Download or read book Exact Closed Form Formulas and Saddlepoint Approximation Methods for Pricing VIX Derivatives Under Alternative Stochastic Volatility Models written by Yue Wang and published by . This book was released on 2013 with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Saddlepoint Approximation Methods for Pricing Derivatives on Discrete Realized Variance written by Wendong Zheng and published by . This book was released on 2013 with total page 36 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider the saddlepoint approximation methods for pricing derivatives whose payoffs depend on the discrete realized variance of the underlying price process of a risky asset. Most of the earlier pricing models of variance products and volatility derivatives use the quadratic variation approximation as the continuous limit of the discrete realized variance. However, the corresponding discretization gap may become significant for short-maturity derivatives. Under L evy models and stochastic volatility models with jumps, we manage to obtain the saddlepoint approximation formulas for pricing variance products and volatility derivatives using the small time asymptotic approximation of the Laplace transform of the discrete realized variance. As an alternative approach, we also develop the conditional saddlepoint approximation method based on a given simulated stochastic variance path via Monte Carlo simulation. This analytic-simulation approach reduces the dimensionality of the simulation of the discrete variance derivatives; and in some cases, the simulation procedure of the realized variance can be effectively performed using an appropriate exact simulation method. We examine numerical accuracy and reliability of various types of the saddlepoint approximation techniques when applied to pricing derivatives on discrete realized variance under different types of asset price processes. The limitations of the saddlepoint approximation methods in pricing variance products and volatility derivatives are also discussed.
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
Download or read book VIX Derivatives Valuation and Estimation Based on Closed Form Series Expansions written by Zhe Zhao and published by . This book was released on 2018 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: A new valuation and calibration method for VIX futures and VIX options is proposed. The method is based on a closed-form Hermite series expansion for a stochastic volatility model with the stochastic variance process driven by an affine drift term. We implement the methodology for the Heston and the mean-reverting CEV stochastic volatility models. A calibration exercise to real market data shows that the method is efficient, accurate, and suitable for practical implementation.
Download or read book Alternative Models for Stochastic Volatility Corrections for Equity and Interest Rate Derivatives written by Tianyu Liang and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: ABSTRACT: A lot of attention has been paid to the stochastic volatility model where the volatility is randomly fluctuating driven by an additional Brownian motion. In our work, we change the mean level in the mean-reverting process from a constant to a function of the underlying process. We apply our models to the pricing of both equity and interest rate derivatives. Throughout the thesis, a singular perturbation method is employed to derive closed-form formulas up to first order asymptotic solutions. We also implement multiplicative noise to arithmetic Ornstein-Uhlenbeck process to produce a wider variety of effects. Calibration and Monte Carlo simulation results show that the proposed model outperform Fouque's original stochastic volatility model during some particular window in history. A more efficient numerical scheme, the heterogeneous multi-scale method (HMM), is introduced to simulate the multi-scale differential equations discussed over the chapters.
Download or read book An Analytical Formula for VIX Futures and Its Applications 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: In this study we present a closed-form, exact solution for the pricing of VIX futures in a stochastic volatility model with simultaneous jumps in both the asset price and volatility processes. The newly derived formula is then used to show that the well-known convexity correction approximations can sometimes lead to large errors. Utilizing the newly derived formula, we also conduct an empirical study, the results of which demonstrate that the Heston stochastic volatility model is a good candidate for the pricing of VIX futures. While incorporating jumps into the underlying price can further improve the pricing of VIX futures, adding jumps to the volatility process appears to contribute little improvement for pricing VIX futures.
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.
Download or read book Pricing Options Under Simultaneous Stochastic Volatility and Jumps written by Moawia Alghalith and published by . This book was released on 2019 with total page 8 pages. Available in PDF, EPUB and Kindle. Book excerpt: We overcome the limitations of the previous literature in the European options pricing. In doing so, we provide a closed-form formula that doesn't require any numerical/computational methods. The formula is as simple as the classical Black-Scholes pricing formula. In addition, we simultaneously include jumps and stochastic volatility.
Download or read book Solving Asset Pricing Models with Stochastic Volatility written by Oliver De Groot and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Pricing Derivatives in Stochastic Volatility Models Using the Finite Difference Method written by and published by . This book was released on 2001 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The Heston stochastic volatility model is one extension of the Black-Scholes model which describes the money markets more accurately so that more realistic prices for derivative products are obtained. From the stochastic differential equation of the underlying financial product a partial differential equation (p.d.e.) for the value function of an option can be derived. This p.d.e. can be solved with the finite difference method (f.d.m.). The stability and consistency of the method is examined. Furthermore a boundary condition is proposed to reduce the numerical error. Finally a non uniform structured grid is derived which is fairly optimal for the numerical result in the most interesting point.
Download or read book Stochastic volatility and the pricing of financial derivatives written by Antoine Petrus Cornelius van der Ploeg and published by Rozenberg Publishers. This book was released on 2006 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Implied Remaining Variance in Derivative Pricing written by Peter Carr and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we give a way to calculate a option implied volatility curve in closed form via the well known quadratic root formula. The closed form expression has 3 free parameters, which parsimoniously govern the assumed dynamics of implied volatility under forward swap measure. Preliminary empirical work suggests the curve fits the swaptions market well (though not perfectly). Unlike previous models of stochastic implied volatility, the current model has no implications for the dynamics of instantaneous volatility.
Download or read book Pricing Vix Options with Stochastic Volatility and Random Jumps written by Guanghua Lian and published by . This book was released on 2012 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This study presents an analytical exact solution for the price of VIX options under stochastic volatility model with simultaneous jumps in the asset price and volatility processes. We shall demonstrate that our new pricing formula can be used to efficiently compute the numerical values of a VIX option. While we also show that the numerical results obtained from our formula consistently match those obtained from Monte Carlo simulation perfectly as a verification of the correctness of our formula, numerical evidence is offered to illustrate that the correctness of the formula proposed in Lin & Chang (2009) is in serious doubt. Moreover, some important and distinct properties of VIX options (e.g., put-call parity, hedging ratios) are also examined and discussed.
Download or read book Derivatives Written on a Power of the Stock Price written by Angelika Esser and published by . This book was released on 2003 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper deals with the pricing of di%0Berent types of power options which are of practical relevance. First of all, a general valuation technique is developed that can be applied to several non-negative payo%0Bs being a function of the terminal stock price only. The key trick in the derivation is to forget about a choice of numeraires in the first place, but to use an appropriate density to change the measure. This allows for an elegant pricing equation in terms of artificial probabilities similar to the Black Scholes formula. However, the probabilities are not necessarily martingale measures. We gain economic intuition in this change of measure when we distinguish between complete and incomplete markets. In a complete market we derive that the measure corresponds to a traded numeraire portfolio, which means that it is a martingale measure whereas in incomplete markets this is in general no longer true. Furthermore, the method is applied to two examples. First, the standard Black Scholes framework is considered as an example for a complete market where the corresponding numeraire portfolio is computed explicitly. Second, we deal with stochastic volatility models as an example for an incomplete market. Closed-form solutions are derived for quot;power optionsquot; (written on a power of the underlying stock) with and without cap, and quot;powered optionsquot; (where the option payo%0B is raised to some power). There are two key contributions of this paper. First, we gain economic insight in the relationship between change of measure and change of numeraire in complete and incomplete markets. Second, we derive closed-form solutions for several types of power options under stochastic volatility.
Download or read book Pricing Derivatives Under L vy Models written by Andrey Itkin and published by . This book was released on 2017 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Derivative Evaluation Using Recombining Trees Under Stochastic Volatility written by Enrico Moretto and published by . This book was released on 2013 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt: Heston (1993) presents a method to derive a closed-form solution for derivative pricing when the volatility of the underlying asset follows stochastic dynamics. His approach works well for European derivatives but, unfortunately, does not readily extend to the pricing of more complex contracts. In this paper we propose an alternative stochastic volatility model which retains many features of Heston model, but is better suited for an easy discretization through recombining trees, in the spirit of Nelson and Ramaswamy (1990). After having discussed the theoretical properties of the model we construct its discretized counterpart through a recombining multinomial tree. We apply the model to the USD/EURO exchange rate market, evaluating both American and barrier options.
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.
