Student-like models for risky asset with dependence

Castelli, Francesco; Leonenko, Nikolai; Shchestyuk, Nataliya

doi:10.1080/07362994.2016.1266945

Cited by 6 publications

(13 citation statements)

References 22 publications

(37 reference statements)

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“…Price of the bond evolves according to formula for continuous rates. The price of the underlying traded asset S t is a strong solution of the following stochastic dierential equation [2]:…”

Section: Student-like Models With Fractal Activity Timementioning

confidence: 99%

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Risk Modelling Approaches for Student-like Models with Fractal Activity Time

Solomanchuk¹,

Shchestyuk²

2022

MMJ

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The paper focuses on value at risk (V@R) measuring for Student-like models of markets with fractal activity time (FAT). The fractal activity time models were introduced by Heyde to try to encompass the empirically found characteristics of real data and elaborated on for Variance Gamma, normal inverse Gaussian and skewed Student distributions. But problem of evaluating an value at risk for this model was not researched. It is worth to mention that if we use normal or symmetric Student`s models than V@R can be computed using standard statistical packages. For calculating V@R for Student-like models we need Monte Carlo method and the iterative scheme for simulating N scenarios of stock prices. We model stock prices as a diffusion processes with the fractal activity time and for modeling increments of fractal activity time we use another diffusion process, which has a given marginal inverse gamma distribution. The aim of the paper is to perform and compare V@R Monte Carlo approach and Markowitz approach for Student-like models in terms of portfolio risk. For this purpose we propose procedure of calculating V@R for two types of investor portfolios. The first one is uniform portfolio, where d assets are equally distributed. The second is optimal Markowitz portfolio, for which variance of return is the smallest out of all other portfolios with the same mean return. The programmed model which was built using R-statistics can be used as to the simulations for any asset and for construct optimal portfolios for any given amount of assets and then can be used for understanding how this optimal portfolio behaves compared to other portfolios for Student-like models of markets with fractal activity time. Also we present numerical results for evaluating V@R for both types of investor portfolio. We show that optimal Markovitz portfolio demonstrates in the most of cases the smallest possible Value at Risk comparing with other portfolios. Thus, for making investor decisions under uncertainty we recommend to apply portfolio optimization and value at risk approach jointly.

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“…Price of the bond evolves according to formula for continuous rates. The price of the underlying traded asset S t is a strong solution of the following stochastic dierential equation [2]:…”

Section: Student-like Models With Fractal Activity Timementioning

confidence: 99%

“…For this aim we describe time-changed processes for Studentlike models with depends. This section is based on the papers [2], [8], [9], [10], [11] where models of the generalized diusion process with "market" time are presented and discussed.…”

Section: Introductionmentioning

confidence: 99%

Risk Modelling Approaches for Student-like Models with Fractal Activity Time

Solomanchuk¹,

Shchestyuk²

2022

MMJ

View full text Add to dashboard Cite

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“…The price of underlying traded assets S(t) is the strong solution of the following stochastic differential equation (SDE) [4]:…”

Section: Stochastic Ito Processes and Its Simulationmentioning

confidence: 99%

“…where µ, and ✓ are constants, " -white noise with normal standard distribution, and ⌧ is a stationary process of active time, with inverse gamma distribution, which was modeled earlier (see [4], [5], [6], [7]).…”

Section: Stochastic Ito Processes and Its Simulationmentioning

confidence: 99%

“…Forth section contents the main result and describes time-changed processes and its simulation. This section is based on the papers [4][5][6][7] where models of the generalized diffusion process with "market" time are presented. In proposed iterative scheme for stock prices we use modeling market time increments as a diffusion processes with a given marginal inverse gamma distribution.…”

Section: Introductionmentioning

confidence: 99%

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Simulating Stochastic Diffusion Processes and Processes with “Market” Time

Boluh

Shchestyuk

2021

MMJ

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The paper focuses on modelling, simulation techniques and numerical methods concerned stochastic processes in subject such as financial mathematics and financial engineering. The main result of this work is simulation of a stochastic process with new market active time using Monte Carlo techniques.The processes with market time is a new vision of how stock price behavior can be modeled so that the nature of the process is more real. The iterative scheme for computer modelling of this process was proposed.It includes the modeling of diffusion processes with a given marginal inverse gamma distribution. Graphs of simulation of the Ornstein-Uhlenbeck random walk for different parameters, a simulation of the diffusion process with a gamma-inverse distribution and simulation of the process with market active time are presented.To simulate stochastic processes, an iterative scheme was used: xk+1 = xk + a(xk, tk) ∆t + b(xk, tk) √ (∆t) εk,, where εk each time a new generation with a normal random number distribution.Next, the tools of programming languages for generating random numbers (evenly distributed, normally distributed) are investigated. Simulation (simulation) of stochastic diffusion processes is carried out; calculation errors and acceleration of convergence are calculated, Euler and Milstein schemes. At the next stage, diffusion processes with a given distribution function, namely with an inverse gamma distribution, were modelled. The final stage was the modelling of stock prices with a new "market" time, the growth of which is a diffusion process with inverse gamma distribution. In the proposed iterative scheme of stock prices, we use the modelling of market time gains as diffusion processes with a given marginal gamma-inverse distribution.The errors of calculations are evaluated using the Milstein scheme. The programmed model can be used to predict future values of time series and for option pricing.

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Option Pricing and Stochastic Optimization

Shchestyuk¹

2022

Springer Proceedings in Mathematics &Amp; Statistics

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In this paper will be demonstrated that the link between optimal option value, risk measuring and risk managing is especially close, and it is given by stochastic optimization.Post the financial crisis of 2008 it has been clear that risk considerations must enter into the valuation of derivatives, previously performed in an entirely "risk neutral world". The average value-at-risk is related to the problem of investor [1], which we use for option pricing. Let

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Student-like models for risky asset with dependence

Cited by 6 publications

References 22 publications

Risk Modelling Approaches for Student-like Models with Fractal Activity Time

Risk Modelling Approaches for Student-like Models with Fractal Activity Time

Simulating Stochastic Diffusion Processes and Processes with “Market” Time

Option Pricing and Stochastic Optimization

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