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Analytical Solvability And Exact Simulation Of Stochastic Volatility Models With Jumps
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Book Synopsis Analytical Solvability and Exact Simulation of Stochastic Volatility Models with Jumps by : Pingping Jiang
Download or read book Analytical Solvability and Exact Simulation of Stochastic Volatility Models with Jumps written by Pingping Jiang and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Modelling and Simulation of Stochastic Volatility in Finance by : Christian Kahl
Download or read book Modelling and Simulation of Stochastic Volatility in Finance written by Christian Kahl and published by Universal-Publishers. This book was released on 2008 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Solvable Local and Stochastic Volatility Models by : Pierre Henry-Labordere
Download or read book Solvable Local and Stochastic Volatility Models written by Pierre Henry-Labordere and published by . This book was released on 2010 with total page 21 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we provide an extensive classification of one- and two-dimensional diffusion processes which admit an exact solution to the Kolmogorov (and hence Black-Scholes) equation (in terms of hypergeometric functions). By identifying the one-dimensional solvable processes with the class of integrable superpotentials introduced recently in supersymmetric quantum mechanics, we obtain new analytical solutions. In particular, by applying supersymmetric transformations on a known solvable diffusion process (such as the Natanzon process for which the solution is given by a hypergeometric function), we obtain a hierarchy of new solutions.For two-dimensional processes, more precisely stochastic volatility models, the classification is achieved for a specific class called gauge-free models including the Heston model, the 3/2-model and the geometric Brownian model. We then present a new exact stochastic volatility model belonging to this class.
Book Synopsis Exact Simulation of the Wishart Stochastic Volatility Model by : Marco Huerner
Download or read book Exact Simulation of the Wishart Stochastic Volatility Model written by Marco Huerner and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
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 Local Stochastic Volatility with Jumps by : Stefano Pagliarani
Download or read book Local Stochastic Volatility with Jumps written by Stefano Pagliarani and published by . This book was released on 2016 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present new approximation formulas for local stochastic volatility models, possibly including Lévy jumps. Our main result is an expansion of the characteristic function which is worked out in the Fourier space. Combined with standard Fourier methods, our result provides efficient and accurate formulas for the prices and the Greeks of plain vanilla options. We finally provide numerical results to illustrate the accuracy with real market data.
Book Synopsis Stochastic Volatility Modeling by : Lorenzo Bergomi
Download or read book Stochastic Volatility Modeling written by Lorenzo Bergomi and published by CRC Press. This book was released on 2015-12-16 with total page 520 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Stochastic Volatility and Jumps by : Katja Ignatieva
Download or read book Stochastic Volatility and Jumps written by Katja Ignatieva and published by . This book was released on 2009 with total page 42 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper analyzes exponentially affine and non-affine stochastic volatility models with jumps in returns and volatility. Markov Chain Monte Carlo (MCMC) technique is applied within a Bayesian inference to estimate model parameters and latent variables using daily returns from the Samp;P 500 stock index. There are two approaches to overcome the problem of misspecification of the square root stochastic volatility model. The first approach proposed by Christo ersen, Jacobs and Mimouni (2008) suggests to investigate some non-affine alternatives of the volatility process. The second approach consists in examining more heavily parametrized models by adding jumps to the return and possibly to the volatility process. The aim of this paper is to combine both model frameworks and to test whether the class of affine models is outperformed by the class of non-affine models if we include jumps into the stochastic processes. We conclude that the non-affine model structure have promising statistical properties and are worth further investigations. Further, we find affine models with jump components that perform similar to the non affine models without jump components. Since non affine models yield economically unrealistic parameter estimates, and research is rather developed for the affine model structures we have a tendency to prefer the affine jump diffusion models.
Book Synopsis Affine Stochastic Volatility Models by : Yannick Dillschneider
Download or read book Affine Stochastic Volatility Models written by Yannick Dillschneider and published by . This book was released on 2019 with total page 13 pages. Available in PDF, EPUB and Kindle. Book excerpt: Affine jump diffusion models in general and affine stochastic volatility models in particular are important modeling tools in finance. Their popularity resides in their exibility coupled with their analytical tractability, especially with respect to characteristic functions and polynomial moments. Within a generic affine jump diffusion model and a generic stochastic volatility model, nested in the former as a special case, this paper collects explicit expressions for various characteristic functions and polynomial moments.
Book Synopsis Efficient Simulation of the Heston Stochastic Volatility Model by : Leif B. G. Andersen
Download or read book Efficient Simulation of the Heston Stochastic Volatility Model written by Leif B. G. Andersen and published by . This book was released on 2007 with total page 38 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic volatility models are increasingly important in practical derivatives pricing applications, yet relatively little work has been undertaken in the development of practical Monte Carlo simulation methods for this class of models. This paper considers several new algorithms for time-discretization and Monte Carlo simulation of Heston-type stochastic volatility models. The algorithms are based on a careful analysis of the properties of affine stochastic volatility diffusions, and are straightforward and quick to implement and execute. Tests on realistic model parameterizations reveal that the computational efficiency and robustness of the simulation schemes proposed in the paper compare very favorably to existing methods.
Book Synopsis Log-Normal Stochastic Volatility Model by : Artur Sepp
Download or read book Log-Normal Stochastic Volatility Model written by Artur Sepp and published by . This book was released on 2016 with total page 76 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis A Comparison of Biased Simulation Schemes for Stochastic Volatility Models by : Roger Lord
Download or read book A Comparison of Biased Simulation Schemes for Stochastic Volatility Models written by Roger Lord and published by . This book was released on 2008 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis The 4/2 Stochastic Volatility Model by : Martino Grasselli
Download or read book The 4/2 Stochastic Volatility Model written by Martino Grasselli and published by . This book was released on 2016 with total page 21 pages. Available in PDF, EPUB and Kindle. Book excerpt: We introduce a new stochastic volatility model that includes, as special instances, the Heston (1993) and the 3/2 model of Heston (1997) and Platen (1997). Our model exhibits important features: first, instantaneous volatility can be uniformly bounded away from zero, and second, our model is mathematically and computationally tractable, thereby enabling an efficient pricing procedure. This called for using the Lie symmetries theory for PDEs; doing so allowed us to extend known results on Bessel processes. Finally, we provide an exact simulation scheme for the model; this is useful in view of the numerical applications.
Book Synopsis A New Class of Stochastic Volatility Models with Jumps by : Mikhail Chernov
Download or read book A New Class of Stochastic Volatility Models with Jumps written by Mikhail Chernov and published by . This book was released on 2012 with total page 37 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this paper is to propose a new class of jump diffusions which feature both stochastic volatility and random intensity jumps. Previous studies have focused primarily on pure jump processes with constant intensity and log-normal jumps or constant jump intensity combined with a one factor stochastic volatility model. We introduce several generalizations which can better accommodate several empirical features of returns data. In their most general form we introduce a class of processes which nests jump-diffusions previously considered in empirical work and includes the affine class of random intensity models studied by Bates (1998) and Duffie, Pan and Singleton (1998) but also allows for non-affine random intensity jump components. We attain the generality of our specification through a generic Levy process characterization of the jump component. The processes we introduce share the desirable feature with the affine class that they yield analytically tractable and explicit option pricing formula. The non-affine class of processes we study include specifications where the random intensity jump component depends on the size of the previous jump which represent an alternative to affine random intensity jump processes which feature correlation between the stochastic volatility and jump component. We also allow for and experiment with different empirical specifications of the jump size distributions. We use two types of data sets. One involves the Samp;P500 and the other comprises of 100 years of daily Dow Jones index. The former is a return series often used in the literature and allows us to compare our results with previous studies. The latter has the advantage to provide a long time series and enhances the possibility of estimating the jump component more precisely. The non-affine random intensity jump processes are more parsimonious than the affine class and appear to fit the data much better.
Book Synopsis Simulation Based Estimation of Stochastic Volatility Type Models by : Christian Mücher
Download or read book Simulation Based Estimation of Stochastic Volatility Type Models written by Christian Mücher and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Stochastic Volatility Models with Jumps and High Frequency Data by : Jonas Kau
Download or read book Stochastic Volatility Models with Jumps and High Frequency Data written by Jonas Kau and published by . This book was released on 2009 with total page 163 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Mathematical Modeling and Methods of Option Pricing by : Lishang Jiang
Download or read book Mathematical Modeling and Methods of Option Pricing written by Lishang Jiang and published by World Scientific. This book was released on 2005 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the perspective of partial differential equations (PDE), this book introduces the Black-Scholes-Merton's option pricing theory. A unified approach is used to model various types of option pricing as PDE problems, to derive pricing formulas as their solutions, and to design efficient algorithms from the numerical calculation of PDEs.
