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Pricing Exotic Variance Swaps Under 3 2 Stochastic Volatility Models
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Book Synopsis Pricing Exotic Variance Swaps Under 3/2-Stochastic Volatility Models by : Chi Yuen
Download or read book Pricing Exotic Variance Swaps Under 3/2-Stochastic Volatility Models written by Chi Yuen and published by . This book was released on 2015 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider pricing of various types of exotic discrete variance swaps, like the gamma swaps and corridor swaps, under the 3/2-stochastic volatility models with jumps. The class of stochastic volatility models (SVM) that use a constant-elasticity-of-variance (CEV) process for the instantaneous variance exhibit nice analytical tractability when the CEV parameter takes just a few special values (namely, 0, 1/2, 1 and 3/2). The popular Heston model corresponds to the choice of the CEV parameter to be 1/2. However, the stochastic volatility dynamics derived from the Heston model fails to agree with empirical findings from actual market data. The choice of 3/2 for the CEV parameter in the SVM shows better agreement with empirical studies while it maintains a good level of analytical tractability. By using the partial integro-differential equation formulation, we manage to derive quasi-closed form pricing formulas for the fair strike values of various types of discrete variance swaps. Pricing properties of these exotic discrete variance swaps under different market conditions are explored.
Book Synopsis Pricing Models of Volatility Products and Exotic Variance Derivatives by : Yue Kuen Kwok
Download or read book Pricing Models of Volatility Products and Exotic Variance Derivatives written by Yue Kuen Kwok and published by CRC Press. This book was released on 2022-05-08 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: Pricing Models of Volatility Products and Exotic Variance Derivatives summarizes most of the recent research results in pricing models of derivatives on discrete realized variance and VIX. The book begins with the presentation of volatility trading and uses of variance derivatives. It then moves on to discuss the robust replication strategy of variance swaps using portfolio of options, which is one of the major milestones in pricing theory of variance derivatives. The replication procedure provides the theoretical foundation of the construction of VIX. This book provides sound arguments for formulating the pricing models of variance derivatives and establishes formal proofs of various technical results. Illustrative numerical examples are included to show accuracy and effectiveness of analytic and approximation methods. Features Useful for practitioners and quants in the financial industry who need to make choices between various pricing models of variance derivatives Fabulous resource for researchers interested in pricing and hedging issues of variance derivatives and VIX products Can be used as a university textbook in a topic course on pricing variance derivatives
Book Synopsis Pricing Models of Volatility Products and Exotic Variance Derivatives by : Yue Kuen Kwok
Download or read book Pricing Models of Volatility Products and Exotic Variance Derivatives written by Yue Kuen Kwok and published by CRC Press. This book was released on 2022-05-08 with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt: Pricing Models of Volatility Products and Exotic Variance Derivatives summarizes most of the recent research results in pricing models of derivatives on discrete realized variance and VIX. The book begins with the presentation of volatility trading and uses of variance derivatives. It then moves on to discuss the robust replication strategy of variance swaps using portfolio of options, which is one of the major milestones in pricing theory of variance derivatives. The replication procedure provides the theoretical foundation of the construction of VIX. This book provides sound arguments for formulating the pricing models of variance derivatives and establishes formal proofs of various technical results. Illustrative numerical examples are included to show accuracy and effectiveness of analytic and approximation methods. Features Useful for practitioners and quants in the financial industry who need to make choices between various pricing models of variance derivatives Fabulous resource for researchers interested in pricing and hedging issues of variance derivatives and VIX products Can be used as a university textbook in a topic course on pricing variance derivatives
Book Synopsis Variance and Volatility Swaps and Futures Pricing for Stochastic Volatility Models by : Anatoliy V. Swishchuk
Download or read book Variance and Volatility Swaps and Futures Pricing for Stochastic Volatility Models written by Anatoliy V. Swishchuk and published by . This book was released on 2017 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this chapter, we consider volatility swap, variance swap and VIX future pricing under different stochastic volatility models and jump diffusion models which are commonly used in financial market. We use convexity correction approximation technique and Laplace transform method to evaluate volatility strikes and estimate VIX future prices. In empirical study, we use Markov chain Monte Carlo algorithm for model calibration based on S&P 500 historical data, evaluate the effect of adding jumps into asset price processes on volatility derivatives pricing, and compare the performance of different pricing approaches.
Book Synopsis Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities by : Anatoli? Vital?evich Svishchuk
Download or read book Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities written by Anatoli? Vital?evich Svishchuk and published by World Scientific. This book was released on 2013 with total page 326 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities is devoted to the modeling and pricing of various kinds of swaps, such as those for variance, volatility, covariance, correlation, for financial and energy markets with different stochastic volatilities, which include CIR process, regime-switching, delayed, mean-reverting, multi-factor, fractional, Levy-based, semi-Markov and COGARCH(1,1). One of the main methods used in this book is change of time method. The book outlines how the change of time method works for different kinds of models and problems arising in financial and energy markets and the associated problems in modeling and pricing of a variety of swaps. The book also contains a study of a new model, the delayed Heston model, which improves the volatility surface fitting as compared with the classical Heston model. The author calculates variance and volatility swaps for this model and provides hedging techniques. The book considers content on the pricing of variance and volatility swaps and option pricing formula for mean-reverting models in energy markets. Some topics such as forward and futures in energy markets priced by multi-factor Levy models and generalization of Black-76 formula with Markov-modulated volatility are part of the book as well, and it includes many numerical examples such as S&P60 Canada Index, S&P500 Index and AECO Natural Gas Index.
Book Synopsis A Closed-Form Exact Solution for Pricing Variance Swaps With Stochastic Volatility by : Song-Ping Zhu
Download or read book A Closed-Form Exact Solution for Pricing Variance Swaps With Stochastic Volatility written by Song-Ping Zhu and published by . This book was released on 2012 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we present a highly efficient approach to price variance swaps with discrete sampling times. We have found a closed-form exact solution for the partial differential equation (PDE) system based on the Heston's two-factor stochastic volatility model embedded in the framework proposed by Little and Pant. In comparison with the previous approximation models based on the assumption of continuous sampling time, the current research of working out a closed-form exact solution for variance swaps with discrete sampling times at least serves for two major purposes: (i) to verify the degree of validity of using a continuous-sampling-time approximation for variance swaps of relatively short sampling period; (ii) to demonstrate that significant errors can result from still adopting such an assumption for a variance swap with small sampling frequencies or long tenor. Other key features of our new solution approach include the following: (1) with the newly found analytic solution, all the hedging ratios of a variance swap can also be analytically derived; (2) numerical values can be very efficiently computed from the newly found analytic formula.
Book Synopsis A Unified Valuation Framework for Variance Swaps Under Non-Affine Stochastic Volatility Models by : Alex Badescu
Download or read book A Unified Valuation Framework for Variance Swaps Under Non-Affine Stochastic Volatility Models written by Alex Badescu and published by . This book was released on 2017 with total page 38 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this article, we investigate the pricing and convergence of general non-affine non-Gaussian GARCH-based variance swap prices. Explicit solutions for fair strike prices under two different sampling schemes are derived using the extended Girsanov principle as our pricing kernel candidate. Following standard assumptions on the time-varying GARCH parameters, we show that these quantities converge to discretely and continuously sampled variance swaps constructed based on the weak diffusion limit of the underlying GARCH model. An empirical study which relies on a joint estimation using both historical returns and VIX data indicates that an asymmetric heavier-tailed distribution is more appropriate for modelling the GARCH innovations. Finally, we provide several numerical exercises to support our theoretical convergence results in which we investigate the effect of the quadratic variation approximation for the realized variance, as well as the impact of discrete versus continuous-time modelling of asset returns.
Book Synopsis Pricing Variance Swaps Under Stochastic Volatility and Stochastic Interest Rate by : Jiling Cao
Download or read book Pricing Variance Swaps Under Stochastic Volatility and Stochastic Interest Rate written by Jiling Cao and published by . This book was released on 2014 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we investigate the effects of imposing stochastic interest rate driven by the Cox-Ingersoll-Ross process along with the Heston stochastic volatility model for pricing variance swaps with discrete sampling times. A dimension reduction mechanism based on the framework of Little and Pant is applied which later reduces to solving sets of one-dimensional partial differential equation. A close form exact solution to the fair delivery price of a variance swap is obtained via derivation of characteristic functions. Practical implementation of this hybrid model is demonstrated through numerical simulations.
Book Synopsis On the Valuation of Variance Swaps with Stochastic Volatility by : Song-Ping Zhu
Download or read book On the Valuation of Variance Swaps with Stochastic Volatility written by Song-Ping Zhu and published by . This book was released on 2011 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper is an extension to a recent paper Zhu and Lian (2009), in which a closed-form exact solution was presented for the price of variance swaps with a particular definition of the realized variance. Here, we further demonstrate that our approach is quite versatile and can be used for other definitions of the realized variance as well. In particular, we present a closed-form formula for the price of a variance swap with the realized variance in the payoff function being defined as a logarithmic return of the underlying asset at some pre-specified discretely sampling points. The simple formula presented here is a result of successfully finding an exact solution of the partial differential equation (PDE) system based on the Heston's (1993) two-factor stochastic volatility model. A distinguishable feature of this new solution is that the computational time involved in pricing variance swaps with discretely sampling time has been substantially improved.
Book Synopsis A General Framework for Discretely Sampled Realized Variance Derivatives in Stochastic Volatility Models with Jumps by : Zhenyu Cui
Download or read book A General Framework for Discretely Sampled Realized Variance Derivatives in Stochastic Volatility Models with Jumps written by Zhenyu Cui and published by . This book was released on 2018 with total page 43 pages. Available in PDF, EPUB and Kindle. Book excerpt: After the recent financial crisis, the market for volatility derivatives has expanded rapidly to meet the demand from investors, risk managers and speculators seeking diversification of the volatility risk. In this paper, we develop a novel and efficient transform-based method to price swaps and options related to discretely-sampled realized variance under a general class of stochastic volatility models with jumps. We utilize frame duality and density projection method combined with a novel continuous-time Markov chain (CTMC) weak approximation scheme of the underlying variance process. Contracts considered include discrete variance swaps, discrete variance options, and discrete volatility options. Models considered include several popular stochastic volatility models with a general jump size distribution: Heston, Scott, Hull-White, Stein-Stein, alpha-Hypergeometric, 3/2 and 4/2 models. Our framework encompasses and extends the current literature on discretely sampled volatility derivatives, and provides highly efficient and accurate valuation methods. Numerical experiments confirm our findings.
Book Synopsis Closed-Form Partial Transform of Triple Joint Density for Pricing Exotic Options and Variance Derivatives Under the 3/2 Model by : Wendong Zheng
Download or read book Closed-Form Partial Transform of Triple Joint Density for Pricing Exotic Options and Variance Derivatives Under the 3/2 Model written by Wendong Zheng and published by . This book was released on 2014 with total page 35 pages. Available in PDF, EPUB and Kindle. Book excerpt: Most of the empirical studies on stochastic volatility dynamics favor the 3/2 specification over the square-root (CIR) process in the Heston model. In the context of option pricing, the 3/2 stochastic volatility model is reported to be able to capture the volatility skew evolution better than the Heston model. In this article, we make a thorough investigation on the analytic tractability of the 3/2 stochastic volatility model by proposing a closed-form formula for the partial transform of the triple joint transition density (X,I,V) which stand for the log asset price, the quadratic variation (continuous realized variance) and the instantaneous variance, respectively. Two different approaches are presented for deriving the key result. In the first approach, we obtain the partial transform by utilizing the exponential affine structure of the pair (X,I) and solving the governing PDE that involves V only. The second approach is more probabilistic and it makes use of the change of measure and conditioning techniques. The closed-form partial transform enables us to deduce a variety of marginal transition density functions or characteristic functions that are crucial in pricing discretely sampled variance derivatives and exotic options that depend on both the asset price and quadratic variation. Various applications and numerical examples on pricing exotic derivatives with forward start or discrete monitoring features are given to demonstrate our unified pricing framework based on the closed-form partial transform under the 3/2 model.
Book Synopsis Pricing Bermudan Variance Swaptions Using Multinomial Trees by : Honglei Zhao
Download or read book Pricing Bermudan Variance Swaptions Using Multinomial Trees written by Honglei Zhao and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In a recent study, we present a tree methodology to evaluate the expected generalized realized variance in a general stochastic volatility model. This provides an efficient way of calculating the fair value of the strike for variance swaps. In this article, we expand the methodology to price nonlinear derivatives written on realized variance. Particularly we introduce a new option contract a Bermudan variance swaption, defined as an option on variance swap with early exercise dates. Within the same framework, we also show how to value forward-start variance swaps, VIX futures and VIX options. Numerical tests show that the methodology introduced is efficient and accurate.
Book Synopsis Exotic Options and Hybrids by : Mohamed Bouzoubaa
Download or read book Exotic Options and Hybrids written by Mohamed Bouzoubaa and published by John Wiley & Sons. This book was released on 2010-05-17 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: The recent financial crisis brought to light many of the misunderstandings and misuses of exotic derivatives. With market participants on both the buy and sell-side having been found guilty of not understanding the products they were dealing with, never before has there been a greater need for clarification and explanation. Exotic Options and Hybrids is a practical guide to structuring, pricing and hedging complex exotic options and hybrid derivatives that will serve readers through the recent crisis, the road to recovery, the next bull market and beyond. Written by experienced practitioners, it focuses on the three main parts of a derivative’s life: the structuring of a product, its pricing and its hedging. Divided into four parts, the book covers a multitude of structures, encompassing many of the most up-to-date and promising products from exotic equity derivatives and structured notes to hybrid derivatives and dynamic strategies. Based on a realistic setting from the heart of the business, inside a derivatives operation, the practical and intuitive discussions of these aspects make these exotic concepts truly accessible. Adoptions of real trades are examined in detail, and all of the numerous examples are carefully selected so as to highlight interesting and significant aspects of the business. The introduction of payoff structures is accompanied by scenario analysis, diagrams and lifelike sample term sheets. Readers learn how to spot where the risks lie to pave the way for sound valuation and hedging of such products. There are also questions and accompanying discussions dispersed in the text, each exploited to illustrate one or more concepts from the context in which they are set. The applications, the strengths and the limitations of various models are highlighted, in relevance to the products and their risks, rather than the model implementations. Models are de-mystified in separately dedicated sections, but their implications are alluded to throughout the book in an intuitive and non-mathematical manner. By discussing exotic options and hybrids in a practical, non-mathematical and highly intuitive setting, this book will blast through the misunderstanding of exotic derivatives, enabling practitioners to fully understand and correctly structure, price and hedge theses products effectively, and stand strong as the only book in its class to make these “exotic” concepts truly accessible.
Book Synopsis Variance Swap with Mean Reversion, Multifactor Stochastic Volatility and Jumps by : Chi Seng Pun
Download or read book Variance Swap with Mean Reversion, Multifactor Stochastic Volatility and Jumps written by Chi Seng Pun and published by . This book was released on 2020 with total page 38 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper examines variance swap pricing using a model that integrates three major features of financial assets, namely the mean reversion in asset price, multi-factor stochastic volatility (SV) and simultaneous jumps in prices and volatility factors. Closed-form solutions are derived for vanilla variance swaps and gamma swaps while the solutions for corridor variance swaps and conditional variance swaps are expressed in a one-dimensional Fourier integral. The numerical tests confirm that the derived solution is accurate and efficient. Furthermore, empirical studies have shown that multi-factor SV models better capture the implied volatility surface from option data. The empirical results of this paper also show that the additional volatility factor contributes significantly to the price of variance swaps. Hence, the results favor multi-factor SV models for pricing variance swaps consistent with the implied volatility surface.
Book Synopsis Variance Swap Premium Under Stochastic Volatility and Self-exciting Jumps by : Ke Chen (Economist)
Download or read book Variance Swap Premium Under Stochastic Volatility and Self-exciting Jumps written by Ke Chen (Economist) and published by . This book was released on 2013 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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 Pricing Variance Swaps Under Stochastic Volatility and Stochastic Interest Rate by : Teh Raihana Nazirah binti Roslan
Download or read book Pricing Variance Swaps Under Stochastic Volatility and Stochastic Interest Rate written by Teh Raihana Nazirah binti Roslan and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Author supplied keywords: Variance swaps; Realized variance; Heston-CIR model; Volatility derivatives; Stochastic volatility; Stochastic interest rate.
