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Closed Form Approximation Of Timer Option Prices Under General Stochastic Volatility Models
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Book Synopsis Closed-Form Approximation of Timer Option Prices Under General Stochastic Volatility Models by : Minqiang Li
Download or read book Closed-Form Approximation of Timer Option Prices Under General Stochastic Volatility Models written by Minqiang Li and published by . This book was released on 2013 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt: We develop an asymptotic expansion technique for pricing timer options under general stochastic volatility models around small volatility of variance. Closed-form approximation formulas have been obtained for the Heston model and the 3/2-model. The approximation has an easy-to-understand Black-Scholes-like form and many other attractive properties. Numerical analysis shows that the approximation formulas are very fast and accurate.
Book Synopsis Problems and Solutions in Mathematical Finance, Volume 2 by : Eric Chin
Download or read book Problems and Solutions in Mathematical Finance, Volume 2 written by Eric Chin and published by John Wiley & Sons. This book was released on 2017-01-04 with total page 1042 pages. Available in PDF, EPUB and Kindle. Book excerpt: Detailed guidance on the mathematics behind equity derivatives Problems and Solutions in Mathematical Finance Volume II is an innovative reference for quantitative practitioners and students, providing guidance through a range of mathematical problems encountered in the finance industry. This volume focuses solely on equity derivatives problems, beginning with basic problems in derivatives securities before moving on to more advanced applications, including the construction of volatility surfaces to price exotic options. By providing a methodology for solving theoretical and practical problems, whilst explaining the limitations of financial models, this book helps readers to develop the skills they need to advance their careers. The text covers a wide range of derivatives pricing, such as European, American, Asian, Barrier and other exotic options. Extensive appendices provide a summary of important formulae from calculus, theory of probability, and differential equations, for the convenience of readers. As Volume II of the four-volume Problems and Solutions in Mathematical Finance series, this book provides clear explanation of the mathematics behind equity derivatives, in order to help readers gain a deeper understanding of their mechanics and a firmer grasp of the calculations. Review the fundamentals of equity derivatives Work through problems from basic securities to advanced exotics pricing Examine numerical methods and detailed derivations of closed-form solutions Utilise formulae for probability, differential equations, and more Mathematical finance relies on mathematical models, numerical methods, computational algorithms and simulations to make trading, hedging, and investment decisions. For the practitioners and graduate students of quantitative finance, Problems and Solutions in Mathematical Finance Volume II provides essential guidance principally towards the subject of equity derivatives.
Book Synopsis Option Prices in Stochastic Volatility Models by : Giulia Terenzi
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.
Book Synopsis Analytical Approximations of Option Prices in Stochastic Volatility Models by :
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:
Book Synopsis A Simple Calibration Procedure of Stochastic Volatility Models with Jumps by Short Term Asymptotics by : Alexey Medvedev
Download or read book A Simple Calibration Procedure of Stochastic Volatility Models with Jumps by Short Term Asymptotics written by Alexey Medvedev and published by . This book was released on 2011 with total page 56 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we develop approximating formulas for European options prices based on short term asymptotics, i.e. when time-to-maturity tends to zero. The analysis is performed in a general setting where stochastic volatility and jumps drive the dynamics of stock returns. In a numerical study we show that the closed form approximation is accurate for a broad range of option parameters typically encountered in practice. An empirical application illustrates its use in calibrating observed smiles of Samp;P 500 index options, and in getting new insight into the dependence of the volatility of volatility and jump size distribution on the spot volatility. We test the consistency of the calibration by showing that the shape of the volatility of volatility inferred from option prices agrees with its estimate from the time series of spot volatilities inferred from the same observed option prices.
Book Synopsis A Mean-Reverting Stochastic Volatility Option-Pricing Model with an Analytic Solution by : Henrik Andersson
Download or read book A Mean-Reverting Stochastic Volatility Option-Pricing Model with an Analytic Solution written by Henrik Andersson and published by . This book was released on 2002 with total page 45 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we derive a closed form approximation to a stochastic volatility option-pricing model and propose a variant of EGARCH for parameter estimation. The model thereby provides a consistent approach to the problem of option pricing and parameter estimation. Using Swedish stocks, the model provides a good fit to the heteroscedasticity prevalent in the time-series. The stochastic volatility model also prices options on the underlying stock more accurately than the traditional Black-Scholes formula. This result holds for both historic and implied volatility. A large part of the volatility smile that is observed for options of different maturity and exercise prices is thereby explained.
Book Synopsis Pricing Options Under Simultaneous Stochastic Volatility and Jumps by : Moawia Alghalith
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.
Book Synopsis A General Formula for Option Prices in a Stochastic Volatility Model by : Stephen Chin
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:
Book Synopsis Stochastic Volatility Models: Option Price Approximation, Asymptotics and Maximum Likelihood Estimation by : Jian Yang
Download or read book Stochastic Volatility Models: Option Price Approximation, Asymptotics and Maximum Likelihood Estimation written by Jian Yang and published by . This book was released on 2006 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Simple New Formula for Options with Stochastic Volatility by : Steven L. Heston
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.
Book Synopsis Option Pricing with Stochastic Volatility by : Bogdan Negrea
Download or read book Option Pricing with Stochastic Volatility written by Bogdan Negrea and published by . This book was released on 2002 with total page 52 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Black and Scholes (1973) option pricing model was developed starting from the hypothesis of constant volatility. However, many empirical studies, have argued that the mentioned hypothesis is subject to debate. A few authors, among who - Stein and Stein (1991), Heston (1993), Bates (1996) and Bakshi et al.(1997, 2000) - suggested the use of the Fourier transform for the density of the underlying return or for the risk-neutral probabilities, in order to evaluate the fair price of an option. In this paper we propose a stochastic valuation model using the Fourier transform for option price. This model can be used for the valuation of European options, characterized by two state variables: the price of the underlying asset and its volatility. We model the stochastic processes described by the two variables and we obtain a partial derivatives equation of which the solution is the price of the derivative. We propose a solution to this partial derivatives equation using the Fourier transform. When we apply the Fourier transform, we demonstrate that a second order partial derivatives equation is solved as an ordinary differential equation. We consider a correlation between the underlying asset price and its volatility and two sources of risk: return and volatility. The first part of the paper describes the hypotheses of the model. After describing the Fourier transforms, we propose a formula for the valuation of European options with stochastic volatility. In the second part, we present a few empirical results on the pricing of CAC 40 index call options.
Book Synopsis Option Pricing with Long Memory Stochastic Volatility Models by : Zhigang Tong
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.
Book Synopsis A General Framework for the Derivation of Asset Price Bounds by : Oleg Bondarenko
Download or read book A General Framework for the Derivation of Asset Price Bounds written by Oleg Bondarenko and published by . This book was released on 2015 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present a generalization of Cochrane and Saá-Requejo's good-deal bounds which allows to include in a flexible way the implications of a given stochastic discount factor model. Furthermore, a useful application to stochastic volatility models of option pricing is provided where closed-form solutions for the bounds are obtained. A calibration exercise demonstrates that our benchmark good-deal pricing results in much tighter bounds. Finally, a discussion of methodological and economic issues is also provided.
Book Synopsis Closed-Form Approximations for Spread Option Prices and Greeks by : Minqiang Li
Download or read book Closed-Form Approximations for Spread Option Prices and Greeks written by Minqiang Li and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We develop a new closed-form approximation method for pricing spread options. Numerical analysis shows that our method is more accurate than existing analytical approximations. Our method is also extremely fast, with computing time more than two orders of magnitude shorter than one-dimensional numerical integration. We also develop closed-form proximations for the greeks of spread options. In addition, we analyze the price sensitivities of spread options and provide lower and upper bounds for digital spread options. Our method enables the accurate pricing of a bulk volume of spread options with different specifications in real time, which offers traders a potential edge in financial markets. The closed-form approximations of greeks serve as valuable tools in financial applications such as dynamic hedging and Value-at-Risk calculations. The availability of a closed-form formula for spread options also helps us understand and design real and financial contracts with embedded spread-option-like features.
Book Synopsis A Closed-form GARCH Option Pricing Model by : Steven L. Heston
Download or read book A Closed-form GARCH Option Pricing Model written by Steven L. Heston and published by . This book was released on 1997 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Long Range Stochastic Volatility with Two Scales in Option Pricing by : Li Kong
Download or read book Long Range Stochastic Volatility with Two Scales in Option Pricing written by Li Kong and published by . This book was released on 2012 with total page 79 pages. Available in PDF, EPUB and Kindle. Book excerpt: We exploit a general framework, a martingale approach method, to estimate the derivative price for different stochastic volatility models. This method is a very useful tool for handling non-markovian volatility models. With this method, we get the order of the approximation error by evaluating the orders of three error correction terms. We also summarize some challenges in using the martingale approach method to evaluate the derivative prices. We propose two stochastic volatility models. Our goal is to get the analytical solution for the derivative prices implied by the models. Another goal is to obtain an explicit model for the implied volatility and in particular how it depends on time to maturity. The first model we propose involves the increments of a standard Brownian Motion for a short time increment. The second model involves fractional Brownian Motion(fBm) and two scales. By using fBm in our model, we naturally incorporate a long-range dependence feature of the volatility process. In addition, the implied volatility corresponding to our second model capture a feature of the volatility as observed in the paper Maturity cycles in implied volatility by Fouque, which analyzed the S & P 500 option price data and observed that for long dated options the implied volatility is approximately affine in the reciprocal of time to maturity, while for short dated options the implied volatility is approximately affine in the reciprocal of square root of time to maturity. The leading term in the implied volatility also matches the case when we have time-dependent volatility in the Black-Scholes equation.
Book Synopsis Fast Hilbert Transform Algorithms for Pricing Discrete Timer Options Under Stochastic Volatility Models by : Pingping Zeng
Download or read book Fast Hilbert Transform Algorithms for Pricing Discrete Timer Options Under Stochastic Volatility Models written by Pingping Zeng and published by . This book was released on 2015 with total page 25 pages. Available in PDF, EPUB and Kindle. Book excerpt: Timer options are barrier style options in the volatility space. A typical timer option is similar to its European vanilla counterpart, except with uncertain expiration date. The finite-maturity timer option expires either when the accumulated realized variance of the underlying asset has reached a pre-specified level or on the mandated expiration date, whichever comes earlier. The challenge in the pricing procedure is the incorporation of the barrier feature in terms of the accumulated realized variance instead of the usual knock-out feature of hitting a barrier by the underlying asset price. We construct efficient and accurate fast Hilbert transform algorithms for pricing finite-maturity discrete timer options under different types of stochastic volatility processes. The stochastic volatility processes nest some popular stochastic volatility models, like the Heston model and 3/2 stochastic volatility model. The barrier feature associated with the accumulated realized variance can be incorporated effectively into the fast Hilbert transform procedure with the computational convenience of avoiding the nuisance of recovering the option values in the real domain at each monitoring time instant in order to check for the expiry condition. Our numerical tests demonstrate high level of accuracy of the fast Hilbert transform algorithms. We also explore the pricing properties of the timer options with respect to various parameters, like volatility of variance, correlation coefficient between the asset price process and instantaneous variance process, sampling frequency, and variance budget.
