Applying the Barycentric Jacobi Spectral Method to Price Options with Transaction Costs in a Fractional Black-Scholes Framework

Nteumagne, Bienvenue Feugang; Pindza, Edson; Maré, Eben

doi:10.4236/jmf.2014.41004

Cited by 4 publications

(4 citation statements)

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“…In most of these applications, the barycentric interpolation at the polynomial zeros is at the heart of the proposed methods. Without aiming at being exhaustive, apart from the applications mentioned in [4] for the particular case of Gauss-Legendre, we add the spectral Gegenbauer method of [10] (where hundreds of nodes and weights are used), the Jacobi pseudospectral methods applied to optimal control problems of reference [35] and fractional differential equations of [36], for which the use of asymptotic methods allows for an application of the method for high accuracies (see Remark 14 of [35] and Remark 4.5 of [36]) or the Jacobi spectral methods for the Black-Scholes model of option pricing [28] which uses hundreds of nodes. Finally we recall that for the computation of the barycentric interpolation formulas, it is important to rely on methods which are efficient for large or moderately large degrees, both for Lagrange [40] as well as for Hermite-Fejér [43] interpolation.…”

Section: 4mentioning

confidence: 99%

Noniterative Computation of Gauss--Jacobi Quadrature

Gil¹,

Segura²,

Temme³

2019

SIAM J. Sci. Comput.

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Asymptotic approximations to the zeros of Jacobi polynomials are given, with methods to obtain the coefficients in the expansions. These approximations can be used as standalone methods for the non-iterative computation of the nodes of Gauss-Jacobi quadratures of high degree (n ≥ 100). We also provide asymptotic approximations for functions related to the first order derivative of Jacobi polynomials which are used for computing the weights of the Gauss-Jacobi quadrature. The performance of the asymptotic approximations is illustrated with numerical examples, and it is shown that nearly double precision relative accuracy is obtained both for the nodes and the weights when n ≥ 100 and −1 < α, β ≤ 5. For smaller degrees the approximations are also useful as they provide 10 −12 relative accuracy for the nodes when n ≥ 20, and just one Newton step would be sufficient to guarantee double precision accuracy in that cases.

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Section: 4mentioning

confidence: 99%

Noniterative Computation of Gauss--Jacobi Quadrature

Gil¹,

Segura²,

Temme³

2019

SIAM J. Sci. Comput.

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“…In the discrete binary setting, transaction costs are imposed by trading on selected nodes of the fractional binary trees, see Rostek (2009). In continuous time, the transaction costs are proportional to the value of the transactions in the underlying stock; see, for example, Biagini et al (2008) and Nteumagné, Pindza, and Maré (2014). Our approach is different in that we assume that the arbitrage tax is determined by the rate of transaction volume acceleration of the hedging portfolio which we refer to as the velocity of hedging.…”

Section: 𝑛−1 𝑘=0mentioning

confidence: 99%

Pricing Derivatives in Hermite Markets

Stoyanov

Rachev

Mittnik

et al. 2019

Int. J. Theor. Appl. Finan.

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We present a new framework for Hermite fractional financial markets, generalizing the fractional Brownian motion and fractional Rosenblatt markets. Considering pure and mixed Hermite markets, we introduce a strategy-specific arbitrage tax on the rate of transaction volume acceleration of the hedging portfolio as the prices of risky assets change, allowing us to transform Hermite markets with arbitrage opportunities to markets with no arbitrage opportunities within the class of Markov trading strategies.

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“…For a given differential equation, the set of all its infinitesimal symmetries form an infinitesimal group (Lie algebra in the modern terminology) [19]. Perhaps the most useful property of a symmetry of a differential equation is that it transforms a solution into another one.…”

Section: Introductionmentioning

confidence: 99%

“…We therefore exploit the results by Leland [11] for the derivation of our model. Earlier in 2014, we achieved an accurate numerical solution of our model with exponential convergence ( [19]) by applying spectral methods to analyze the problem. In this paper, we use Lie symmetries to determine the exact solution, which sets us a step closer to the calibration of our model.…”

Section: Introductionmentioning

confidence: 99%

Symmetry Analysis of Options Pricing with Transactions Costs Driven by Fractional Brownian Noises

Bf¹,

Pindza²,

Maré³

2017

J Appl Computat Math

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We provide a closed-form solution for the European and Asian option pricing models when the source of randomness is a fractional Brownian motion as opposed to the geometric Brownian motion. In addition to the source of randomness, transaction costs are considered to be non-negligible. For the case of the European option, proportional transaction costs hide in the volatility and do not change the form of the model. The construction of the solution is based on the symmetries of the model. The model for Asian options has an additional parameter that makes the volatility time-dependent, which complicates the solution process. However, we are still able to obtain solutions using Lie symmetry methods. Citation: Nteumagn BF, Pindza E, Mare E (2017) Symmetry Analysis of Options Pricing with Transactions Costs Driven by Fractional BrownianNoises. J Appl Computat Math 6: 356. doi: 10.4172/2168-9679.1000356 Page 2 of 8Volume 6 of the option pricing equation relies heavily on the random process followed by the underlying stock price. In the case of the celebrated Black-Scholes equation, one assumes that the stock follows the classical geometric Brownian motion [5,24]. This process assumes independent increments, which makes it different from the fractional Brownian motion (fBm) where there is a serial correlation in the increments. This correlation allows for predictability in the model. The similarity between the two processes is the fact that they are both geometric.It has been shown [8] that random changes can be accurately measured by the use of a parameter known as the Hurst parameter. The construction of a dam on the Nile river in the early 1950s saw the birth of the Hurst exponent. The measure of the water levels of increase or decrease was known to follow a random walk. In other words, the next level of increase or decrease is independent of the previous one. However, Edwin Hurst, hydrologist in the Nile dam project, showed that the long term memory of the system indicates an autocorrelation of the time series of water levels, and the rate at which these decrease as the lag between pairs of values increase [8]. The movement of stock price is similar to this model. We therefore exploit the results by Leland [11] for the derivation of our model. Earlier in 2014, we achieved an accurate numerical solution of our model with exponential convergence ([19]) by applying spectral methods to analyze the problem. In this paper, we use Lie symmetries to determine the exact solution, which sets us a step closer to the calibration of our model.We first give some preliminaries on Lie symmetries of differential equations in Section 2. Secondly, we determine the symmetries of the European and Asian option pricing models and obtain their analytical solutions in Section 3. In Section 4 we apply the Lie symmetry technique to reduce the PDEs and provide solutions. Finally, we provide concluding remarks in Section 5. Lie Symmetries and Equivalence TransformationsOur aim in this section is to provide a brief introduction to ...

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Applying the Barycentric Jacobi Spectral Method to Price Options with Transaction Costs in a Fractional Black-Scholes Framework

Cited by 4 publications

References 19 publications

Noniterative Computation of Gauss--Jacobi Quadrature

Noniterative Computation of Gauss--Jacobi Quadrature

Pricing Derivatives in Hermite Markets

Symmetry Analysis of Options Pricing with Transactions Costs Driven by Fractional Brownian Noises

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