scholarly journals Pricing, Risk and Volatility in Subordinated Market Models

Risks ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 124
Author(s):  
Jean-Philippe Aguilar ◽  
Justin Lars Kirkby ◽  
Jan Korbel
Keyword(s):  
Long Memory ◽  
Portfolio Performance ◽  
Market Models ◽  
Volatility Clustering ◽  
Fourier Techniques ◽  
Delta Hedging ◽  
Analytical Formulas ◽  
Time Changes ◽  
Complex Phenomena ◽  
Pricing Risk

We consider several market models, where time is subordinated to a stochastic process. These models are based on various time changes in the Lévy processes driving asset returns, or on fractional extensions of the diffusion equation; they were introduced to capture complex phenomena such as volatility clustering or long memory. After recalling recent results on option pricing in subordinated market models, we establish several analytical formulas for market sensitivities and portfolio performance in this class of models, and discuss some useful approximations when options are not far from the money. We also provide some tools for volatility modelling and delta hedging, as well as comparisons with numerical Fourier techniques.

scholarly journals The Fractional View of Complexity

Entropy ◽  
2019 ◽  
Vol 21 (12) ◽  
pp. 1217
Author(s):  
António M. Lopes ◽  
J.A. Tenreiro Machado
Keyword(s):  
Differential Equations ◽  
Spatial Heterogeneity ◽  
Fractional Differential Equations ◽  
Fractal Analysis ◽  
Long Memory ◽  
Fractional Differential ◽  
Complex Phenomena

Fractal analysis and fractional differential equations have been proven as useful tools for describing the dynamics of complex phenomena characterized by long memory and spatial heterogeneity [...]

scholarly journals A Bertrand Duopoly Game with Long-Memory Effects

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
Baogui Xin ◽  
Fangrui Cao ◽  
Wei Peng ◽  
Abdelalim A. Elsadany
Keyword(s):  
Fractional Order ◽  
Long Memory ◽  
Nash Equilibria ◽  
Numerical Solutions ◽  
Short Term Memory ◽  
Memory Effects ◽  
Short Term ◽  
Bifurcation And Chaos ◽  
Long Short Term Memory ◽  
Complex Phenomena

Reconsidering the Bertrand duopoly game based on the concept of long short-term memory, we construct a fractional-order Bertrand duopoly game by extending the integer-order game to its corresponding fractional-order form. We build such a Bertrand duopoly game, in which both players can make their decisions with long-memory effects. Then, we investigate its Nash equilibria, local stability, and numerical solutions. Using the bifurcation diagram, the phase portrait, time series, and the 0-1 test for chaos, we numerically validate these results and illustrate its complex phenomena, such as bifurcation and chaos.

scholarly journals News-Driven Expectations and Volatility Clustering

2020 ◽  
Vol 13 (1) ◽  
pp. 17
Author(s):  
Sabiou M. Inoua
Keyword(s):  
Long Memory ◽  
Rational Expectations ◽  
Linear Models ◽  
Autoregressive Process ◽  
Volatility Clustering ◽  
Agent Based ◽  
Market Participants ◽  
Long Run ◽  
Highly Nonlinear ◽  
Empirical Regularities

Financial volatility obeys two fascinating empirical regularities that apply to various assets, on various markets, and on various time scales: it is fat-tailed (more precisely power-law distributed) and it tends to be clustered in time. Many interesting models have been proposed to account for these regularities, notably agent-based models, which mimic the two empirical laws through a complex mix of nonlinear mechanisms such as traders switching between trading strategies in highly nonlinear way. This paper explains the two regularities simply in terms of traders’ attitudes towards news, an explanation that follows from the very traditional dichotomy of financial market participants, investors versus speculators, whose behaviors are reduced to their simplest forms. Long-run investors’ valuations of an asset are assumed to follow a news-driven random walk, thus capturing the investors’ persistent, long memory of fundamental news. Short-term speculators’ anticipated returns, on the other hand, are assumed to follow a news-driven autoregressive process, capturing their shorter memory of fundamental news, and, by the same token, the feedback intrinsic to the short-sighted, trend-following (or herding) mindset of speculators. These simple, linear models of traders’ expectations explain the two financial regularities in a generic and robust way. Rational expectations, the dominant model of traders’ expectations, is not assumed here, owing to the famous no-speculation, no-trade results.

scholarly journals Advanced materials modelling via fractional calculus: challenges and perspectives

2020 ◽  
Vol 378 (2172) ◽  
pp. 20200050
Author(s):  
Giuseppe Failla ◽  
Massimiliano Zingales
Keyword(s):  
Fractional Calculus ◽  
Long Memory ◽  
Experimental Studies ◽  
Advanced Materials ◽  
Fractional Operators ◽  
Theme Issue ◽  
Materials Modelling ◽  
Local Media ◽  
Non Local ◽  
Complex Phenomena

Fractional calculus is now a well-established tool in engineering science, with very promising applications in materials modelling. Indeed, several studies have shown that fractional operators can successfully describe complex long-memory and multiscale phenomena in materials, which can hardly be captured by standard mathematical approaches as, for instance, classical differential calculus. Furthermore, fractional calculus has recently proved to be an excellent framework for modelling non-conventional fractal and non-local media, opening valuable prospects on future engineered materials. The theme issue gathers cutting-edge theoretical, computational and experimental studies on advanced materials modelling via fractional calculus, with a focus on complex phenomena and non-conventional media. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

Numerical modeling of wave propagation in a cracked solid

2019 ◽  
Vol 24 (9) ◽  
pp. 2895-2913
Author(s):  
Quriaky Gomez ◽  
Oana Ciobanu ◽  
Ioan R Ionescu
Keyword(s):  
Wave Propagation ◽  
Wave Speed ◽  
Crack Density ◽  
Pulse Amplitude ◽  
Theoretical Formula ◽  
Density Parameter ◽  
Static Elasticity ◽  
Analytical Formulas ◽  
Complex Phenomena ◽  
Damaged Materials

In this study, we investigate the dynamics of (damaged) materials with a nonlinear microstructure (microcracks in frictional contact) using a discontinuous Galerkin method. Although the propagation of a plane wave is associated with a complex phenomenon and the stress field loses its homogeneity, the loading pulse has an overall front wave at each moment. Thus, a macroscopic behavior can be extracted and compared with reference solutions based on analytical formulas deduced from the effective (static) elasticity models of a cracked solid. The influences of the mesh and of the microcrack pattern have been tested to choose an optimal numerical setting. We analyzed the sensitivity of the damaged pulse with respect to microcrack density, wavelength, and microcrack orientation. For small values of crack density parameter, the theoretical formula and the computed speed of the damaged pulse are very close, but for larger values there is an important gap between them. For large ratios of wavelength over crack length, the wave speed depends only on the crack density parameter. However, if this ratio is of the order of unity, the wave speed, pulse duration, and pulse amplitude are very sensitive to the wavelength. For more complex phenomena, namely blast propagation in a cracked material, we discuss how the microcrack orientation affects wave propagation and scattering.

Volatility clustering in agent based market models

2003 ◽  
Vol 324 (1-2) ◽  
pp. 6-16 ◽  
Author(s):  
Irene Giardina ◽  
Jean-Philippe Bouchaud
Keyword(s):  
Market Models ◽  
Volatility Clustering ◽  
Agent Based

scholarly journals Option Pricing under the Subordinated Market Models

2022 ◽  
Vol 2022 ◽  
pp. 1-8
Author(s):  
Longjin Lv ◽  
Changjuan Zheng ◽  
Luna Wang
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Long Memory ◽  
Stock Price ◽  
Fractional Diffusion ◽  
Option Prices ◽  
European Options ◽  
Self Similarity ◽  
Market Models ◽  
Proposed Model

This paper aims to study option pricing problem under the subordinated Brownian motion. Firstly, we prove that the subordinated Brownian motion controlled by the fractional diffusion equation has many financial properties, such as self-similarity, leptokurtic, and long memory, which indicate that the fractional calculus can describe the financial data well. Then, we investigate the option pricing under the assumption that the stock price is driven by the subordinated Brownian motion. The closed-form pricing formula for European options is derived. In the comparison with the classic Black–Sholes model, we find the option prices become higher, and the “volatility smiles” phenomenon happens in the proposed model. Finally, an empirical analysis is performed to show the validity of these results.

scholarly journals Fractional Black–Scholes option pricing, volatility calibration and implied Hurst exponents in South African context

2017 ◽  
Vol 20 (1) ◽  
Author(s):  
Emlyn Flint ◽  
Eben Maré
Keyword(s):  
South African ◽  
Long Memory ◽  
Hurst Exponent ◽  
Implied Volatility ◽  
Contingent Claims ◽  
Deterministic Function ◽  
Volatility Model ◽  
Black Scholes ◽  
Delta Hedging ◽  
Equity Index

Background: Contingent claims on underlying assets are typically priced under a framework that assumes, inter alia, that the log returns of the underlying asset are normally distributed. However, many researchers have shown that this assumption is violated in practice. Such violations include the statistical properties of heavy tails, volatility clustering, leptokurtosis and long memory. This paper considers the pricing of contingent claims when the underlying is assumed to display long memory, an issue that has heretofore not received much attention.Aim: We address several theoretical and practical issues in option pricing and implied volatility calibration in a fractional Black–Scholes market. We introduce a novel eight-parameter fractional Black–Scholes-inspired (FBSI) model for the implied volatility surface, and consider in depth the issue of calibration. One of the main benefits of such a model is that it allows one to decompose implied volatility into an independent long-memory component – captured by an implied Hurst exponent – and a conditional implied volatility component. Such a decomposition has useful applications in the areas of derivatives trading, risk management, delta hedging and dynamic asset allocation.Setting: The proposed FBSI volatility model is calibrated to South African equity index options data as well as South African Rand/American Dollar currency options data. However, given the focus on the theoretical development of the model, the results in this paper are applicable across all financial markets.Methods: The FBSI model essentially combines a deterministic function form of the 1-year implied volatility skew with a separate deterministic function for the implied Hurst exponent, thus allowing one to model both observed implied volatility surfaces as well as decompose them into independent volatility and long-memory components respectively. Calibration of the model makes use of a quasi-explicit weighted least-squares optimisation routine.Results: It is shown that a fractional Black–Scholes model always admits a non-constant implied volatility term structure when the Hurst exponent is not 0.5, and that 1-year implied volatility is independent of the Hurst exponent and equivalent to fractional volatility. Furthermore, we show that the FBSI model fits the equity index implied volatility data very well but that a more flexible Hurst exponent parameterisation is required to fit accurately the currency implied volatility data.Conclusion: The FBSI model is an arbitrage-free deterministic volatility model that can accurately model equity index implied volatility. It also provides one with an estimate of the implied Hurst exponent, which could be very useful in derivatives trading and delta hedging.

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