Exact Simulation Of The Wishart Stochastic Volatility Model
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| Author | : Marco Huerner |
| Publisher | : |
| Total Pages | : |
| Release | : 2014 |
| Genre | : |
| ISBN | : |
This thesis deals with the simulation of the Wishart stochastic volatility model (WSVM) which is a matrix generalization of the famous Heston model. Lately, an exact sampling scheme has been introduced. Its theoretical foundations are given in two papers. First, Ahdida and Alfonsi [2] find a methodology to simulate exactly the Wishart process for a general parameter space. Second, Kang and Kang [22] complete the scheme by proposing an expression for the conditional Laplace transform of the risky asset given the final state of the variance process. The thesis has two principle goals. First, we merge the theoretical foundations necessary to understand the exact sampling methodology and collect the corresponding proofs. Thereby, we build the basics for consecutive theoretical work, especially with respect to a necessary discussion of the numerical properties of the model. Second, we provide a prototype computational implementation. This implementation intends to be a first Monte Carlo framework for numerical experiments, testing purposes and further algorithmic improvements. It provides the tool to address future computation related research tasks. The current version is written in the MATLAB language m, and C.
| Author | : Alexander van Haastrecht |
| Publisher | : |
| Total Pages | : 35 |
| Release | : 2011 |
| Genre | : |
| ISBN | : |
We deal with several efficient discretization methods for the simulation of the Heston stochastic volatility model. The resulting schemes can be used to calculate all kind of options and corresponding sensitivities, in particular the exotic options that cannot be valued with closed-form solutions. We focus on to the (computational) efficiency of the simulation schemes: though the Broadie and Kaya (2006) paper provided an exact simulation method for the Heston dynamics, we argue why its practical use might be limited. Instead we consider efficient approximations of the exact scheme, which try to exploit certain distributional features of the underlying variance process. The resulting methods are fast, highly accurate and easy to implement. We conclude by numerically comparing our new schemes to the exact scheme of Broadie and Kaya, the almost exact scheme of Smith, the Kahl-Jackel scheme, the Full Truncation scheme of Lord et al. and the Quadratic Exponential scheme of Andersen.
| Author | : Pingping Jiang |
| Publisher | : |
| Total Pages | : 0 |
| Release | : 2021 |
| Genre | : |
| ISBN | : |
We perform a thorough investigation on the analytical solvability of general stochastic volatility (SV) models with Levy jumps and propose a unified, accurate, and efficient almost exact simulation method to price various financial derivatives. Our theoretical results lay a foundation for a range of valuation, calibration, and econometric problems. Our almost exact simulation method is applicable to a broad class of models and enables effective pricing of path-dependent financial derivatives, whereas the traditional exact simulation method is always tailor-made for some specific models and is generally time-consuming, which limits its use in the case of path-dependent financial derivatives. More specifically, by combining a decomposition technique with a change of measure approach, we first develop a simple probabilistic method to derive a unified formula for the conditional characteristic function of the log-asset price under general SV models with Levy jumps and show under which conditions this new formula admits a closed-form expression. The conditional and unconditional joint characteristic functions of the log-asset price and the integrated variance can be easily obtained as byproducts. Second, we take advantage of our main theoretical result, the Hilbert transform method, the interpolation technique, and the dimension reduction technique to construct unified and efficient almost exact simulation schemes. Finally, we apply our almost exact simulation method to price European options, discretely monitored weighted variance swaps, and discretely monitored variance options under a wide variety of SV models with Levy jumps. Extensive numerical examples demonstrate the high level of accuracy and efficiency of our almost exact simulation method in terms of bias, root-mean-squared error (RMS error), and CPU time.
| Author | : Jan Baldeaux |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 432 |
| Release | : 2013-08-13 |
| Genre | : Mathematics |
| ISBN | : 3319007475 |
This research monograph provides an introduction to tractable multidimensional diffusion models, where transition densities, Laplace transforms, Fourier transforms, fundamental solutions or functionals can be obtained in explicit form. The book also provides an introduction to the use of Lie symmetry group methods for diffusions, which allows to compute a wide range of functionals. Besides the well-known methodology on affine diffusions it presents a novel approach to affine processes with applications in finance. Numerical methods, including Monte Carlo and quadrature methods, are discussed together with supporting material on stochastic processes. Applications in finance, for instance, on credit risk and credit valuation adjustment are included in the book. The functionals of multidimensional diffusions analyzed in this book are significant for many areas of application beyond finance. The book is aimed at a wide readership, and develops an intuitive and rigorous understanding of the mathematics underlying the derivation of explicit formulas for functionals of multidimensional diffusions.
| Author | : Giuseppe Maddaloni |
| Publisher | : |
| Total Pages | : |
| Release | : 1998 |
| Genre | : |
| ISBN | : |
| Author | : Christian Kahl |
| Publisher | : Universal-Publishers |
| Total Pages | : 219 |
| Release | : 2008 |
| Genre | : Business & Economics |
| ISBN | : 1581123833 |
The famous Black-Scholes model was the starting point of a new financial industry and has been a very important pillar of all options trading since. One of its core assumptions is that the volatility of the underlying asset is constant. It was realised early that one has to specify a dynamic on the volatility itself to get closer to market behaviour. There are mainly two aspects making this fact apparent. Considering historical evolution of volatility by analysing time series data one observes erratic behaviour over time. Secondly, backing out implied volatility from daily traded plain vanilla options, the volatility changes with strike. The most common realisations of this phenomenon are the implied volatility smile or skew. The natural question arises how to extend the Black-Scholes model appropriately. Within this book the concept of stochastic volatility is analysed and discussed with special regard to the numerical problems occurring either in calibrating the model to the market implied volatility surface or in the numerical simulation of the two-dimensional system of stochastic differential equations required to price non-vanilla financial derivatives. We introduce a new stochastic volatility model, the so-called Hyp-Hyp model, and use Watanabe's calculus to find an analytical approximation to the model implied volatility. Further, the class of affine diffusion models, such as Heston, is analysed in view of using the characteristic function and Fourier inversion techniques to value European derivatives.
| Author | : Roger Lord |
| Publisher | : |
| Total Pages | : 30 |
| Release | : 2008 |
| Genre | : |
| ISBN | : |
Using an Euler discretisation to simulate a mean-reverting CEV process gives rise to the problem that while the process itself is guaranteed to be nonnegative, the discretisation is not. Although an exact and efficient simulation algorithm exists for this process, at present this is not the case for the CEV-SV stochastic volatility model, with the Heston model as a special case, where the variance is modelled as a mean-reverting CEV process. Consequently, when using an Euler discretisation, one must carefully think about how to fix negative variances. Our contribution is threefold. Firstly, we unify all Euler fixes into a single general framework. Secondly, we introduce the new full truncation scheme, tailored to minimise the positive bias found when pricing European options. Thirdly and finally, we numerically compare all Euler fixes to recent quasi-second order schemes of Kahl and Jauml;ckel and Ninomiya and Victoir, as well as to the exact scheme of Broadie and Kaya. The choice of fix is found to be extremely important. The full truncation scheme outperforms all considered biased schemes in terms of bias and root-mean-squared error.
| Author | : Luc Bauwens |
| Publisher | : John Wiley & Sons |
| Total Pages | : 566 |
| Release | : 2012-03-22 |
| Genre | : Business & Economics |
| ISBN | : 1118272056 |
A complete guide to the theory and practice of volatility models in financial engineering Volatility has become a hot topic in this era of instant communications, spawning a great deal of research in empirical finance and time series econometrics. Providing an overview of the most recent advances, Handbook of Volatility Models and Their Applications explores key concepts and topics essential for modeling the volatility of financial time series, both univariate and multivariate, parametric and non-parametric, high-frequency and low-frequency. Featuring contributions from international experts in the field, the book features numerous examples and applications from real-world projects and cutting-edge research, showing step by step how to use various methods accurately and efficiently when assessing volatility rates. Following a comprehensive introduction to the topic, readers are provided with three distinct sections that unify the statistical and practical aspects of volatility: Autoregressive Conditional Heteroskedasticity and Stochastic Volatility presents ARCH and stochastic volatility models, with a focus on recent research topics including mean, volatility, and skewness spillovers in equity markets Other Models and Methods presents alternative approaches, such as multiplicative error models, nonparametric and semi-parametric models, and copula-based models of (co)volatilities Realized Volatility explores issues of the measurement of volatility by realized variances and covariances, guiding readers on how to successfully model and forecast these measures Handbook of Volatility Models and Their Applications is an essential reference for academics and practitioners in finance, business, and econometrics who work with volatility models in their everyday work. The book also serves as a supplement for courses on risk management and volatility at the upper-undergraduate and graduate levels.
| Author | : Mr.Noureddine Krichene |
| Publisher | : International Monetary Fund |
| Total Pages | : 30 |
| Release | : 2003-06-01 |
| Genre | : Business & Economics |
| ISBN | : 1451854846 |
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.
| Author | : Lorenzo Bergomi |
| Publisher | : CRC Press |
| Total Pages | : 520 |
| Release | : 2015-12-16 |
| Genre | : Business & Economics |
| ISBN | : 1482244071 |
Packed with insights, Lorenzo Bergomi's Stochastic Volatility Modeling explains how stochastic volatility is used to address issues arising in the modeling of derivatives, including:Which trading issues do we tackle with stochastic volatility? How do we design models and assess their relevance? How do we tell which models are usable and when does c
