Download or read book Application of Stochastic Volatility Models in Option Pricing written by Pascal Debus and published by GRIN Verlag. This book was released on 2013-09-09 with total page 59 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bachelorarbeit aus dem Jahr 2010 im Fachbereich BWL - Investition und Finanzierung, Note: 1,2, EBS Universität für Wirtschaft und Recht, Sprache: Deutsch, Abstract: The Black-Scholes (or Black-Scholes-Merton) Model has become the standard model for the pricing of options and can surely be seen as one of the main reasons for the growth of the derivative market after the model ́s introduction in 1973. As a consequence, the inventors of the model, Robert Merton, Myron Scholes, and without doubt also Fischer Black, if he had not died in 1995, were awarded the Nobel prize for economics in 1997. The model, however, makes some strict assumptions that must hold true for accurate pricing of an option. The most important one is constant volatility, whereas empirical evidence shows that volatility is heteroscedastic. This leads to increased mispricing of options especially in the case of out of the money options as well as to a phenomenon known as volatility smile. As a consequence, researchers introduced various approaches to expand the model by allowing the volatility to be non-constant and to follow a sto-chastic process. It is the objective of this thesis to investigate if the pricing accuracy of the Black-Scholes model can be significantly improved by applying a stochastic volatility model.

Download or read book Option Prices in Stochastic Volatility Models written by Giulia Terenzi and published by . This book was released on 2018 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study option pricing problems in stochastic volatility models. In the first part of this thesis we focus on American options in the Heston model. We first give an analytical characterization of the value function of an American option as the unique solution of the associated (degenerate) parabolic obstacle problem. Our approach is based on variational inequalities in suitable weighted Sobolev spaces and extends recent results of Daskalopoulos and Feehan (2011, 2016) and Feehan and Pop (2015). We also investigate the properties of the American value function. In particular, we prove that, under suitable assumptions on the payoff, the value function is nondecreasing with respect to the volatility variable. Then, we focus on an American put option and we extend some results which are well known in the Black and Scholes world. In particular, we prove the strict convexity of the value function in the continuation region, some properties of the free boundary function, the Early Exercise Price formula and a weak form of the smooth fit principle. This is done mostly by using probabilistic techniques.In the second part we deal with the numerical computation of European and American option prices in jump-diffusion stochastic volatility models. We first focus on the Bates-Hull-White model, i.e. the Bates model with a stochastic interest rate. We consider a backward hybrid algorithm which uses a Markov chain approximation (in particular, a “multiple jumps” tree) in the direction of the volatility and the interest rate and a (deterministic) finite-difference approach in order to handle the underlying asset price process. Moreover, we provide a simulation scheme to be used for Monte Carlo evaluations. Numerical results show the reliability and the efficiency of the proposed methods.Finally, we analyze the rate of convergence of the hybrid algorithm applied to general jump-diffusion models. We study first order weak convergence of Markov chains to diffusions under quite general assumptions. Then, we prove the convergence of the algorithm, by studying the stability and the consistency of the hybrid scheme, in a sense that allows us to exploit the probabilistic features of the Markov chain approximation.

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

Download or read book Analytical Comparisons of Option Prices in Stochastic Volatility Models written by Vicky Henderson and published by . This book was released on 2002 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Option Pricing with Long Memory Stochastic Volatility Models written by Zhigang Tong and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we propose two continuous time stochastic volatility models with long memory that generalize two existing models. More importantly, we provide analytical formulae that allow us to study option prices numerically, rather than by means of simulation. We are not aware about analytical results in continuous time long memory case. In both models, we allow for the non-zero correlation between the stochastic volatility and stock price processes. We numerically study the effects of long memory on the option prices. We show that the fractional integration parameter has the opposite effect to that of volatility of volatility parameter in short memory models. We also find that long memory models have the potential to accommodate the short term options and the decay of volatility skew better than the corresponding short memory stochastic volatility models.

Download or read book Analytical Approximations of Option Prices in Stochastic Volatility Models written by and published by . This book was released on 2007 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Option Pricing with Long Memory Stochastic Volatility Models written by Zhigang Tong and published by LAP Lambert Academic Publishing. This book was released on 2013 with total page 184 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is now known that long memory stochastic volatility models can capture the well-documented evidence of volatility persistence. However, due to the complex structures of the long memory processes, the analytical formulas for option prices are not available yet. In this book, we propose two fractional continuous time stochastic volatility models which are built on the popular short memory stochastic volatility models. Using the tools from stochastic calculus, fractional calculus and Fourier transform, we derive the (approximate) analytical solutions for option prices. We also numerically study the effects of long memory on option prices. We show that the fractional integration parameter has the opposite effect to that of volatility of volatility parameter. We also find that long memory models can accommodate the short term options and the decay of volatility skew better than the corresponding short memory models. These findings would appeal to the researchers and practitioners in the areas of quantitative finance.

Download or read book Empirical Performance of Option Pricing Models with Stochastic Local Volatility written by Greg Orosi and published by . This book was released on 2014 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt: We examine the empirical performance of several stochastic local volatility models that are the extensions of the Heston stochastic volatility model. Our results indicate that the stochastic volatility model with quadratic local volatility significantly outperforms the stochastic volatility model with CEV type local volatility. Moreover, we compare the performance of these models to several other benchmarks and find that the quadratic local volatility model compares well to the best performing option pricing models reported in the current literature for European-style S&P500 index options. Our results also indicate that the model with quadratic local volatility reproduces the characteristics of the implied volatility surface more accurately than the Heston model. Finally, we demonstrate that capturing the shape of the implied volatility surface is necessary to price binary options accurately.

Download or read book Option Valuation Under Stochastic Volatility written by Alan L. Lewis and published by . This book was released on 2000 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Preliminary View of Calculating Call Option Prices Utilizing Stochastic Volatility Models written by Karl Shen and published by . This book was released on 2009 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: We will begin with a review of key financial topics and outline many of the crucial ideas utilized in the latter half of the paper. Formal notation for important variables will also be established. Then, a derivation of the Black-Scholes equation will lead to a discussion of its shortcomings, and the introduction of stochastic volatility models. Chapter 2 will focus on a variation of the CIR Model using stock price in the volatility driving process, and its behavior to a greater degree. The key area of discussion will be to approximate a hedging function for European call option prices by Taylor Expansion. We will apply this estimation to real data, and analyze the behavior of the price correction. Then make conclusions about whether stock price has any positive effects on the model.

Download or read book Option Pricing Models and Volatility Using Excel VBA written by Fabrice D. Rouah and published by John Wiley & Sons. This book was released on 2012-06-15 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: This comprehensive guide offers traders, quants, and students the tools and techniques for using advanced models for pricing options. The accompanying website includes data files, such as options prices, stock prices, or index prices, as well as all of the codes needed to use the option and volatility models described in the book. Praise for Option Pricing Models & Volatility Using Excel-VBA "Excel is already a great pedagogical tool for teaching option valuation and risk management. But the VBA routines in this book elevate Excel to an industrial-strength financial engineering toolbox. I have no doubt that it will become hugely successful as a reference for option traders and risk managers." —Peter Christoffersen, Associate Professor of Finance, Desautels Faculty of Management, McGill University "This book is filled with methodology and techniques on how to implement option pricing and volatility models in VBA. The book takes an in-depth look into how to implement the Heston and Heston and Nandi models and includes an entire chapter on parameter estimation, but this is just the tip of the iceberg. Everyone interested in derivatives should have this book in their personal library." —Espen Gaarder Haug, option trader, philosopher, and author of Derivatives Models on Models "I am impressed. This is an important book because it is the first book to cover the modern generation of option models, including stochastic volatility and GARCH." —Steven L. Heston, Assistant Professor of Finance, R.H. Smith School of Business, University of Maryland

Download or read book A Comparison of Option Prices Under Different Pricing Measures in a Stochastic Volatility Model with Correlation written by Vicky Henderson and published by . This book was released on 2004 with total page 34 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk.As an example, and as our main motivation, we investigate option pricing under the class of q-optimal pricing measures. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and variance-optimal pricing measures. If the Sharpe ratio is deterministic, the comparison collapses to the well known result that option prices computed under these three pricing measures are the same.As a concrete example, we specialise to a variant of the Heston model for which the Sharpe ratio is increasing in volatility. For this example we are able to deduce option prices are decreasing in the parameter q. Numerical solution of the pricing pde corroborates the theory and shows the magnitude of the differences in option price due to varying q. Choice of quot;qquot; is shown to influence the level of the implied volatility smile for options of varying maturity.

Download or read book A General Formula for Option Prices in a Stochastic Volatility Model written by Stephen Chin and published by . This book was released on 2009 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Simple New Formula for Options with Stochastic Volatility written by Steven L. Heston and published by . This book was released on 1998 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper shows a relationship between bond pricing models and option pricing models with stochastic volatility. It exploits this relationship to find a new stochastic volatility model with a closed-form solution for European option prices. The model allows nonzero correlation between volatility and spot asset returns. When the correlation is unity the model contains the Black-Scholes [1973] model and Cox's [1975] constant elasticity of variance model as special cases. The option formula preserves the Black-Scholes property that changes in volatility are equivalent to changes in option expiration.

Download or read book Volatility Surface and Term Structure written by Kin Keung Lai and published by Routledge. This book was released on 2013-09-11 with total page 102 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides different financial models based on options to predict underlying asset price and design the risk hedging strategies. Authors of the book have made theoretical innovation to these models to enable the models to be applicable to real market. The book also introduces risk management and hedging strategies based on different criterions. These strategies provide practical guide for real option trading. This book studies the classical stochastic volatility and deterministic volatility models. For the former, the classical Heston model is integrated with volatility term structure. The correlation of Heston model is considered to be variable. For the latter, the local volatility model is improved from experience of financial practice. The improved local volatility surface is then used for price forecasting. VaR and CVaR are employed as standard criterions for risk management. The options trading strategies are also designed combining different types of options and they have been proven to be profitable in real market. This book is a combination of theory and practice. Users will find the applications of these financial models in real market to be effective and efficient.

Download or read book The Black Scholes and Heston Models for Option Pricing written by Ziqun Ye and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic volatility models on option pricing have received much study following the discovery of the non-at implied surface following the crash of the stock markets in 1987. The most widely used stochastic volatility model is introduced by Heston (1993) because of its ability to generate volatility satisfying the market observations, being non-negative and mean-reverting, and also providing a closed-form solution for the European options. However, little research has been done on Heston model used to price early-exercise options. This presumably is largely due to the absence of a closed-form solution and the increase in computational requirement that complicates the required calibration exercise. This thesis examines the performance of the Heston model versus the Black-Scholes model for the American Style equity option of Microsoft and the index option of S&P 100 index. We employ a finite difference method combined with a Projected Successive Over-relaxation method for pricing an American put option under the Black-Scholes model, while an Alternating Direction Implicit method is utilized to decompose a multi-dimensional partial differential equation into several one dimensional steps under the Heston model. For the calibration of the Heston model, we apply a two step procedure where in the first step we apply an indirect inference method to historical stock prices to estimate diffusion parameters under a probability measure and then use a least squares method to estimate the instantaneous volatility and the market risk premium which are used to switch from working under the probability measure to working under the risk-neutral measure. We find that option price is positively related with the value of the mean reverting speed and the long-term variance. It is not sensitive to the market price of risk and it is negatively related with the risk free rate and the volatility of volatility. By comparing the European put option and the American put option under the Heston model, we observe that their implied volatility generally follow similar patterns. However, there are still some interesting observations that can be made from the comparison of the two put options. First, for the out-of-the-money category, the American and European options have rather comparable implied volatilities with the American options' implied volatility being slightly bigger than the European options. While for the in-the-money category, the implied volatility of the European options is notably higher than the American options and its value exceeds the implied volatility of the American options. We also assess the performance of the Heston model by comparing its result with the result from the Black-Scholes model. We observe that overall the Heston model performs better than the Black-Scholes model. In particular, the Heston model has tendency of underpricing the in-the-money option and overpricing the out-of-the-money option. Whereas, the Black-Scholes model is inclined to underprice both the in-the-money option and the out-of-the-money option.b.

Download or read book Analytically Tractable Stochastic Stock Price Models written by Archil Gulisashvili and published by Springer. This book was released on 2012-09-05 with total page 0 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.