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Workshop on Statistical Inference for Lévy Processes with Applications to Finance July 15 - 17, 2009 Programme
Yacine Ait-Sahalia (Princeton University) 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) 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 Lévyprocesses 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 L¶evy and di®usion models are compared. We close by comparing discrete and continous time. Fabienne Comte (Université René Descartes - Paris 5) Nonparametric estimation for pure jump Lévy 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 Lévy density of a pure jump Lévy 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 Lévy 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 Lévy 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 Lévy process to the dynamics of a financial asset. Ernst Eberlein (University of Freiburg) Correlation based calibration of Lévy interest rate models The Lévy interest rate theory is discussed by introducing the Lévy forward rate and the Lévy 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 Lévy forward rate model. Mark Podolskij ((ETH Zürich)) 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 (Université René Descartes- Paris 5) Nonparametric estimation for pure jump Lévy processes based on high frequency data. The talk is based on a joint work with Fabienne Comte. We study nonparametric estimation of the Lévy density for pure jump Lévy 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-Universität Marburg) 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 Lévy $\alpha$-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)
In multivariate Lévy processes, the Lévy 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 Lévy processes. Analogously to the concept of
tail dependence we call this probability jump tail dependence. In
contrast to copulas, Lévy 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 Lévy
copula alone and that it is independent of the jump distributions of the
marginal Lévy processes. For the estimation of jump tail dependence we
derive asymptotical relations between jump tail dependence in the Lévy
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.
Presentation Shota Gugushvili (EURANDOM) Given a discrete time sample $X_1,... X_n$ from a Lévy
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 Implied Lévy 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
Lévy framework and therefore based on distributions that match more
closely historical returns. We introduce the concept of implied Lévy
volatility, hereby extending the intuitive Black-Scholes implied
volatility into a more general context. The Lévy models are obtained by
replacing the Wiener distribution modeling the diffusion part of the
log-return process by a more empirically founded Lévy distribution. The
Lévy space volatility model will arise by multiplying volatility with
the underlying Lévy process, whereas the Lévy time volatility model will
arise by multiplying volatility squared with time. Lévy 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 Lévy 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 Lévy volatility curves
and vice versa. It is shown that any smiling or smirking Black-Scholes
volatility curve can be transformed into a atter Lévy volatility curve
under a well chosen parameter set. This gives some evidence to the fact
that the implied Lévy models could lead to atter volatility curve for
more practical datasets. Hence, implied Lévy 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 Lévy 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) Hilmar Mai (Humboldt-Universität Berlin) 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. José Manuel Corcuera (University
of Barcelona) Completeness and hedging in a Lévy bond market We consider bond markets with one factor where the noise
is a Lévy 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: Lévy processes, martingale measure, hedging,
incomplete markets Mathematics Subject Classification 2000: 60H30, 60G46, 91B28 References Antonis Papapantoleon
(Technische Universität Berlin) A new approach to LIBOR modeling 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 Reiß (Humboldt-Universität
Berlin) Nonparametric estimation for Levy processes from
low-frequency observations We suppose that a Lévy process is observed at
discrete time points. A rather general construction of minimum-distance
estimators is shown to give consistent estimators of the Lévy-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) 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. Klüppelberg, 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 Lévy 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 Lévy 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 Lévy measure and prove the asymptotic efficiency for some
estimators. Mathias Vetter (Ruhr-Universität 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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