence of self-financing trading strategies leading to the desired result. Next, we introduce, in each small market, proportional transaction costs, and we show that a slight modification of the previous trading strategies leads to a SAA when the transaction costs converge fast enough to 0.

show abstract

General Conditions of Weak Convergence of Discrete-Time Multiplicative Scheme to Asset Price with Memory

Mishura

Ralchenko

Shklyar

2020

Risks

View full text Add to dashboard Cite

We present general conditions for the weak convergence of a discrete-time additive scheme to a stochastic process with memory in the space D [ 0 , T ] . Then we investigate the convergence of the related multiplicative scheme to a process that can be interpreted as an asset price with memory. As an example, we study an additive scheme that converges to fractional Brownian motion, which is based on the Cholesky decomposition of its covariance matrix. The second example is a scheme converging to the Riemann–Liouville fractional Brownian motion. The multiplicative counterparts for these two schemes are also considered. As an auxiliary result of independent interest, we obtain sufficient conditions for monotonicity along diagonals in the Cholesky decomposition of the covariance matrix of a stationary Gaussian process.

show abstract

Asymptotic proportion of arbitrage points in fractional binary markets

Cited by 3 publications

References 17 publications

Critical Transaction Costs and 1-Step Asymptotic Arbitrage in Fractional Binary Markets

Critical Transaction Costs and 1-Step Asymptotic Arbitrage in Fractional Binary Markets

Strong asymptotic arbitrage in the large fractional binary market

General Conditions of Weak Convergence of Discrete-Time Multiplicative Scheme to Asset Price with Memory

Contact Info

Product

Resources

About