European Institute for Statistics, Probability, Stochastic Operations Research and its Applications

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Workshop on

Statistical Inference for LÚvy Processes with Applications to Finance

 July 15 - 17, 2009

  Wednesday, July 15
10:00 - 11:00 Registration
11:00 - 11:15 Opening
11:15 - 12:00 N. Bingham, Multivariate elliptic processes
12:00 - 12:45 J. M. Corcuera, Completeness and hedging in a LÚvy bond market
12:45 - 14:00 Lunch
14:00 - 14:45 F. Comte, Nonparametric estimation for pure jump LÚvy processes with fixed sample step
14:45 - 15:30 V. Genon-Catalot, Nonparametric estimation for pure jump LÚvy processes based on high frequency data
15:30  -16:00 Coffee/tea break
16:00 - 16:25 F. Guillaume, Implied LÚvy volatility
16:25 - 16:50 P.Dobranszky, Historical Calibration of the Equivalent Martingale Measure
16:50 - 17:15 C. Mancini, Coefficients reconstruction of a Markov model with jumps, given discrete      observations, and interest rate modeling
17:15 - 17:30 Coffee/tea break
17:30 - 17:55 M. Vetter, Limit theorems for bipower variation of semimartingales
17:55 - 18:20 P. Posedel,Joint analysis and estimation of stock prices and trading volume in Barndorff-Nielsen and Shephard stochastic volatility models
19:00 Dinner
  Thursday, July 16
10:00 - 10:45 E. Eberlein, Correlation based calibration in LÚvy interest rate models
10:45 - 11:15 Coffee/tea break
11:15 - 12:00 M. Rei▀, Nonparametric estimation for Levy processes from low-frequency observations
12:00 - 12:25 E. Veerman, Semiparametrics for compound Poisson distributions
12:25 - 12:50 S. Gugushvili, Nonparametric estimation of the characteristic triplet of a discretely observed LÚvy process
12:50 - 14:00 Lunch
14:00 - 14:45 Y. A´t-Sahalia, Analyzing the Spectrum of Asset Returns: Jump and Volatility Components in High Frequency Data
14:45 - 15:30 A. Szimayer,  Semiparametric Continuous Time GARCH Models: An Estimation Function Approach
15:30  -16:00 Coffee/tea break
16:00 - 16:25 R. Laeven   Non-parametric estimation for multivariate LÚvy processes
16:25 - 16:50 D. Belomestny   Spectral estimation of the fractional order of a Levy process
16:50 - 17:15 H. Mai   Maximum-Likelihood-Estimation for LÚvy-driven Ornstein-Uhlenbeck processes
17:15 - 17:30 Coffee/tea break
17:30 - 17:55 O. Grothe, Estimating Jump Tail Dependence in LÚvy Copula Models
17:55 - 18:20 G. Germano, Continuous-time random walks, fractional calculus and stochastic integrals: a model of high-frequency financial time series
  Friday, July 17
10:00 - 10:45 R. Cont
10:45 - 11:15 Coffee/tea break
11:15 - 12:00 M. Podolskij, Inference for discretely observed semimartingales plus noise
12:00 - 12:45 A. Papapantoleon, A new approach to LIBOR modeling
12:45 - 14:00 Lunch
14:00 - 14:45 H. Geman
14:45 - 15:00 Closing


Yacine Ait-Sahalia  (Princeton University)
Joint work with Jean Jacod

Analyzing the Spectrum of Asset Returns: Jump and Volatility Components in High Frequency Data

This paper describes a simple yet powerful methodology to decompose asset returns sampled at high frequency into their base components (continuous, small jumps, large jumps), determine the relative magnitude of the components, and analyze the finer characteristics of these components such as the degree of activity of the jumps.


Denis Belomestny (WIAS)

Spectral estimation of the fractional order of a Levy process

In this talk I consider the problem of estimating the fractional order of a LÚvy process from low frequency historical and options data. An estimation methodology will be developed which allows us to treat both estimation and calibration problems in a unified way. The corresponding procedure consists of two steps: the estimation of a conditional characteristic function and the weighted least squares estimation of the fractional order in spectral domain. While the second step is identical for both calibration and estimation, the first one depends on the problem at hand. Minimax rates of convergence for the fractional order estimate will be derived, the asymptotic normality will be proved and a data-driven algorithm based on aggregation will be proposed. The performance of the estimator in both estimation and calibration setups will be illustrated by a simulation study.


Nick Bingham (Imperial College London)

Multivariate elliptic processes

The talk begins with the relevant distribution theory, allowing us to model e.g. asset return or log-price distributions in d dimensions, if we have a portfolio of d assets: multivariate elliptic distributions, in»nite divisibility, self-decomposability, type G. We then turn to dynamics. First we use LÚvyprocesses in d dimensions and processes of Ornstein-Uhlenbeck type. The idea is to use the stochastic representation of elliptic distributions to give a model for return or log-price processes which does not su«er from the curse of dimensionality. Secondly, we use ergodic di«usions, again in d dimensions. Examples are given, and the LÂevy and di«usion models are compared. We close by comparing discrete and continous time.


Fabienne Comte (UniversitÚ RenÚ Descartes -  Paris 5)

Nonparametric estimation for pure jump LÚvy processes with fixed sample step

This talk is about a work in collaboration with V. Genon-Catalot. It is concerned with nonparametric estimation of the LÚvy density of a pure jump LÚvy process. The sample path is observed at n discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvolution methods and deduced from appropriate estimators of the characteristic function and its first derivative. We obtain a bound for the L^2-risk, under general assumptions on the model. Then we propose a penalty function that allows to build an adaptive estimator. The risk bound for the adaptive estimator is obtained under additional assumptions on the LÚvy density. Examples of models fitting in our framework are described and rates of convergence of the estimator are discussed.


Peter Dobranszky (Finalyse NV, BNP Paribas Fortis, Katholieke Universiteit Leuven)

Historical Calibration of the Equivalent Martingale Measure

There is a wide-ranging literature on financial asset price models that exhibit both jumps and stochastic volatility features. Maybe the most general class of these models is the class of time-changed LÚvy processes. Most of these models imply incomplete markets meaning there exist payoff patterns that cannot be replicated by self-financing dynamic trading strategies. However, assuming arbitrage freeness, the existence of a risk-neutral equivalent martingale measure that can be used for derivatives pricing is guaranteed. But, since the market is incomplete, the riskneutral equivalent martingale measure is not unique. While the various equivalent martingale measures may agree on the price of quoted hedging instruments, they may disagree on the price of exotic derivatives. In practice, the equivalent martingale measure, which determines also the price of derivatives, is chosen by selecting an underlying model and by the way how this model is calibrated. The model is usually chosen based on expert intuitions, while its calibration is usually done by fitting the model to vanilla option prices quoted on a single trading day. We will call this way of calibration to single-day calibration. In case if the financial asset price returns would follow time-homogenous processes, the single day calibration would work fine. However, it has been observed by several authors that the dynamics of financial asset prices are exposed to some persistent latent variables too. Such variables are the stochastic volatility and stochastic skewness factors. The problem with the single-day calibration is that we may miss the real dynamics of the underlying asset price. For instance, on the days when a mean-reverting latent variable is around its equilibrium state, the mean-reversion rate can not be estimated robustly. Alternatively, we propose to do a historical calibration of the equivalent martingale measure. By the word historical we do not mean the combination of the historical probability measure and the current risk-neutral measure. Rather, we assume that the risk-neutral measure is fixed not only through strikes and maturities, but also through trading days. We calibrate our model globally to a whole history of vanilla option prices. This means that we have to calibrate some model parameters that are valid for each trading day and we have to calibrate a set of state variables separately for each trading day. However, the high number of parameters and variables that need to be calibrated prevents the use of conventional calibration procedures. In our proposal we separate the calibration of model parameters and state variables and we carry out a two-level calibration procedure. We present this calibration procedure. By simulating option prices on several trading days, then calibrating the model either to a single trading day or to several trading days, we show that the historical calibration is more stable than the single-day calibration and it is also unbiased. As an application of the historical calibration procedure we fit a multi-factor time-changed LÚvy process to the dynamics of a financial asset.


Ernst Eberlein (University of Freiburg)
Joint work with Maximilian Beinhofer and Arend Janssen

Correlation based calibration of LÚvy interest rate models

The LÚvy interest rate theory is discussed by introducing the LÚvy forward rate and the LÚvy Libor model as well as its multi-currency extension. Explicit formulas for the correlations of zero coupon bond prices and interest rates with varying maturities are derived. Based on a data set of daily German government bond prices we calibrate the LÚvy forward rate model.


Mark Podolskij ((ETH ZŘrich))

Inference for discretely observed semimartingales plus noise

In this talk we discuss some methods which enables us to estimate certain characteristics of semimartingales in the high frequency setting with noise. We present the laws of large numbers and show the associated central limit theorems.


Valentine Genon-Catalot (UniversitÚ RenÚ Descartes- Paris 5)

Nonparametric estimation for pure jump LÚvy processes based on high frequency data.

The talk is based on a joint work with Fabienne Comte. We study nonparametric estimation of the LÚvy density for pure jump LÚvy processes based on discrete observations with sampling interval tending to 0 while the total length time where observations are taken tends to infinity. We use a deconvolution approach to build an adaptive nonparametric estimator and provide a bound for the L2-risk. Then, we use a direct approach to construct an estimator on a given compact interval. We discuss rates of convergence and give numerical simulation results on examples.

Guido Germano (Philipps-Universitńt Marburg)
Joint work Mauro Politi, Philipps-Universitńt Marburg, Enrico Scalas, UniversitÓ del Piemonte Orientale, Amedeo Avogadro. RenÚ L. Schilling, Technische Universitńt Dresden

Continuous-time random walks, fractional calculus and stochastic integrals: a model of high-frequency financial time series

The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications in finance, but also in insurance, economics, and the natural sciences. It is particularly well suited as a phenomenologic model of high-frequency financial time series, though there are also other ones for this purpose, e.g. autoregressive processes (GARCH-ACD). We focus on uncoupled CTRWs with a symmetric LÚvy $\alpha$-stable distribution of tick-by-tick log-returns and a one-parameter Mittag-Leffler geometric stable distribution of intertrade durations. Remarkably these distributions have fat tails and lead to a process with unbounded quadratic variation, whose probability density, in the diffusive limit of vanishing scale parameters, satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation; both generalize the standard diffusion equation solved by the probability density of the Wiener process, providing a phenomenologic model of anomalous diffusion. We define a class of stochastic integrals driven by a CTRW, which includes the It\=o and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale and, as a consequence of the martingale transform theorem, its It\=o integral is a martingale too. A CTRW and stochastic integrals driven by a CTRW can be easily computed by Monte Carlo simulation. There is an efficient and accurate numerical method to generate the random numbers for this sort of CTRW, leading to a stochastic solution of the FDE, which is almost as easy and fast to compute as for a normal compound Poisson process corresponding to standard diffusion. The relations between a CTRW, its quadratic variation, its Stratonovich integral and its It\=o integral are highlighted by numerical calculations. We provide an analytic expression for the probability density function of the quadratic variation of the stochastic process described by the FDE, and check it by Monte Carlo. We discuss the application of this model to the pricing of options that are short to maturity, within a time horizon from a few hours to a few days, and the related problem ob obtaining the CTRW parameters from historical high-frequency time series.


Oliver Grothe (University of Cologne)

Estimating Jump Tail Dependence in LÚvy Copula Models

In multivariate LÚvy processes, the LÚvy Copula determines the dependence of concurrent jumps. In this talk we focus on the dependence of extreme jumps in such models: We analyze the conditional probability of observing a large jump in one component, given a large jump in another component of the LÚvy processes. Analogously to the concept of tail dependence we call this probability jump tail dependence. In contrast to copulas, LÚvy copulas do not correspond to probability distribution functions. Nevertheless, we are able to show that the probability given by jump tail dependence is determined by the LÚvy copula alone and that it is independent of the jump distributions of the marginal LÚvy processes. For the estimation of jump tail dependence we derive asymptotical relations between jump tail dependence in the LÚvy copula and tail dependence of the marginal distributions of the processes. In a simulation study we illustrate different effects of jump tail dependence. We analyze the estimation of jump tail dependence for different frequencies and validate bootstrap estimators for the variance of the estimator.


Shota Gugushvili (EURANDOM)

Nonparametric estimation of the characteristic triplet of a discretely observed LÚvy process

Given a discrete time sample $X_1,... X_n$ from a LÚvy process $X=(X_t)_{t\geq 0}$ of a finite jump activity, we study the problem of nonparametric estimation of the characteristic triplet $(\gamma,\sigma^2,\rho)$ corresponding to the process $X.$ Based on Fourier inversion and kernel smoothing, we propose estimators of $\gamma,\sigma^2$ and $\rho$ and study their asymptotic behaviour. The obtained results include derivation of upper bounds on the mean square error of the estimators of $\gamma$ and $\sigma^2$ and an upper bound on the mean integrated square error of an estimator of $\rho.$


Florence Guillaume
Joint work with JosÚ Manuel Corcuera, Peter Leoni, Wim Schoutens

Implied LÚvy volatility

The concept of implied volatility under the Black-Scholes model is one of the key points of its success and its widespread use since it allows to perfectly match model prices and market prices. In fact it gives another, more convenient and robust, way of quoting plain vanilla European option prices. Rather than quoting the premium in the relevant currency, the options are quoted in terms of Black-Scholes implied volatility. Over the years, option traders have developed an intuition in this quantity. As it turns out, this model parameter depends on the characteristics of the contract. More precisely, it depends on the strike price and the remaining lifetime of the option. The precise functional form is called the volatility surface and follows its own dynamics in the market. This model parameter needs to be adjusted separately for each individual contract given the inadequacy of the underlying Black-Scholes model. By analyzing empirical historical data, it is not hard to see that stock returns tend to be more skewed and have fatter tails than those the normal distribution can provide. Hence blind trust in a single implied volatility number and all the numbers derived from that, like deltas and other hedge parameters could be dangerous. Here a similar concept is developed but now under a LÚvy framework and therefore based on distributions that match more closely historical returns. We introduce the concept of implied LÚvy volatility, hereby extending the intuitive Black-Scholes implied volatility into a more general context. The LÚvy models are obtained by replacing the Wiener distribution modeling the diffusion part of the log-return process by a more empirically founded LÚvy distribution. The LÚvy space volatility model will arise by multiplying volatility with the underlying LÚvy process, whereas the LÚvy time volatility model will arise by multiplying volatility squared with time. LÚvy implied time and space volatility are introduced and a study of the resulting skew-adjustment is made. The price and Greeks of vanilla options are computed by making use of the COS method proposed by Fang and Oosterlee. This method rests on Fourier-cosine series expansions and can be applied for any model if the characteristic function of the log-price process at maturity T is available. By switching from the Black-Scholes world to the LÚvy world, we introduce additional degrees of freedom (i.e. parameters that can be set freely) which can be used in order to minimize the curvature of the volatility surface. We look how Black-Scholes curves are translated into implied LÚvy volatility curves and vice versa. It is shown that any smiling or smirking Black-Scholes volatility curve can be transformed into a atter LÚvy volatility curve under a well chosen parameter set. This gives some evidence to the fact that the implied LÚvy models could lead to atter volatility curve for more practical datasets. Hence, implied LÚvy volatility model can be of a particular interest for practitioners facing the problem of pricing barrier options since for the Black-Scholes model, it is not clear which volatility one should use (the one of the barrier or the one of the strike). Model performance is studied by analyzing delta-hedging strategies for short term ATM vanilla under the Normal Inverse Gaussian and the Meixner model, both qualitatively and on historical time-series of the S&P500. The LÚvy degrees of freedom can thus be determined such that the absolute value of the mean and the square root of the variance of the daily hedging error are minimized. It is shown that using the historical optimal parameters leads to a significant reduction of the variance of the hedging error (amounting to more than 50 percents), which is particularly attractive for option hedging.


Roger J.A. Laeven (Tilburg University, CentER and EURANDOM)

Non-parametric estimation for multivariate LÚvy processes

This paper proposes two non-parametric estimators for the dependence function of a multivariate LÚvy process, and derives the estimators’ properties. In addition, an independence test is constructed. The estimators and test are applicable despite the presence of a continuous Brownian component in the process. Finally, the estimators and test are implemented on Monte Carlo simulations and on asset returns data.


Hilmar Mai (Humboldt-Universitńt Berlin)

Maximum-Likelihood-Estimation for LÚvy-driven Ornstein-Uhlenbeck processes

We develop a maximum likelihood approach for estimating the coefficient of a LÚvy driven Ornstein-Uhlenbeck process. In order to do this we prove that the laws of the Ornstein-Uhlenbeck processes corresponding to different coefficients are mutually absolutely continuous if and only if the background driving LÚvy process has a diffusive component. Then, we give conditions such that the maximum likelihood estimator exists uniquely and is strongly consistent as well as asymptotically normal. To obtain these results we show that the class of Ornstein-Uhlenbeck processes corresponding to different coefficients forms a curved exponential family of stochastic processes.


Cecilia Mancini (University of Florence)

Coefficients reconstruction of a Markov model with jumps, given discrete observations, and interest rate modeling

We reconstruct the level-dependent diffusion coefficient of a univariate semimartingale with jumps which is observed discretely. The consistency and asymptotic normality of our estimator are provided in presence of both finite and infinite activity (finite variation) jumps. Our results rely on kernel estimation, using the properties of the local time of the data generating process, and the fact that it is possible to disentangle the discontinuous part of the state variable through those squared increments between observations not exceeding a suitable threshold function. We also reconstruct the drift and the jump intensity coefficients when they are level-dependent and jumps have finite activity, through consistent and asymptotically normal estimators. Simulated experiments show that the newly proposed estimators are better performing in finite samples than alternative estimators, and this allows us to reexamine the estimation of a univariate model for the short term interest rate, for which we find less jumps and more variance due to the diffusion part than previous studies.


JosÚ Manuel Corcuera (University of Barcelona)

Completeness and hedging in a LÚvy bond market

We consider bond markets with one factor where the noise is a LÚvy process. By beginning with the dynamics of the short rates under the historical probability we consider the bonds as derivatives valued under certain risk neutral probability. The completeness problem is analyzed by using new representation theorems for martingales with jumps with a view towards hedging.

Keywords: LÚvy processes, martingale measure, hedging, incomplete markets Mathematics Subject

Classification 2000: 60H30, 60G46, 91B28

[1] Corcuera, J.M., Nualart, D., Schoutens, W. Completion of a LÚvy Market by Power-Jump-Assets. Finance and Stochastics 9(1), 109-127, (2003).
[2] Corcuera, J.M.; Nualart, D.; Schoutens, W. Moment derivatives and LÚvy-type market completion. In Option pricing and Advanced LÚvy models, A. Kyprianou, W. Scoutens & P. Wilmott (eds.), pp 169-193. Wilmott Collection. Wiley, Chichester (2005).
[3] Corcuera, J.M.; Guerra, J.; Nualart, D.; Schoutens, W. Optimal investment in a LÚvy market Applied Mathematics and Optimization,53: 279-309, (2006).
[4] Corcuera, J.M.; Guerra, J. Dynamic complex hedging in additive markets Quantitative Finance, (2009), to appear.


Antonis Papapantoleon (Technische Universitńt Berlin)

A new approach to LIBOR modeling
Joint article with Martin Keller-Ressel and Josef Teichmann

LIBOR market models are the favorite models of practitioners for the pricing of interest rate derivatives, however they suffer from severe intractability problems due to the random terms that enter the SDEs during the construction of the model. As a result, if the driving process is continuous then caplets can be priced in closed form, but not swaptions or other multi-LIBOR products; in case the driving process has jumps, then even caplets cannot be priced in closed form. In both cases, the calibration of the model to cap and swaption market data is very difficult and requires some short of approximation (e.g. \frozen drift" approximation). On the other hand, modeling forward prices produces a very tractable model, but negative LIBOR rates can occur, which contradicts any economic intuition. In this work we propose a new approach to modeling LIBOR rates based on a ne factor processes.We construct suitable martingales that stay greater than one for all times, utilizing the Markov property of a ne processes. Then, we model LIBOR rates in a framework that produces positive LIBOR rates in an analytically tractable model; in particular, LIBOR rates have affine stochastic dynamics under any forward measure. Hence, this model unifies the advantages of LIBOR market models and forward price models. We derive Fourier transform valuation formulas for caplets and swaptions, hence the calibration of the model is very easy. Furthermore, when the driving process is the CIR process, closed form valuation formulas - using the 2-distribution function - are derived for caps and swaptions.


Petra Posedel (University of Zagreb)

Joint analysis and estimation of stock prices and trading volume in Barndorff-Nielsen and Shephard stochastic volatility models

We introduce a variant of the Barndorff-Nielsen and Shephard stochastic volatility model where the non Gaussian Ornstein-Uhlenbeck process describes  some measure of trading intensity like trading volume or number of trades instead of unobservable instantaneous variance. We develop an explicit estimator based on martingale estimating functions in a bivariate model that is not a diffusion, but admits jumps. It is assumed that both the quantities are observed on a discrete grid of fixed width, and the observation horizon tends to infinity. We show that the estimator is consistent and asymptotically normal and give explicit expressions of the asymptotic covariance matrix. Our method is illustrated by a finite sample experiment and a statistical analysis on the International Business Machines Corporation (IBM) stock from the New York Stock Exchange (NYSE) and the Microsoft Corporation (MSFT) stock from Nasdaq during a history of five years.


Markus Rei▀ (Humboldt-Universitńt Berlin)
Joint work with M. Neumann

Nonparametric estimation for Levy processes from low-frequency observations

We suppose that a LÚvy process is observed at discrete time points. A rather general construction of minimum-distance estimators is shown to give consistent estimators of the LÚvy-Khinchine characteristics as the number of observations tends to infinity, keeping the observation distance fixed. For a specific $C^2$-criterion this estimator is rate-optimal. The connection with deconvolution and inverse problems is explained. A key step in the proof is a uniform control on the deviations of the empirical characteristic function on the whole real line.


Alexander Szimayer (University of Bonn)
Joint work with Boris Buchmann

Semiparametric Continuous Time GARCH Models: An Estimation Function Approach

The ARCH model of Engle (1982) and its generalisation of Bollerslev (1986) are popular models in financial econometrics where they are designed to capture some of the distinctive features of asset price, exchange rate, and other series. So-called stylised facts characterise financial returns data as heavytailed, uncorrelated, but not independent, with time-varying volatility and a long range dependence effect evident in volatility, this last also being manifest as a persistence in volatility. KlŘppelberg, et al. (2004) suggest an extension of the (G)ARCH concept to continuous time processes. The COGARCH (continuous time GARCH) model, is based on a single background driving LÚvy process, and generalises the essential features of discrete time GARCH processes. In this paper we propose an estimation procedure for the COGARCH based on martingale estimation functions. The structural COGARCH parameters are estimated, but not the characteristics of the driving LÚvy process, classifying our approach as semiparametric. For estimating the COGARCH, we offer an alternative to the methods of moments investigated by Haug et al. (2007), and further, our development parallels the work of Li and Turtle (2000) for the discrete GARCH process.


Enno Veerman (Universiteit van Amsterdam)

Semiparametrics for compound Poisson distributions

In this talk we consider the semiparametric model of all compound Poisson distributions. We construct one-dimensional submodels with a density with respect to some dominating measure and calculate the corresponding score functions. Next we show that the tangent space of score functions is dense in the maximal tangent set $L^0_2(P)$, in other words, we prove that the compound Poisson model is non-parametric. Using this we then investigate the pathwise differentiability of functionals of the LÚvy measure and prove the asymptotic efficiency for some estimators.

Mathias Vetter (Ruhr-Universitńt Bochum)

Limit theorems for bipower variation of semimartingales

This talk presents limit theorems for certain functionals of semimartingales observed at high frequency. In particular, we extend results from Jacod to the case of bipower variation, showing under standard assumptions that one obtains a limiting variable, which is in general different from the case of a continuous semimartingale. In a second step a truncated version of bipower variation is constructed, which has a similar asymptotic behaviour as standard bipower variation for a continuous semimartingale and thus provides a feasible central limit theorem for the estimation of the integrated volatility even when the semimartingale exhibits jumps.



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