A Lognormal Type Stochastic Volatility Model With Quadratic Drift PDF Download

Author: Peter Carr
Publisher:
ISBN:
Category :
Languages : en
Pages : 26

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Book Description
This paper presents a novel one-factor stochastic volatility model where the instantaneous volatility of the asset log-return is a diffusion with a quadratic drift and a linear dispersion function. The instantaneous volatility mean reverts around a constant level, with a speed of mean reversion that is affine in the instantaneous volatility level. The steady-state distribution of the instantaneous volatility belongs to the class of Generalized Inverse Gaussian distributions. We show that the quadratic term in the drift is crucial to avoid moment explosions and to preserve the martingale property of the stock price process. Using a conveniently chosen change of measure, we relate the model to the class of polynomial diffusions. This remarkable relation allows us to develop a highly accurate option price approximation technique based on orthogonal polynomial expansions.

Author: Artur Sepp
Publisher:
ISBN:
Category :
Languages : en
Pages : 76

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Book Description
While empirical studies have established that the log-normal stochastic volatility (SV) model is superior to its alternatives, the model does not allow for the analytical solutions available for affine models. To circumvent this, we show that the joint moment generating function (MGF) of the log-price and the quadratic variance (QV) under the log-normal SV model can be decomposed into a leading term, which is given by an exponential-affine form, and a residual term, whose estimate depends on the higher order moments of the volatility process. We prove that the second-order leading term is theoretically consistent with the expected values and covariance matrix of the log-price and the quadratic variance. We further extend this approach to the log-normal SV model with jumps. We use Fourier inversion techniques to value vanilla options on the equity and the QV and, by comparison to Monte Carlo simulations, we show that the second-order leading term is precise for the valuation of vanilla options. We generalize the affine decomposition to other non-affine stochastic volatility models with polynomial drift and volatility functions, and with jumps in the volatility process.

Author: Majed Sidani
Publisher:
ISBN:
Category :
Languages : en
Pages : 8

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Book Description
I derive a closed form formula for pricing a European style call option assuming stochastic volatility and normal dynamics for the underlying. The model differs from the Heston model in the underlying dynamics only - normal as opposed to lognormal.

Author: Neil Shephard
Publisher: Oxford University Press, USA
ISBN: 0199257205
Category : Business & Economics
Languages : en
Pages : 534

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Book Description
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.

Author: Lorenzo Bergomi
Publisher: CRC Press
ISBN: 1482244071
Category : Business & Economics
Languages : en
Pages : 520

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Book Description
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

Author: Jun Yu
Publisher:
ISBN:
Category :
Languages : en
Pages : 33

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Book Description
This paper proposes a class of nonlinear stochastic volatility models based on the Box-Cox transformation which offers an alternative to the one introduced in Andersen (1994). The proposed class encompasses many parametric stochastic volatility models that have appeared in the literature, including the well known lognormal stochastic volatility model, and has an advantage in the ease with which different specifications on stochastic volatility can be tested. In addition, the functional form of transformation which induces marginal normality of volatility is obtained as a byproduct of this general way of modeling stochastic volatility. The efficient method of moments approach is used to estimate model parameters. Empirical results reveal that the lognormal stochastic volatility model is rejected for daily index return data but not for daily individual stock return data. As a consequence, the stock volatility can be well described by the lognormal distribution as its marginal distribution, consistent with the result found in a recent literature (cf Andersen et al (2001a)). However, the index volatility does not follow the lognormal distribution as its marginal distribution.

Author: Christian Kahl
Publisher: Universal-Publishers
ISBN: 1581123833
Category : Business & Economics
Languages : en
Pages : 219

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Book Description
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: Makoto Takahashi
Publisher: Springer Nature
ISBN: 981990935X
Category : Business & Economics
Languages : en
Pages : 120

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Book Description
This treatise delves into the latest advancements in stochastic volatility models, highlighting the utilization of Markov chain Monte Carlo simulations for estimating model parameters and forecasting the volatility and quantiles of financial asset returns. The modeling of financial time series volatility constitutes a crucial aspect of finance, as it plays a vital role in predicting return distributions and managing risks. Among the various econometric models available, the stochastic volatility model has been a popular choice, particularly in comparison to other models, such as GARCH models, as it has demonstrated superior performance in previous empirical studies in terms of fit, forecasting volatility, and evaluating tail risk measures such as Value-at-Risk and Expected Shortfall. The book also explores an extension of the basic stochastic volatility model, incorporating a skewed return error distribution and a realized volatility measurement equation. The concept of realized volatility, a newly established estimator of volatility using intraday returns data, is introduced, and a comprehensive description of the resulting realized stochastic volatility model is provided. The text contains a thorough explanation of several efficient sampling algorithms for latent log volatilities, as well as an illustration of parameter estimation and volatility prediction through empirical studies utilizing various asset return data, including the yen/US dollar exchange rate, the Dow Jones Industrial Average, and the Nikkei 225 stock index. This publication is highly recommended for readers with an interest in the latest developments in stochastic volatility models and realized stochastic volatility models, particularly in regards to financial risk management.

Author: Luc Bauwens
Publisher: John Wiley & Sons
ISBN: 0470872519
Category : Business & Economics
Languages : en
Pages : 566

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Book Description
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: Michail Korniotis
Publisher:
ISBN: 9781109509427
Category :
Languages : en
Pages : 179

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Book Description
We propose a multi-factor stochastic model that can be used as a modeling tool in several areas of applied mathematics. Our modeling efforts are focused on addressing the basic characteristics of quantities that represent random rate of change. These characteristics include properties of their evolution pattern, cross-factor correlation, and the stochastic nature of their diffusion parameter. At the same time, we address the question of solutions implied by the model, as well as the model's tractability. The model is introduced in a general mathematical context, prior to any specific problem consideration. We choose this approach to stress the model's functional independence of any particular application. Within this framework, we are able to represent the evolution of quantities sensitive to random rates of change, as solutions of partial differential equations. We obtain solutions of the resulting partial differential equations by adopting a two-step solution method. The first step approximates the solution using perturbation methods. This procedure specifies the two leading terms as solutions of simpler differential problems. The second step allows us to derive explicit solutions for the terms using the eigenfunction expansion method. A computer algorithm for the solution was also built. This allowed the calibration of the model parameters and a comparison of fitness with existing models. The usefulness and flexibility of the model is demonstrated by considering applications in three areas of applied mathematics: Interest rate, credit risk, and mortality modeling. We comment on how our model generalizes existing models in these areas and its advantages over previously proposed models.