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Analysis Of Stochastic Pdes Arising From Large Portfolios Of Stochastic Volatility Models
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Book Synopsis Analysis of Stochastic PDEs Arising from Large Portfolios of Stochastic Volatility Models by : Nikolaos Kolliopoulos
Download or read book Analysis of Stochastic PDEs Arising from Large Portfolios of Stochastic Volatility Models written by Nikolaos Kolliopoulos and published by . This book was released on 2018 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Parameter Estimation in Stochastic Volatility Models by : Jaya P. N. Bishwal
Download or read book Parameter Estimation in Stochastic Volatility Models written by Jaya P. N. Bishwal and published by Springer Nature. This book was released on 2022-08-06 with total page 634 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book develops alternative methods to estimate the unknown parameters in stochastic volatility models, offering a new approach to test model accuracy. While there is ample research to document stochastic differential equation models driven by Brownian motion based on discrete observations of the underlying diffusion process, these traditional methods often fail to estimate the unknown parameters in the unobserved volatility processes. This text studies the second order rate of weak convergence to normality to obtain refined inference results like confidence interval, as well as nontraditional continuous time stochastic volatility models driven by fractional Levy processes. By incorporating jumps and long memory into the volatility process, these new methods will help better predict option pricing and stock market crash risk. Some simulation algorithms for numerical experiments are provided.
Book Synopsis Modeling Stochastic Volatility with Application to Stock Returns by : Mr.Noureddine Krichene
Download or read book Modeling Stochastic Volatility with Application to Stock Returns written by Mr.Noureddine Krichene and published by International Monetary Fund. This book was released on 2003-06-01 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt: A stochastic volatility model where volatility was driven solely by a latent variable called news was estimated for three stock indices. A Markov chain Monte Carlo algorithm was used for estimating Bayesian parameters and filtering volatilities. Volatility persistence being close to one was consistent with both volatility clustering and mean reversion. Filtering showed highly volatile markets, reflecting frequent pertinent news. Diagnostics showed no model failure, although specification improvements were always possible. The model corroborated stylized findings in volatility modeling and has potential value for market participants in asset pricing and risk management, as well as for policymakers in the design of macroeconomic policies conducive to less volatile financial markets.
Book Synopsis Analytically Tractable Stochastic Stock Price Models by : Archil Gulisashvili
Download or read book Analytically Tractable Stochastic Stock Price Models written by Archil Gulisashvili and published by Springer Science & Business Media. This book was released on 2012-09-04 with total page 371 pages. Available in PDF, EPUB and Kindle. Book excerpt: Asymptotic analysis of stochastic stock price models is the central topic of the present volume. Special examples of such models are stochastic volatility models, that have been developed as an answer to certain imperfections in a celebrated Black-Scholes model of option pricing. In a stock price model with stochastic volatility, the random behavior of the volatility is described by a stochastic process. For instance, in the Hull-White model the volatility process is a geometric Brownian motion, the Stein-Stein model uses an Ornstein-Uhlenbeck process as the stochastic volatility, and in the Heston model a Cox-Ingersoll-Ross process governs the behavior of the volatility. One of the author's main goals is to provide sharp asymptotic formulas with error estimates for distribution densities of stock prices, option pricing functions, and implied volatilities in various stochastic volatility models. The author also establishes sharp asymptotic formulas for the implied volatility at extreme strikes in general stochastic stock price models. The present volume is addressed to researchers and graduate students working in the area of financial mathematics, analysis, or probability theory. The reader is expected to be familiar with elements of classical analysis, stochastic analysis and probability theory.
Book Synopsis Effective Media Analysis for Stochastic Volatility Models by :
Download or read book Effective Media Analysis for Stochastic Volatility Models written by and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Topics in McKean-Vlasov Equations: Rank-Based Dynamics and Markovian Projection with Applications in Finance and Stochastic Control by : Jiacheng Zhang
Download or read book Topics in McKean-Vlasov Equations: Rank-Based Dynamics and Markovian Projection with Applications in Finance and Stochastic Control written by Jiacheng Zhang and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we consider McKean-Vlasov stochastic differential equations (SDEs), arising from various fields, such as the large-system limit of mean field games, particle systems with mean field interactions, financial mathematics, optimal control, game theory and mathematical physics. We study three aspects of the equations: as limits of interacting particle systems, the existence and uniqueness for them and the connection between the time-marginal distribution and the law of the process.Firstly, in the setting of rank-based models, we use the mean field limit and the Gaussian fluctuations to characterize the dynamics of observables which capture the diversity of a financial market. The results can be used to study the performance of functionally generated portfolios over short-term and medium-term horizons.Secondly, we study the McKean-Vlasov SDE arising from the calibration of local stochastic volatility models in finance. Despite the limited theoretical understanding, we give the strong existence result of stationary solutions for these SDEs, as well as their strong uniqueness in an important special case.Thirdly, we consider conditional McKean-Vlasov stochastic differential equations where the conditional time-marginals of the solutions satisfy non-linear stochastic partial differential equations (SPDEs) of the second order and the laws of the conditional time-marginals follow Fokker-Planck equations (FPEs) on the space of probability measures. We establish connections between the SDEs, SPDEs and the FPEs. This provides a useful tool to obtain Markovian controls in the context of controlled McKean-Vlasov dynamics.
Book Synopsis Multivariate Stochastic Volatility Models and Large Deviation Principles by : Archil Gulisashvili
Download or read book Multivariate Stochastic Volatility Models and Large Deviation Principles written by Archil Gulisashvili and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We establish a comprehensive sample path large deviation principle (LDP) for log-price processes associated with multivariate time-inhomogeneous stochastic volatility models. Examples of models for which the new LDP holds include Gaussian models, non-Gaussian fractional models, mixed models, models with reflection, and models in which the volatility process is a solution to a Volterra type stochastic integral equation. The sample path and small-noise LDPs for log-price processes are used to obtain large deviation style asymptotic formulas for the distribution function of the first exit time of a log-price process from an open set, multidimensional binary barrier options, call options, Asian options, and the implied volatility. Such formulas capture leading order asymptotics of the above-mentioned important quantities arising in the theory of stochastic volatility models. We also prove a sample path LDP for solutions to Volterra type stochastic integral equations with predictable coefficients depending on auxiliary stochastic processes.
Book Synopsis Stochastic Volatility by : Neil Shephard
Download or read book Stochastic Volatility written by Neil Shephard and published by Oxford University Press, USA. This book was released on 2005 with total page 534 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic volatility is the main concept used in the fields of financial economics and mathematical finance to deal with time-varying volatility in financial markets. This work brings together some of the main papers that have influenced this field, andshows that the development of this subject has been highly multidisciplinary.
Book Synopsis Rough PDEs for Local Stochastic Volatility Models by : Peter Bank
Download or read book Rough PDEs for Local Stochastic Volatility Models written by Peter Bank and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work, we introduce a novel pricing methodology in general, possibly non-Markovian local stochastic volatility (LSV) models. We observe that by conditioning the LSV dynamics on the Brownian motion that drives the volatility, one obtains a time-inhomogeneous Markov process. Using tools from rough path theory, we describe how to precisely understand the conditional LSV dynamics and reveal their Markovian nature. The latter allows us to connect the conditional dynamics to so-called rough partial differential equations (RPDEs), through a Feynman-Kac type of formula. In terms of European pricing, conditional on realizations of one Brownian motion, we can compute conditional option prices by solving the corresponding linear RPDEs, and then average over all samples to find unconditional prices. Our approach depends only minimally on the specification of the volatility, making it applicable for a wide range of classical and rough LSV models, and it establishes a PDE pricing method for non-Markovian models. Finally, we present a first glimpse at numerical methods for RPDEs and apply them to price European options in several rough LSV models.
Book Synopsis Pricing Derivatives in Stochastic Volatility Models Using the Finite Difference Method by :
Download or read book Pricing Derivatives in Stochastic Volatility Models Using the Finite Difference Method written by and published by . This book was released on 2001 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The Heston stochastic volatility model is one extension of the Black-Scholes model which describes the money markets more accurately so that more realistic prices for derivative products are obtained. From the stochastic differential equation of the underlying financial product a partial differential equation (p.d.e.) for the value function of an option can be derived. This p.d.e. can be solved with the finite difference method (f.d.m.). The stability and consistency of the method is examined. Furthermore a boundary condition is proposed to reduce the numerical error. Finally a non uniform structured grid is derived which is fairly optimal for the numerical result in the most interesting point.
Book Synopsis An Analysis of the Method of Moment Estimators for a Stochastic Volatility Model by : Anna Griffel
Download or read book An Analysis of the Method of Moment Estimators for a Stochastic Volatility Model written by Anna Griffel and published by . This book was released on 1995 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis 'Effective' Parameters for Stochastic Volatility Models by : Zaizhi Wang
Download or read book 'Effective' Parameters for Stochastic Volatility Models written by Zaizhi Wang and published by . This book was released on 2008 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper tackles the issue of approximated formula for stochastic model with time dependent model parameters, using an averaging principle. The idea lies in finding a similar model but with constant parameters that is the closest to our initial process, along the same lines as results proven by Gyouml;ngy (1986) for general stochastic processes. We extend previous results found by Piterbarg (2005) for the particular case of SABR model (Hagan (2002)). The resulting formula can be evaluated very quickly solving the implied Riccati equations. We compare the approximation with exact solution of the corresponding partial differential equation using an ADI method. Numerical results show that the approximation works well for short term maturities.
Book Synopsis Turbo-Charged Local Stochastic Volatility Models by : Ghislain Vong
Download or read book Turbo-Charged Local Stochastic Volatility Models written by Ghislain Vong and published by . This book was released on 2013 with total page 12 pages. Available in PDF, EPUB and Kindle. Book excerpt: This article presents an alternative formulation of the standard Local Stochastic Volatility model (LSV) widely used for the pricing and risk-management of FX derivatives and to a lesser extent of equity derivatives. In the standard model, calibration is achieved by solving a non-linear two-factor Kolmogorov forward PDE, where a minimum number of vol points is required to achieve convergence of a finite difference implementation. In contrast, we propose to model the volatility process by a Markov chain defined over an arbitrary small number of states, so that calibration and pricing are achieved by solving a coupled system of one-factor PDEs. The practical benefits are twofolds: existing one-factor PDE solvers can be recycled to model stochastic volatility, while the reduction in number of discretisation points implies a speedup in execution time that enables real-time risk-management of large derivatives position.
Book Synopsis Comment on Jacquier, Polson and Rossi's "Bayesian Analysis of Stochastic Volatility Models by : Daniel B. Nelson
Download or read book Comment on Jacquier, Polson and Rossi's "Bayesian Analysis of Stochastic Volatility Models written by Daniel B. Nelson and published by . This book was released on 1994 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Large Deviations for Rough and Complete Stochastic Volatility Models by : Chloe Alice Lacombe
Download or read book Large Deviations for Rough and Complete Stochastic Volatility Models written by Chloe Alice Lacombe and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Computationally Efficient Multi-asset Stochastic Volatility Modeling by : Yizhou Fang
Download or read book Computationally Efficient Multi-asset Stochastic Volatility Modeling written by Yizhou Fang and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic volatility (SV) models are popular in financial modeling, because they capture the inherent uncertainty of the asset volatility. Since assets are observed to co-move together, multi-asset SV (mSV) models are more appealing than combining single-asset SV models in portfolio analysis and risk management. However, the latent volatility process renders the observed data likelihood intractable. Therefore, parameter inference typically requires computationally intensive methods to integrate the latent volatilities out, so that it is computationally challenging to estimate the model parameters. This three-part thesis is concerned with mSV modeling that is both conceptually and computationally scalable to large financial portfolios. In Part I, we explore the potential of substituting the latent volatility by an observable market proxy. For more than 20 years of out-of-sample predictions, we find that modeling the Standard and Poor's 500 (SPX) index by a simple framework of Seemingly Unrelated Regressions (SUR) with VIX volatility proxy is comparable to the benchmark Heston model with latent volatility, at a fraction of the computational cost. In Part II, we propose a new mSV model structured around a common volatility factor, which also can be proxied by an observable process. Unlike existing mSV models, the number of parameters in ours scales linearly instead of quadratically in the number of assets -- a desirable property for parameter inference of high-dimensional portfolios. Empirical evidence suggests that the common-factor volatility structure has considerable benefits for option pricing compared to a richer class of unconstrained models. In Part III, we propose an approximate method of parameter inference for mSV models based on the Kalman filter. A large-scale simulation study indicates that the approximation is orders of magnitude faster than exact inference methods, while retaining comparable accuracy.
Book Synopsis Analysis of Stochastic Volatility in High-Frequency Financial Data by :
Download or read book Analysis of Stochastic Volatility in High-Frequency Financial Data written by and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
