A Closed-Form Exact Solution for Pricing Variance Swaps With Stochastic Volatility

A Closed-Form Exact Solution for Pricing Variance Swaps With Stochastic Volatility
Author: Song-Ping Zhu
Publisher:
Total Pages: 0
Release: 2012
Genre:
ISBN:
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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.

On the Valuation of Variance Swaps with Stochastic Volatility

On the Valuation of Variance Swaps with Stochastic Volatility
Author: Song-Ping Zhu
Publisher:
Total Pages: 0
Release: 2011
Genre:
ISBN:
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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.

Pricing Variance Swaps Under Stochastic Volatility and Stochastic Interest Rate

Pricing Variance Swaps Under Stochastic Volatility and Stochastic Interest Rate
Author: Jiling Cao
Publisher:
Total Pages: 16
Release: 2014
Genre:
ISBN:
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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.

Pricing Models of Volatility Products and Exotic Variance Derivatives

Pricing Models of Volatility Products and Exotic Variance Derivatives
Author: Yue Kuen Kwok
Publisher: CRC Press
Total Pages: 402
Release: 2022-05-08
Genre: Mathematics
ISBN: 1000584275
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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

Modeling And Pricing Of Swaps For Financial And Energy Markets With Stochastic Volatilities

Modeling And Pricing Of Swaps For Financial And Energy Markets With Stochastic Volatilities
Author: Anatoliy Swishchuk
Publisher: World Scientific
Total Pages: 326
Release: 2013-06-03
Genre: Business & Economics
ISBN: 9814440140
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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.

Pricing Models of Volatility Products and Exotic Variance Derivatives

Pricing Models of Volatility Products and Exotic Variance Derivatives
Author: Yue Kuen Kwok
Publisher: CRC Press
Total Pages: 283
Release: 2022-05-08
Genre: Business & Economics
ISBN: 1000584259
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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

Variance Swap with Mean Reversion, Multifactor Stochastic Volatility and Jumps

Variance Swap with Mean Reversion, Multifactor Stochastic Volatility and Jumps
Author: Chi Seng Pun
Publisher:
Total Pages: 38
Release: 2020
Genre:
ISBN:
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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.

Pricing Exotic Variance Swaps Under 3/2-Stochastic Volatility Models

Pricing Exotic Variance Swaps Under 3/2-Stochastic Volatility Models
Author: Chi Yuen
Publisher:
Total Pages: 26
Release: 2015
Genre:
ISBN:
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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.

Recent Developments in Data Science and Business Analytics

Recent Developments in Data Science and Business Analytics
Author: Madjid Tavana
Publisher: Springer
Total Pages: 494
Release: 2018-03-27
Genre: Business & Economics
ISBN: 3319727451
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This edited volume is brought out from the contributions of the research papers presented in the International Conference on Data Science and Business Analytics (ICDSBA- 2017), which was held during September 23-25 2017 in ChangSha, China. As we all know, the field of data science and business analytics is emerging at the intersection of the fields of mathematics, statistics, operations research, information systems, computer science and engineering. Data science and business analytics is an interdisciplinary field about processes and systems to extract knowledge or insights from data. Data science and business analytics employ techniques and theories drawn from many fields including signal processing, probability models, machine learning, statistical learning, data mining, database, data engineering, pattern recognition, visualization, descriptive analytics, predictive analytics, prescriptive analytics, uncertainty modeling, big data, data warehousing, data compression, computer programming, business intelligence, computational intelligence, and high performance computing among others. The volume contains 55 contributions from diverse areas of Data Science and Business Analytics, which has been categorized into five sections, namely: i) Marketing and Supply Chain Analytics; ii) Logistics and Operations Analytics; iii) Financial Analytics. iv) Predictive Modeling and Data Analytics; v) Communications and Information Systems Analytics. The readers shall not only receive the theoretical knowledge about this upcoming area but also cutting edge applications of this domains.

Exact Pricing with Stochastic Volatility and Jumps

Exact Pricing with Stochastic Volatility and Jumps
Author: Fernanda D'Ippoliti
Publisher:
Total Pages: 25
Release: 2014
Genre:
ISBN:
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A stochastic volatility jump-diffusion model for pricing derivatives with jumps in both spot returns and volatility dynamics is presented. This model admits, in the spirit of Heston, a closed-form solution for European-style options. The structure of the model is also suitable to obtain the fair delivery price of variance swaps. To evaluate derivatives whose value does not admit a closed-form expression, a methodology based on an "exact algorithm'', in the sense that no discretization of equations is required, is developed and applied to barrier options. Goodness of pricing algorithm is tested using DJ Euro Stoxx 50 market data for European options. Finally, the algorithm is applied to compute prices and Greeks of barrier options.