Exact and numerical solution of Black–Scholes matrix equation

López, Juan Carlos Cortés; Jódar, L.; Sala, Rosa Martínez; Sevilla‐Peris, Pablo

doi:10.1016/j.amc.2003.11.018

Cited by 6 publications

(4 citation statements)

References 7 publications

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“…In an effort to manage these inconsistencies, advances have been made in both the model itself and its respective solution methods -both analytical (Harper, 1994;Forsyth et al, 1999;Jódar et al, 2005;Rodrigo and Mamon, 2006;Bohner and Zheng, 2009;Edeki et al, 2015;Shin and Kim, 2016) and numerical (Cortés et al, 2005;Company et al, 2008;Ankudinova and Ehrhardt, 2008;Černá et al, 2016). However, in spite of such concerted efforts having been employed to improve the performance of the BSM, multiple studies still show that these techniques have not been able to very well capture more realistic aspects that characterize day-to-day transactions such as the presence of transaction costs, high volatility, illiquid markets and large investor preference.…”

Section: Introductionmentioning

confidence: 99%

A Hybrid Artificial Neural Network Model for Option Pricing

Simiyu¹,

Waititu²,

Akinyi³

2019

Journal of Mathematics and Statistics

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In the absence of a well-defined input selection technique associated with the pure ANN models, Option pricing using pure ANN models while relaxing the assumption of constant volatility remains a challenge. The conservative drill espoused has been to make allowances for a large number of input lags with the confidence that the ability of ANN to integrate suppleness and redundancy generates a more robust model. This is to say that the nonexistence of input selection criteria notwithstanding, the models have been developed without due consideration to the effect that the choice of input selection technique would have on model complexity, learning difficulty and performance measures. In this study, we deviate from the conventional techniques applied by the pure ANN option price models and adopt the hybrid model in which the volatility component is handled using some celebrated time series models, with specificity to the ANN-GJR-GARCH model-a hybrid of the ANN and a time series hybrid. The hybrid ANN option pricing model is then framed and tested with the forecasts of the ANN-GJR-GARCH model as a volatility input alongside two other inputstime to maturity and moneyness. Finally, we compare the performance of the hybrid model developed with that of a pure ANN model. Results indicate that the hybrid model outperforms the pure ANN model not only in forecasting but also in the training time and model complexity.

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Section: Introductionmentioning

confidence: 99%

A Hybrid Artificial Neural Network Model for Option Pricing

Simiyu¹,

Waititu²,

Akinyi³

2019

Journal of Mathematics and Statistics

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“…On the other hand, numerous advances have similarly been made on the numerical solution including the use of the: Mellin transform (Cortés et al, 2005), semidiscretization technique (Company et al, 2008), Crank-Niolcon method (Ankudinova and Ehrhardt, 2008), Cubic spline wavelets and multi-wavelet bases method (Černá et al, 2016), the two-step backward differentiation formula in the temporal discretization and a Higher-Order Difference approximation with Identity Expansion (HODIE) scheme (Rao, 2016), among others. A detailed review on the use of these techniques can be found on Al Saedi and Tularam (2018).…”

Section: Introductionmentioning

confidence: 99%

Comparative Analysis of the Artificial Neural Networks Options Pricing Model Under Constant and Time-Variant Volatilities

Simiyu¹,

Waititu²,

Akinyi³

2019

Journal of Mathematics and Statistics

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Option pricing using artificial neural networks (ANN) model while relaxing the assumption of constant volatility still remains a challenge. The conventional practice for pure ANN models has been to either model volatility using the very ANN model and have the model output fed as an input to the ANN option pricing model, or to make allowances for a large number of lags directly as inputs to the option pricing model with the belief that the ability of ANN to incorporate flexibility and redundancy creates a more robust model. This has been done in spite of a well-known fact-that financial time series data harbors a set of characteristics such as volatility clustering, leptokurtosis and leverage effects-features that ANNs in their pure forms have proved inadequate in capturing. Consequently, this study sought to follow the conventional methods employed by other studies and developed two pure ANN option pricing models-one with constant volatility and the other while violating the assumption of constant volatility with an aim of establishing whether significant differences exist in the outputs of the two models. The intraday data for the AAPL stock option for the period between December 2016 and March 2017 with 56,238 data points was used in validating the developed models. Results indicate that the ANN model (with varying volatility) makes better predictions than the model with constant volatility. However, the difference between the performance of the two models is not significant at 0.05 level of significance.

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“…Yet, little attention has been paid to systems of convection-diffusion-reaction equations, although they also arise naturally in several branches of science such as biomechanics, 26 combustion, 27 computer science, 28 ecology, 29 economy, 30 epidemiology, 31 finance, 32 groundwater pollution, 33 heat transfer, 34 neuroscience, 35 physiology, 36 seepage flow, 37 solid mechanics, 38 and turbulence. 39 The reason for this immaturity is the lack of a maximum principle when going from a single transport equation toward systems of coupled equations in the most general form.…”

Section: Introductionmentioning

confidence: 99%

A stabilization technique for coupled convection‐diffusion‐reaction equations

Hernández

Massart

Peerlings

et al. 2018

Numerical Meth Engineering

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Partial differential equations having diffusive, convective, and reactive terms appear in the modeling of a large variety of processes in several branches of science. Often, several species or components interact with each other, rendering strongly coupled systems of convection-diffusion-reaction equations. Exact solutions are available in extremely few cases lacking practical interest due to the simplifications made to render such equations amenable by analytical tools. Then, numerical approximation remains the best strategy for solving these problems. The properties of these systems of equations, particularly the lack of sufficient physical diffusion, cause most traditional numerical methods to fail, with the appearance of violent and nonphysical oscillations, even for the single equation case. For systems of equations, the situation is even harder due to the lack of fundamental principles guiding numerical discretization. Therefore, strategies must be developed in order to obtain physically meaningful and numerically stable approximations. Such stabilization techniques have been extensively developed for the single equation case in contrast to the multiple equations case. This paper presents a perturbation-based stabilization technique for coupled systems of one-dimensional convection-diffusion-reaction equations. Its characteristics are discussed, providing evidence of its versatility and effectiveness through a thorough assessment. KEYWORDS convection-diffusion equation, differential equations, finite element methods, stability Int J Numer Methods Eng. 2018;116:43-65.wileyonlinelibrary.com/journal/nme

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Exact and numerical solution of Black–Scholes matrix equation

Cited by 6 publications

References 7 publications

A Hybrid Artificial Neural Network Model for Option Pricing

A Hybrid Artificial Neural Network Model for Option Pricing

Comparative Analysis of the Artificial Neural Networks Options Pricing Model Under Constant and Time-Variant Volatilities

A stabilization technique for coupled convection‐diffusion‐reaction equations

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