Application of Fractal Processes and Fractional Derivatives in Finance
In recent years, there has been a fast growth in the application of long-memory processes to underlying assets including stock, volatility index, exchange rate, etc. The fractional Brownian motion is the most popular of the long-memory processes and was introduced by Kolmogorov in 1940 and later by...
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MDPI - Multidisciplinary Digital Publishing Institute
2024
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| description | In recent years, there has been a fast growth in the application of long-memory processes to underlying assets including stock, volatility index, exchange rate, etc. The fractional Brownian motion is the most popular of the long-memory processes and was introduced by Kolmogorov in 1940 and later by Mandelbrot in 1965. It has been used in hydrology and climatology as well as finance. The dynamics of the volatility of asset price or asset price itself were modelled as a fractional Brownian motion in finance and are called rough volatility models and the fractional Black–Scholes model, respectively. Fractional diffusion processes are also used to model the dynamics of underlying assets. The option price under the fractional diffusion setting induces fractional partial differential equations involving the fractional derivatives with respect to the time and the space, respectively. Some closed-form solutions might be found via transform methods in some cases of applications, and numerical methods to solve fractional partial differential equations are being developed. This Special Issue focuses on empirical studies as well as option pricing. The empirical studies consist of multifractal analyses of stock market and volatility index. Multifractal analyses include cross-correlation multifractal analysis, multifractal detrended fluctuation analysis, and other fractional analyses. Meanwhile, option pricing focuses on the fractional Black–Scholes models and their variants, including the fuzzy fractional Black–Scholes model, uncertain fractional differential equation, and model with fractional-order feature. |
| format | Online |
| id | doab-20.500.12854ir-139257 |
| institution | Directory of Open Access Books |
| language | eng |
| publishDate | 2024 |
| publishDateRange | 2024 |
| publishDateSort | 2024 |
| publisher | MDPI - Multidisciplinary Digital Publishing Institute |
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| spelling | doab-20.500.12854ir-1392572024-07-04T09:33:05Z Application of Fractal Processes and Fractional Derivatives in Finance Chan, Leung Lung homotopy perturbation method Elaki transform fractional Black–Scholes equation granular differentiability fractional differential equation uncertainty theory currency model currency option pricing convergence rate high-order finite difference method Markov regime-switching jump-diffusion model partial integro-differential equations China’s stock market stock market slump multifractality stock forecast fractional-order particle swarm optimization algorithm mixed fraction Brownian motion Hurst variational iteration method generalized fractional derivative generalized Laplace tranform generalized Mittag–Leffler function fractional Black-Scholes model ELS finite difference scheme technological innovation finance real economy multifractal denoising stock prediction asymmetry Hurst exponent deep learning neural networks time-fractional Black-Scholes PDEs double barriers options numerical methods global market efficiency multifractal detrended fluctuation analysis developed markets emerging markets frontier markets the generalized value at risk (GCoVaR) systemically important banks (SIBs) risk spillover thema EDItEUR::K Economics, Finance, Business and Management thema EDItEUR::K Economics, Finance, Business and Management::KF Finance and accounting In recent years, there has been a fast growth in the application of long-memory processes to underlying assets including stock, volatility index, exchange rate, etc. The fractional Brownian motion is the most popular of the long-memory processes and was introduced by Kolmogorov in 1940 and later by Mandelbrot in 1965. It has been used in hydrology and climatology as well as finance. The dynamics of the volatility of asset price or asset price itself were modelled as a fractional Brownian motion in finance and are called rough volatility models and the fractional Black–Scholes model, respectively. Fractional diffusion processes are also used to model the dynamics of underlying assets. The option price under the fractional diffusion setting induces fractional partial differential equations involving the fractional derivatives with respect to the time and the space, respectively. Some closed-form solutions might be found via transform methods in some cases of applications, and numerical methods to solve fractional partial differential equations are being developed. This Special Issue focuses on empirical studies as well as option pricing. The empirical studies consist of multifractal analyses of stock market and volatility index. Multifractal analyses include cross-correlation multifractal analysis, multifractal detrended fluctuation analysis, and other fractional analyses. Meanwhile, option pricing focuses on the fractional Black–Scholes models and their variants, including the fuzzy fractional Black–Scholes model, uncertain fractional differential equation, and model with fractional-order feature. 2024-07-04T09:33:01Z 2024-07-04T09:33:01Z 2024 book ONIX_20240704_9783725810918_53 9783725810918 9783725810925 https://directory.doabooks.org/handle/20.500.12854/139257 eng application/octet-stream Attribution-NonCommercial-NoDerivatives 4.0 International https://mdpi.com/books/pdfview/book/9252 https://mdpi.com/books/pdfview/book/9252 MDPI - Multidisciplinary Digital Publishing Institute 10.3390/books978-3-7258-1092-5 10.3390/books978-3-7258-1092-5 46cabcaa-dd94-4bfe-87b4-55023c1b36d0 9783725810918 9783725810925 248 open access |
| spellingShingle | homotopy perturbation method Elaki transform fractional Black–Scholes equation granular differentiability fractional differential equation uncertainty theory currency model currency option pricing convergence rate high-order finite difference method Markov regime-switching jump-diffusion model partial integro-differential equations China’s stock market stock market slump multifractality stock forecast fractional-order particle swarm optimization algorithm mixed fraction Brownian motion Hurst variational iteration method generalized fractional derivative generalized Laplace tranform generalized Mittag–Leffler function fractional Black-Scholes model ELS finite difference scheme technological innovation finance real economy multifractal denoising stock prediction asymmetry Hurst exponent deep learning neural networks time-fractional Black-Scholes PDEs double barriers options numerical methods global market efficiency multifractal detrended fluctuation analysis developed markets emerging markets frontier markets the generalized value at risk (GCoVaR) systemically important banks (SIBs) risk spillover thema EDItEUR::K Economics, Finance, Business and Management thema EDItEUR::K Economics, Finance, Business and Management::KF Finance and accounting Application of Fractal Processes and Fractional Derivatives in Finance |
| title | Application of Fractal Processes and Fractional Derivatives in Finance |
| title_full | Application of Fractal Processes and Fractional Derivatives in Finance |
| title_fullStr | Application of Fractal Processes and Fractional Derivatives in Finance |
| title_full_unstemmed | Application of Fractal Processes and Fractional Derivatives in Finance |
| title_short | Application of Fractal Processes and Fractional Derivatives in Finance |
| title_sort | application of fractal processes and fractional derivatives in finance |
| topic | homotopy perturbation method Elaki transform fractional Black–Scholes equation granular differentiability fractional differential equation uncertainty theory currency model currency option pricing convergence rate high-order finite difference method Markov regime-switching jump-diffusion model partial integro-differential equations China’s stock market stock market slump multifractality stock forecast fractional-order particle swarm optimization algorithm mixed fraction Brownian motion Hurst variational iteration method generalized fractional derivative generalized Laplace tranform generalized Mittag–Leffler function fractional Black-Scholes model ELS finite difference scheme technological innovation finance real economy multifractal denoising stock prediction asymmetry Hurst exponent deep learning neural networks time-fractional Black-Scholes PDEs double barriers options numerical methods global market efficiency multifractal detrended fluctuation analysis developed markets emerging markets frontier markets the generalized value at risk (GCoVaR) systemically important banks (SIBs) risk spillover thema EDItEUR::K Economics, Finance, Business and Management thema EDItEUR::K Economics, Finance, Business and Management::KF Finance and accounting |
| topic_facet | homotopy perturbation method Elaki transform fractional Black–Scholes equation granular differentiability fractional differential equation uncertainty theory currency model currency option pricing convergence rate high-order finite difference method Markov regime-switching jump-diffusion model partial integro-differential equations China’s stock market stock market slump multifractality stock forecast fractional-order particle swarm optimization algorithm mixed fraction Brownian motion Hurst variational iteration method generalized fractional derivative generalized Laplace tranform generalized Mittag–Leffler function fractional Black-Scholes model ELS finite difference scheme technological innovation finance real economy multifractal denoising stock prediction asymmetry Hurst exponent deep learning neural networks time-fractional Black-Scholes PDEs double barriers options numerical methods global market efficiency multifractal detrended fluctuation analysis developed markets emerging markets frontier markets the generalized value at risk (GCoVaR) systemically important banks (SIBs) risk spillover thema EDItEUR::K Economics, Finance, Business and Management thema EDItEUR::K Economics, Finance, Business and Management::KF Finance and accounting |
| url | ONIX_20240704_9783725810918_53 |