This longrun volatility persistence is not taken into account by garch models. Levy processes driven by stochastic volatility springerlink. The scope of these models goes from simple exponential levy models to sde with poisson jumps both on the volatility and on the returns. We introduce a pathwise integration for volterra processes driven by l. Modelling energy spot prices by volatility modulated levydriven. Under the car1 model for the spot volatility v t, it has been shown in the study by barndorffnielsen and shephard 2001 that the daily integrated volatility is an arma1,1 process so that its autocorrelation function at lags greater than zero is a decreasing exponential function. Estimation is regarded as the principal challenge in applying these models since they were proposed by barndorffnielsen and shephard j.
Using the url or doi link below will ensure access to this page indefinitely. Second, stochastic volatility is regarded as a key factor for modelling energy spot prices. Multivariate stochastic volatility models based on non. Nonparametric estimation for some models driven by levy processes jose figueroalopez1 2 1statistics and applied probability university of california, santa barbara 2department of statistics purdue university the third erich l. Several calibration methods have been proposed in the literature to deal with these. Stock prices driven by levy processes or other related jump processes have received a great deal of attention in recent years. Brockwell 2001a, b and brockwell and marquardt 2005 introduced a generalization to the levydriven continu. Option pricing and hedging for optimized levy driven. Annualized standard deviation of the change in price or value of a nancial security. L evy processes, poisson random measures, jumpdi usion nancial models, exponential l evy.
Estimation methods for levy based models of asset prices. Filtering and estimation for a class of stochastic volatility models with intractable likelihoods vankov, emilian r. Examples are the modelling of stochastic volatility in the class of models introduced by barndorffnielsen and shephard 2001 and the construction of a class of continuoustime garch models which generalize the cogarch1,1 process of kluppelberg, lindner and maller 2004 and which exhibit properties analogous to those of the discretetime. This is not too surprising as one can actually observe jumps in market data.
Pdf a continuous time garch process driven by a levy. Inference for volatility processes 5 where k is a modi. In this paper we extend option pricing under levy dynamics, by assuming that the volatility of the levy process is stochastic. Levy processes con tribute nonnormality and jumps in the observed part of the model and fractional processes contribute long me mory in unobserved part in the model. Modeling the evolution of a financial price series as forced by a stochastic volatility process has a long history in financial econometrics. Ornstein uhlenbeckoumodels,inwhichthedrivingprocess for a volatility factor is a purejump levy process with nonnegative increments. Inference for levy driven stochastic volatility models via adaptive. The objective of this paper is to study the arbitrage free pricing of variance and volatility swaps for barndorffnielsen and shephard type levy process driven financial markets. The sample autocorrelation function of the series is shown in fig.
Simulation methods for levy driven continuoustime autoregressive moving average carma stochastic volatility models article in journal of business and economic statistics 24october. Stephens department of mathematics, imperial college london, sw7 2az, london, uk d. Levydriven time series models for financial data sciencedirect. These processes are widely used in applications to turbulence, signal processes, biology, and in environmental. This paper studies the parameter estimation problem for ornsteinuhlenbeck stochastic volatility models driven by levy processes. We will focus upon the following model the heston model with a variance gamma model in the log price heston, 1993. We, therefore, develop the analog of the standard stochastic volatility models, when the underlying process is not a standard unit variance brownian motion, but rather a standardized levy process.
We present a general class of stochastic volatility models with jumps where the stochastic variance process follows a levydriven ornsteinuhlenbeck ou process and the jumps in the logprice process follow a levy process. Multivariate continuous time stochastic volatility models driven by a l. The s tochastic volatility models are driven by levy processes is introduced by8, 9 the bates model is simpler but in this model jumps and stochastic volatility are independent. Gradientbased simulated maximum likelihood estimation for. Dynamic sensitivity analysis in levy process driven option. Z p dg tdb t where x t logs t, w t 1, w t 2and b tare independent brownian motions and g tis a gamma.
Maximum quasilikelihood estimation in fractional levy. Volatilityofvolatility risk darien huang ivan shaliastovich september 2014 abstract we show that timevarying volatility of volatility is a signi cant risk factor which a ects both the crosssection and the timeseries of index and vix option returns, above and beyond volatility risk itself. The levy measure of x is then ux 1 x 1 2 z 1 0 exp. Modelling energy spot prices by volatility modulated levy. We study ornsteinuhlenbeck stochastic processes driven by levy processes, and extend them to more general nonornsteinuhlenbeck models. Bns model denotes a connection of jumps and stochastic volatility. Nonparametric estimation for some models driven by levy. The model for the meanreverting time change is then generalized to include nongaussian models that are solutions to ornsteinuhlenbeck equations driven by. Inference for stochastic volatility models driven by levy. Carma processes driven by nongaussian noise tumias primary sources essays in technology and science, 1 no. We present a methodology that allows one to compute option prices.
Levy processdriven asymmetric heteroscedastic option pricing model and empirical analysis gaoxun zhang,1 yi zheng,2 honglei zhang,3 and xinchen xie4 1school of science, southwest university of science and technology, mianyang 621010, china 2department of industrial engineering, school of construction and management engineering, xihua university, chengdu 610039, china 3school of management. Indirect inference for levydriven continuoustime garch models. Models driven by levy processes in this article series quantstart returns to the discussion of pricing derivative securities, a topic which was covered a few years ago on the site through an introduction to stochastic calculus. Modelling energy spot prices by volatility modulated levydriven volterra processes ole e. Driven stochastic volatility models through realized variance measures. Our cogarch continuous time garch model, based on a single background driving levy process, is different from, though related to. Request pdf simulation methods for levydriven carma stochastic volatility models we develop simulation schemes for the new classes of nongaussian pure jump levy processes for stochastic. Inference for levy driven stochastic volatility models via adaptive sequential monte carlo article in scandinavian journal of statistics 381. Markov chain monte carlo estimation of stochastic volatility. In particular, we investigate the means of making the correlation structure in the volatility process more flexible. Within this setup, it is quite straightforward to generate simulations from a levydriven carma stochastic volatility model augmented by. In probability theory, a levy process, named after the french mathematician paul levy, is a stochastic process with independent, stationary increments.
Models with jumps allow for more realistic representations of price dynamics. Typically, bayesian inference from such models is performed using markov chain monte carlo mcmc. One of the major challenges in arbitrage free pricing of swap is to obtain an accurate pricing expression which can be used with good computational accuracy. The advantage of our constructed model over the prevailing is the reflection of infinite jump behaviors in underlying assets and clustering effect in time change volatility. Driven by an arbitrary levy process it exhibits regularly varying tails. Pricing variance and volatility swaps for barndorffnielsen. Statistical estimation of stochastic volatility models typically falls into one of the following two broad categories. In chapter 1, predictionbased estimating functions are used to estimate a model with diffusion type stochastic volatility. Inference for stochastic volatility models driven by levy processes by matthew p. Anh, heyde, and leonenko 2002 proposed a levy driven stochastic volatility model, where the volatility process is of moving average type. The volatility of the continuous component is constant through time. Method of moment estimation in timechanged levy models.
Applications of shorttime asymptotics to the statistical. Inference for levy driven stochastic volatility models via. Levydriven ornsteinuhlenbeck or car1 processes were introduced by barndorffnielsen and shephard 1 as a model for stochastic volatility. We investigate simulation methodology for bayesian inference in levy. We pay attention to stochastic volatility models with jumps cases where the levy jumps are driven by infinite activity processes and finite activity process, respectively.
Financial modelling with ornsteinuhlenbeck processes. Option pricing the price of an option of strike k and maturity t, is a function of market parameters such as an underlying asset price s, a riskfree interest rate r and a market volatility s. Simulation and inference for stochastic volatility models. Multivariate continuous time stochastic volatility models. In the following paper we investigate simulation and inference for a class levy driven stochastic volatility sv models. Levydriven continuoustime arma processes springerlink. Simulation methods for levydriven continuoustime autoregressive moving average carma stochastic volatility models. Siam journal on numerical analysis society for industrial. Pdf we use a discrete time analysis, giving necessary and sufficient. A continuoustime garch process driven by a levy process. It also uses circular maximum likelihood estimation technology to improve the stability of model.
Simulation methods for levydriven carma stochastic. Estimation of continuous time models driven by levy processes. Driven stochastic volatility models through realized variance measures the econometrics journal, vol. Simulation methods for levydriven continuoustime autoregressive moving. Volatility and volatilityofvolatility movements are. Levy processdriven asymmetric heteroscedastic option pricing. Estimation is regarded as the principal challenge in applying these models since they were proposed by barndorffnielsen and shephard. Dec 22, 2010 we investigate simulation methodology for bayesian inference in levy. Tail behavior of multivariate levydriven mixed moving average processes and supou stochastic volatility models advances in applied probability, 43 no. Option pricing with discrete time jump processes halshs.
Simulation methods for levydriven carma stochastic volatility. In this article we develop and assess practical schemes to simulate from l. Intensity of jumps is, on average, homogeneous through time. In the models considered herein, the volatility dynamics are governed by the brockwellstyle carma extension of the. Financial modelling with ornsteinuhlenbeck processes driven. Lehmann symposium, 2007 figueroalopez nonparametric estimation levy models. Option pricing models driven by the spacetime fractional.