Analytical Approximations of Option Prices in Stochastic Volatility Models PDF Download
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Author:
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Languages : en
Pages : 142
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Book Description
Author:
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ISBN:
Category :
Languages : en
Pages : 142
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Author: Jian Yang
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Languages : en
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Author: Hideharu Funahashi
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Languages : en
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This paper extends the Wiener-Ito chaos expansion approach proposed by Funahashi and Kijima (2013) to an equity-interest-rate hybrid model for the pricing of European contingent claims with special emphasis on calibration to the option markets. Our model can capture the volatility skew and smile of option markets, as well as the stochastic nature of interest rates. Further, the proposed method is applicable to widely used option pricing models such as local volatility models, stochastic volatility models, and their combinations with the stochastic nature of interest rates; hence, it is suitable for practical purposes. Through numerical examples, we show that our approximation is quite accurate even for long-maturity and/or high-volatility cases.
Author: Giulia Terenzi
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Languages : en
Pages : 0
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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.
Author: Vicky Henderson
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Category :
Languages : en
Pages :
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Author: Stefano Pagliarani
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Category :
Languages : en
Pages : 0
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Book Description
We present new approximation formulas for local stochastic volatility models, possibly including Lévy jumps. Our main result is an expansion of the characteristic function which is worked out in the Fourier space. Combined with standard Fourier methods, our result provides efficient and accurate formulas for the prices and the Greeks of plain vanilla options. We finally provide numerical results to illustrate the accuracy with real market data.
Author: Christian Kahl
Publisher: Universal-Publishers
ISBN: 1581123833
Category : Business & Economics
Languages : en
Pages : 219
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Book Description
The famous Black-Scholes model was the starting point of a new financial industry and has been a very important pillar of all options trading since. One of its core assumptions is that the volatility of the underlying asset is constant. It was realised early that one has to specify a dynamic on the volatility itself to get closer to market behaviour. There are mainly two aspects making this fact apparent. Considering historical evolution of volatility by analysing time series data one observes erratic behaviour over time. Secondly, backing out implied volatility from daily traded plain vanilla options, the volatility changes with strike. The most common realisations of this phenomenon are the implied volatility smile or skew. The natural question arises how to extend the Black-Scholes model appropriately. Within this book the concept of stochastic volatility is analysed and discussed with special regard to the numerical problems occurring either in calibrating the model to the market implied volatility surface or in the numerical simulation of the two-dimensional system of stochastic differential equations required to price non-vanilla financial derivatives. We introduce a new stochastic volatility model, the so-called Hyp-Hyp model, and use Watanabe's calculus to find an analytical approximation to the model implied volatility. Further, the class of affine diffusion models, such as Heston, is analysed in view of using the characteristic function and Fourier inversion techniques to value European derivatives.
Author: Henrik Andersson
Publisher:
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Languages : en
Pages : 45
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Book Description
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.
Author: Wen Cheng
Publisher: LAP Lambert Academic Publishing
ISBN: 9783846531969
Category :
Languages : en
Pages : 180
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Book Description
Closed-form solutions to Kolmogorov type equations are very important in many application areas, such as derivative pricing, quantum mechanics, statistical physics, etc. However, they are only available for some very special equations. This book considers general second order parabolic equations with coefficients that are dependent on both time and space. It extends the recently developed Dyson-Taylor commutator method for autonomous equations to non-autonomous equations. Closed-form approximations of the Green's functions that are accurate to any prescribed order are obtained. Consequently, the solutions to second order parabolic equations can be obtained by integrating the approximated Green's functions against initial data. For applications, this book considers Local Volatility models and Stochastic Volatility models that appear in option pricing theory, gives explicit formulas for European option prices, and carries out numerical tests for such formulas.
Author: Minqiang Li
Publisher:
ISBN:
Category :
Languages : en
Pages : 44
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Book Description
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.
