Some recent developments in stochastic volatility modelling

Barndorff–Nielsen, Ole E.; Nicolato, Elisa; Shephard, Neil

doi:10.1088/1469-7688/2/1/301

Cited by 96 publications

(70 citation statements)

References 47 publications

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“…In Appendix B, we summarize the simulation methodologies we have used in our numerical study. Note that when estimating the price of the option on the continuous average, we correct the discretization bias inherent in the simulation using our maximum lower bound based on the continuous average as the control variate 5 . The sets of parameter values used for the Lévy models are from the calibrations of Schoutens [62] and Fusai and Meucci [40]; the ASV models' parameter values are from the calibrations of Duffie et al [33] (see also [13], [2] and [68]) and Nicolato and Venardos [57]; the CEV parameter sets are taken from Cai et al [15].…”

Section: Theorem 3 (Fixed and Floating Strike Continuous Asian Optiomentioning

confidence: 99%

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General Optimized Lower and Upper Bounds for Discrete and Continuous Arithmetic Asian Options

Fusai

Kyriakou

2016

Mathematics of OR

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This is the accepted version of the paper.This version of the publication may differ from the final published version. We propose an accurate method for pricing arithmetic Asian options on the discrete or continuous average in a general model setting by means of a lower bound approximation. In particular, we derive analytical expressions for the lower bound in the Fourier domain. This is then recovered by a single univariate inversion and sharpened using an optimization technique. In addition, we derive an upper bound to the error from the lower bound price approximation. Our proposed method can be applied to computing the prices and price sensitivities of Asian options with fixed or floating strike price, discrete or continuous averaging, under a wide range of stochastic dynamic models, including exponential Lévy models, stochastic volatility models, and the constant elasticity of variance diffusion. Our extensive numerical experiments highlight the notable performance and robustness of our optimized lower bound for different test cases. Permanent repository linkKey words : arithmetic Asian options; CEV diffusion; stochastic volatility models; Lévy processes; discrete average; continuous average MSC2000 subject classification : Primary: 91B70, 91B25, 60J25; Secondary: 60H30, 91B24 OR/MS subject classification : Primary: asset pricing, diffusion, Markov processes, stochastic model applications 1. Introduction. We develop accurate analytical pricing formulae for discretely and continuously monitored arithmetic Asian options under general stochastic asset models, including exponential Lévy models, stochastic volatility models, and the constant elasticity of variance diffusion. The payoff of the arithmetic Asian option depends on the arithmetic average price of the underlying asset monitored over a pre-specified period. For more than two decades, much effort has been put into the research on efficient methodologies for computing the price of this option or, in general, expected values of functionals of the average value, under different model assumptions for the underlying. Developing such methods is of considerable practical importance as arithmetic averages see wide application in many fields of finance. Amongst others, we mention uses in computing net present value in project valuation (see [72]), optimal capacity planning under average demand uncertainty for a single firm (see [32]) and stock-swap merger proposals (see [60]). Weighted arithmetic averages also appear in technical analysis and in algorithmic trading; for example, we recall the moving average trading rule and its use from an asset allocation perspective (see [75]). Moving average automatic trading strategies set buying and selling orders depending on the position of the average price for a given period with respect to the current market price (see [50]). Finally, weighted arithmetic average indexes are used as trading benchmarks in pension plans (see [11]).Arithmetic Asian options are very popular among derivatives traders and risk managers. Their...

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Section: Theorem 3 (Fixed and Floating Strike Continuous Asian Optiomentioning

confidence: 99%

“…Barndorff-Nielsen and Shephard [6] and Barndorff-Nielsen et al [5] suggest the so-called BNS model of the form…”

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confidence: 99%

General Optimized Lower and Upper Bounds for Discrete and Continuous Arithmetic Asian Options

Fusai

Kyriakou

2016

Mathematics of OR

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“…"Stochastic volatility" models (Barndorff-Nielsen et al [26], Chib et al [75], Ghysels et al [145], Harvey and Shephard [163], Jacquier et al [174], Shephard [280], Taylor [288]), "implied volatility" models (Day and Lewis [88], Latane and Rendleman [195], Schmalensee and Trippi [272]), "historical volatility" models (Beckers [30], Garman and Klass [140], Kunitomo [190], Parkinson [263], Rogers and Satchell [269]) and "realized volatility" models are examples from the financial econometric literature of estimating volatility of asset returns.…”

Section: Other Methods Of Volatility Modelingmentioning

confidence: 99%

Autoregressive Conditional Heteroscedasticity (ARCH) Models: A Review

Degiannakis

Xekalaki

2004

Quality Technology & Quantitative Management

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Autoregressive Conditional Heteroscedasticity (ARCH) models have successfully been employed in order to predict asset return volatility. Predicting volatility is of great importance in pricing financial derivatives, selecting portfolios, measuring and managing investment risk more accurately. In this paper, a number of univariate and multivariate ARCH models, their estimating methods and the characteristics of financial time series, which are captured by volatility models, are presented. The number of possible conditional volatility formulations is vast. Therefore, a systematic presentation of the models that have been considered in the ARCH literature can be useful in guiding one's choice of a model for exploiting future volatility, with applications in financial markets.

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“…Let us note that the (non decreasing) increments of σ 2 l (t) is referred to (in the field of mathematical finance [29,30]) as the stochastic volatility.…”

Section: Definitionmentioning

confidence: 99%

Multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws

2002

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We define a large class of continuous time multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal Multifractal Random Walk (MRW) [1, 2] and the log-Poisson "product of cynlindrical pulses" [3]. Our construction is based on some "continuous stochastic multiplication" (as introduced in [4]) from coarse to fine scales that can be seen as a continuous interpolation of discrete multiplicative cascades. We prove the stochastic convergence of the defined processes and study their main statistical properties. The question of genericity (universality) of limit multifractal processes is addressed within this new framework. We finally provide a method for numerical simulations and discuss some specific examples.

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Some recent developments in stochastic volatility modelling

Cited by 96 publications

References 47 publications

General Optimized Lower and Upper Bounds for Discrete and Continuous Arithmetic Asian Options

General Optimized Lower and Upper Bounds for Discrete and Continuous Arithmetic Asian Options

Autoregressive Conditional Heteroscedasticity (ARCH) Models: A Review

Multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws

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