Local and Stochastic Volatility Under Stochastic Interest Rates Using Mixture Models and the Multidimensional Fractional FFT. PDF Download
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Author: Mark S. Joshi
Publisher:
ISBN:
Category :
Languages : en
Pages : 20
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Book Description
Stochastic volatility, local volatility and stochastic interest rates are three of the most important extensions to the standard Black-Scholes framework. Although much work has been done on models incorporating one or two of these extensions, very little has been done on the combination of all three. We show how to efficiently calibrate and simulate such a model by utilizing a mixture diffusion based approach, which takes advantage of the multidimensional fractional FFT.
Author: Mark S. Joshi
Publisher:
ISBN:
Category :
Languages : en
Pages : 20
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Book Description
Stochastic volatility, local volatility and stochastic interest rates are three of the most important extensions to the standard Black-Scholes framework. Although much work has been done on models incorporating one or two of these extensions, very little has been done on the combination of all three. We show how to efficiently calibrate and simulate such a model by utilizing a mixture diffusion based approach, which takes advantage of the multidimensional fractional FFT.
Author: Mark S. Joshi
Publisher:
ISBN:
Category :
Languages : en
Pages : 22
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Book Description
A key requirement of any equity hybrid derivatives pricing model is the ability to rapidly and accurately calibrate to vanilla option prices. To this end, we present two methods for calibrating a local volatility model under correlated stochastic interest rates. This is achieved by first fitting a mixture model to market prices, and then determining the local volatility function that is consistent with this mixture model.
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: Eric Benhamou
Publisher:
ISBN:
Category :
Languages : en
Pages : 10
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Book Description
This paper studies the impact of stochastic interest rates for local volatility hybrids. Our research shows that it is possible to explicitly determine the bias between the local volatility of a model with stochastic interest rates and the local volatility of the same model, but with deterministic interest rates as a function between the correlation of the stochastic interest rates and the digital at the local strike. The paper will show that this bias can be expressed in a simpler form under the assumption of a diffusion of the stochastic interest rates, enabling us to compute a fast calibration for a hybrid model with stochastic interest rates. This bias leads to a decrease in the value of the local volatility as a result of the induced volatility caused by the stochastic drift. Numerical results illustrate the importance of the bias and confirm that some stochastic noise arises from the stochastic drift.
Author: Antonio Mele
Publisher: Springer Science & Business Media
ISBN: 1461545331
Category : Business & Economics
Languages : en
Pages : 156
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Book Description
Stochastic Volatility in Financial Markets presents advanced topics in financial econometrics and theoretical finance, and is divided into three main parts. The first part aims at documenting an empirical regularity of financial price changes: the occurrence of sudden and persistent changes of financial markets volatility. This phenomenon, technically termed `stochastic volatility', or `conditional heteroskedasticity', has been well known for at least 20 years; in this part, further, useful theoretical properties of conditionally heteroskedastic models are uncovered. The second part goes beyond the statistical aspects of stochastic volatility models: it constructs and uses new fully articulated, theoretically-sounded financial asset pricing models that allow for the presence of conditional heteroskedasticity. The third part shows how the inclusion of the statistical aspects of stochastic volatility in a rigorous economic scheme can be faced from an empirical standpoint.
Author: Luc Bauwens
Publisher: John Wiley & Sons
ISBN: 0470872519
Category : Business & Economics
Languages : en
Pages : 566
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Book Description
A complete guide to the theory and practice of volatility models in financial engineering Volatility has become a hot topic in this era of instant communications, spawning a great deal of research in empirical finance and time series econometrics. Providing an overview of the most recent advances, Handbook of Volatility Models and Their Applications explores key concepts and topics essential for modeling the volatility of financial time series, both univariate and multivariate, parametric and non-parametric, high-frequency and low-frequency. Featuring contributions from international experts in the field, the book features numerous examples and applications from real-world projects and cutting-edge research, showing step by step how to use various methods accurately and efficiently when assessing volatility rates. Following a comprehensive introduction to the topic, readers are provided with three distinct sections that unify the statistical and practical aspects of volatility: Autoregressive Conditional Heteroskedasticity and Stochastic Volatility presents ARCH and stochastic volatility models, with a focus on recent research topics including mean, volatility, and skewness spillovers in equity markets Other Models and Methods presents alternative approaches, such as multiplicative error models, nonparametric and semi-parametric models, and copula-based models of (co)volatilities Realized Volatility explores issues of the measurement of volatility by realized variances and covariances, guiding readers on how to successfully model and forecast these measures Handbook of Volatility Models and Their Applications is an essential reference for academics and practitioners in finance, business, and econometrics who work with volatility models in their everyday work. The book also serves as a supplement for courses on risk management and volatility at the upper-undergraduate and graduate levels.
Author: Oleg Kovrizhkin
Publisher:
ISBN:
Category :
Languages : en
Pages : 36
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Book Description
We consider the following models:1. Generalization of a local volatility model rolled with a moving average of the spot: dS = mu Sdt + sigma(S/A)SdW$ where A(t) is a moving average of spot S.2. Generalization of Heston pure stochastic volatility model rolled with a moving average of the stochastic volatility: dS = mu Sdt + sigma SdW, dsigma^2 = k(theta - sigma^2)dt + gamma sigma dZ where theta(t) is a moving average of variance sigma^2.3. Generalization of a full stochastic volatility with the process for volatility depending on both sigma and S and rolled with a moving average of S: dS = mu Sdt + sigma SdW, dsigma = a(sigma, S/A)dt + b(sigma, S/A)dZ,corr(dW, dZ) = rho(sigma, S/A)$, where A(t) is a moving average of the spot S. We will generalize these and other ideas further and show that they lead to a 2-factor pure stochastic volatility model: dS = mu Sdt + sigma SdW$, sigma = sigma(v_1, v_2), dv_1 = a_1(v_1, v_2)dt + b_1(v_1, v_2)dZ_1,dv_2 = a_2(v_1, v_2)dt + b_2(v_1, v_2)dZ_2, corr(dW, dZ_1) = rho_1(v_1, v_2), corr(dW, dZ_2) = rho_2(v_1, v_2), corr(dZ_1, dZ_2) = rho_3(v_1, v_2) and give examples of analytically solvable models, applicable for multicurrency models consistent with cross currency pairs dynamics in FX. We also consider jumps and stochastic interest rates.
Author: Bing Hu
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
Author: Mingyang Xu
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
Local volatility model is a relatively simple way to capture volatility skew/smile. In spite of its drawbacks, it remains popular among practitioners for derivative pricing and hedging. For long-dated options or interest rate/equity hybrid products, in order to take into account the effect of stochastic interest rate on equity price volatility stochastic interest rate is often modelled together with stochastic equity price. Similar to local volatility model with deterministic interest rate, a forward Dupire PDE can be derived using Arrow-Debreu price method, which can then be shown to be equivalent to adding an additional correction term on top of Dupire forward PDE with deterministic interest rate. Calibrating a local volatility model by the forward Dupire PDE approach with adaptively mixed grids ensures both calibration accuracy and efficiency. Based on Malliavin calculus an accurate analytic approximation is also derived for the correction term incorporating impacts from both interest rate volatility and correlation, which integrates along a more likely straight line path for better accuracy. Eventually, the hybrid local volatility model can be calibrated in a two-step process, namely, calibrate local volatility model with deterministic interest rate and add adjustment for stochastic interest rate. Due to the lack of analytic solution and path-dependency nature of some products, Monte Carlo is a simple but flexible pricing method. In order to improve its convergence, we develop a scheme to combine merits of different simulation schemes and show its effectiveness.
Author: Cong Yi
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
In this thesis I will present my PhD research work, focusing mainly on financialmodelling of asset?s volatility and the pricing of contingent claims (financial derivatives), which consists of four topics:1. Several changing volatility models are introduced and the pricing of Europeanoptions is derived under these models;2. A general local stochastic volatility model with stochastic interest rates (IR)is studied in the modelling of foreign exchange (FX) rates. The pricing of FXoptions under this model is examined through the use of an asymptotic expansionmethod, based on Watanabe-Yoshida theory. The perfect/partial hedging issuesof FX options in the presence of local stochastic volatility and stochastic IRs arealso considered. Finally, the impact of stochastic volatility on the pricing of FX-IRstructured products (PRDCs) is examined;3. A new method of non-biased Monte Carlo simulation for a stochastic volatilitymodel (Heston Model) is proposed;4. The LIBOR/swap market model with stochastic volatility and jump processesis studied, as well as the pricing of interest rate options under that model. In conclusion, some future research topics are suggested. Key words: Changing Volatility Models, Stochastic Volatility Models, LocalStochastic Volatility Models, Hedging Greeks, Jump Diffusion Models, ImpliedVolatility, Fourier Transform, Asymptotic Expansion, LIBOR Market Model, MonteCarlo Simulation, Saddle Point Approximation.
