Discretely Sampled Research Articles

Discrete Sampling of Extreme Events Modifies Their Statistics.

Extreme value (EV) statistics of correlated systems are widely investigated in many fields, spanning the spectrum from weather forecasting to earthquake prediction. Does the unavoidable discrete sampling of a continuous correlated stochastic process change its EV distribution? We explore this question for correlated random variables modeled via Langevin dynamics for a particle in a potential field. For potentials growing at infinity faster than linearly and for long measurement times, we find that the EV distribution of the discretely sampled process diverges from that of the full continuous dataset and converges to that of independent and identically distributed random variables drawn from the process's equilibrium measure. However, for processes with sublinear potentials, the long-time limit is the EV statistics of the continuously sampled data. We treat processes whose equilibrium measures belong to the three EV attractors: Gumbel, Fréchet, and Weibull. Our Letter shows that the EV statistics can be extremely sensitive to the sampling rate of the data.

Nonparametric Regression Based on Discretely Sampled Curves

Nonparametric Regression Based on Discretely Sampled Curves

Paleoenvironmental conditions and sedimentation dynamics in Central Europe inferred from geochemical data of the loess-paleosol sequence at Süttő (Hungary)

This study provides a detailed geochemical record of one of the best studied loess-paleosol sequences (LPS) in Hungary covering Marine Isotope Stage (MIS) 6 to MIS 2. Süttő is situated in the northwestern Carpathian Basin at the transition between oceanic and continental climate. We investigated weathering intensity, dust provenance and sedimentation dynamics by high-resolution X-ray fluorescence (XRF) scanning of discrete samples in comparison with quantitative XRF and other published proxy data such as grain-size, magnetic susceptibility and micromorphology. Multivariate statistics unravel pedogenesis in terms of decalcification and grain-size dynamics as processes controlling the elemental composition. Furthermore, high-resolution chemostratigraphy indicates geochemical homogeneity for mature MIS 5 paleosols and the modern soil. Together with the A-CN-K plot providing no hint for K-feldspar weathering, semi-arid continental paleoenvironmental conditions are considered as prevailing. Dust provenance is studied by geochemical reference literature data from potential ultimate dust source areas, Th/metal ratios and comparison with the composition of the upper continental crust (UCC). Results indicate a felsic rock composition and a stable dust source, albeit immediate (adjacent Danube floodplains) and ultimate dust sources (surrounding mountain belts) need to be distinguished. According to the Sr-K-Rb plot, the Austroalpine cover nappes area northwest of Süttő might be the dominant ultimate dust source. The weathering-resistant elements Al, Ti and Zr as well as Zr/metal ratios indicate Zr addition and thus strongest sediment sorting and/or recycling during MIS 6. Since Al, Ti and Zr show minimal correlations with grain-size, we suggest a cascade-alike transport pathway from ultimate source to sink involving multiple sediment recycling and sorting cycles in both fluvial and eolian domains. Therefore, this study highlights how high-resolution qualitative geochemical data contributes to a better understanding of paleoenvironmental and sedimentation dynamics at LPS. Application of such a versatile geochemical screening tool is of broad interest to the scientific community working on (terrestrial) paleoenvironmental archives that are commonly discretely sampled.

XRF scanning of discrete samples – A chemostratigraphic approach exemplified for loess-paleosol sequences from the Island of Susak, Croatia

X-ray fluorescence (XRF) scanning of discrete samples provides a time- and cost-efficient alternative to conventional geochemical analyses of discretely sampled paleoenvironmental archives. Loess-paleosol sequences (LPS) as complex terrestrial archives are thus predestinated for benchmarking this new application of the well-established XRF scanning technology. Stratigraphically incomplete outcrops, like those on the Island of Susak (Croatia), are unique paleoenvironmental archives at least for the penultimate and last interglacial/glacial cycles and provide ideal test sites. Available proxy data from previous investigations on Susak ensure validation of conclusions drawn from XRF scanning-derived data.Three representative profiles were studied to examine the potential of XRF scanning-derived element ratios as proxy for environmental processes: (1) Ca/Sr for the strength of ubiquitously present secondary calcification; (2) Rb/K for the intensity of weathering; (3) Ti/Al and Ti/Zr for changes in wind speed with Zr being additionally indicative for sediment recycling; (4) Si/Al for changes in grain-size; and (5) Br and Cl as proxies for sea spray. Furthermore, we established a multivariate statistics-based chemostratigraphy to facilitate profile correlation and to improve lithostratigraphic characterization. In conclusion, this study demonstrates that XRF scanning of discrete samples from LPS is a valuable tool for paleoenvironmental reconstruction.

Conditional Least Squares Estimation for Discretely Sampled Nonergodic Diffusions

Conditional Least Squares Estimation for Discretely Sampled Nonergodic Diffusions

Geophysical Tutorial: The function of interpolation

When interpreting seismic or well data in desktop software, it's easy to forget that the data are discretely sampled in space and time. Even when you look at an apparently continuous log curve, there are really only samples, typically every 0.1524 m (6 inches). When you display seismic data as wiggle traces, there are perhaps samples only every 2 or 4 ms in the time dimension. Some software lets you turn off the interpolation so you can see the discrete data samples, as in Figure 1a .

Valuation of options on discretely sampled variance: a general analytic approximation

The values of options on realized variance are significantly impacted by the discrete sampling of realized variance and may be substantially higher than the values of options on continuously sampled variance (or, quadratic variation). Under arbitrary stochastic volatility dynamics, we analyze the discretization effect and obtain a simple analytical correction term to be applied to the value of options on continuously sampled variance. Our final result is remarkably compact and allows for a straightforward implementation in many of the standard stochastic volatility models proposed in the literature.

A closed-form pricing formula for variance swaps with mean-reverting Gaussian volatility

Although variance swaps have become an important financial derivative to hedge against volatility risks, closed-form formulae have been developed only recently, when the realized variance is defined on discrete sample points and no continuous approximation is adopted to alleviate the mathematical difficulties associated with dealing with the discreteness of the sample data. In this paper, a new closed-form pricing formula for the value of a discretely sampled variance swap is presented under the assumption that the underlying asset prices can be described by a mean-reverting Gaussian volatility model. With the newly found analytical formula, not only can all the hedging ratios of a variance swap be analytically derived, the numerical values of the swap price can be efficiently computed as well. doi: 10.1017/S144618111400011X

A CLOSED-FORM PRICING FORMULA FOR VARIANCE SWAPS WITH MEAN-REVERTING GAUSSIAN VOLATILITY

AbstractAlthough variance swaps have become an important financial derivative to hedge against volatility risks, closed-form formulae have been developed only recently, when the realized variance is defined on discrete sample points and no continuous approximation is adopted to alleviate the mathematical difficulties associated with dealing with the discreteness of the sample data. In this paper, a new closed-form pricing formula for the value of a discretely sampled variance swap is presented under the assumption that the underlying asset prices can be described by a mean-reverting Gaussian volatility model. With the newly found analytical formula, not only can all the hedging ratios of a variance swap be analytically derived, the numerical values of the swap price can be efficiently computed as well.

Volatility Swaps and Variance Options on Discretely Sampled Realized Variance

Volatility swaps and variance options are financial products written on discretely sampled realized variance. Actively traded in over-the-counter markets, these products are priced often by a continuously sampled approximation to simplify the computations. This paper presents an analytical approach to efficiently and accurately price discretely sampled volatility derivatives, under the Heston stochastic volatility model. We first obtain an accurate approximation for the characteristic function of the discretely sampled realized variance. This characteristic function is then applied to derive semi-analytical (up to an inverse Laplace transform) pricing formulae for variance options, volatility swaps and volatility options. We examine with numerical examples the accuracies of the approach in pricing these volatility derivatives. We also test the effect of discrete sampling in pricing volatility derivatives. For realistic contract specifications and model parameters, we find that although continuously sampled variance swaps and options are cheaper than their discretely sampled counterparts, continuously sampled volatility swaps are, however, more expensive than their discretely sampled counterparts.

Prices and Asymptotics for Discrete Variance Swaps

We study the fair strike of a discrete variance swap for a general time-homogeneous stochastic volatility model. In the special cases of Heston, Hull–White and Schöbel–Zhu stochastic volatility models, we give simple explicit expressions (improving Broadie and Jain (2008a). The effect of jumps and discrete sampling on volatility and variance swaps. International Journal of Theoretical and Applied Finance, 11(8), 761–797) in the case of the Heston model). We give conditions on parameters under which the fair strike of a discrete variance swap is higher or lower than that of the continuous variance swap. The interest rate and the correlation between the underlying price and its volatility are key elements in this analysis. We derive asymptotics for the discrete variance swaps and compare our results with those of Broadie and Jain (2008a. The effect of jumps and discrete sampling on volatility and variance swaps. International Journal of Theoretical and Applied Finance, 11(8), 761–797), Jarrow et al. (2013. Discretely sampled variance and volatility swaps versus their continuous approximations. Finance and Stochastics, 17(2), 305–324) and Keller-Ressel and Griessler (2012. Convex order of discrete, continuous and predictable quadratic variation and applications to options on variance. Working paper. Retrieved from http://arxiv.org/abs/1103.2310).

Pricing variance and volatility swaps in a stochastic volatility model with regime switching: discrete observations case

This study presents a set of closed-form exact solutions for pricing discretely sampled variance swaps and volatility swaps, based on the Heston stochastic volatility model with regime switching. In comparison with all the previous studies in the literature, this research, which obtains closed-form exact solutions for variance and volatility swaps with discrete sampling times, serves several purposes. (1) It verifies the degree of validity of Elliott et al.'s [Appl. Math. Finance, 2007, 14(1), 41–62] continuous-sampling-time approximation for variance and volatility swaps of relatively short sampling periods. (2) It examines the effect of ignoring regime switching on pricing variance and volatility swaps. (3) It contributes to bridging the gap between Zhu and Lian's [Math. Finance, 2011, 21(2), 233–256] approach and Elliott et al.'s framework. (4) Finally, it presents a semi-Monte-Carlo simulation for the pricing of other important realized variance based derivatives.

Fourier Transform Algorithms for Pricing and Hedging Discretely Sampled Exotic Variance Products and Volatility Derivatives under Additive Processes

We develop efficient fast Fourier transform algorithms for pricing and hedging discretely sampled variance products and volatility derivatives under additive processes (time-inhomogeneous Levy processes). Our numerical algorithms are non-trivial versions of the Fourier space time stepping method to nonlinear path dependent payoff structures, like those in variance products and volatility derivatives. The exotic path dependency associated with the discretely sampled realized variance is captured in the numerical procedure by updating two path dependent state variables across monitoring dates. The time stepping procedure between successive monitoring dates can be performed using fast Fourier transform calculations without the usual tedious time stepping calculations in typical nite di erence algorithms. We also derive effective numerical procedures that compute the hedge parameters of variance products and volatility derivatives. Numerical tests on pricing various variance products and volatility derivatives were performed that illustrate eciency, accuracy, reliability and robustness of the proposed Fourier transform algorithms.

Pricing of Discretely Sampled Asian Options Under Levy Processes

We develop a new method for pricing options on discretely sampled arithmetic average in exponential L'evy models. The main idea is the reduction to a backward induction procedure for the difference W_n between the Asian option with averaging over n sampling periods and the price of the European option with maturity one period. This allows for an efficient truncation of the state space. At each step of backward induction, W_n is calculated accurately and fast using a piece-wise interpolation or splines, fast convolution and either iFFT or the parabolic inverse Fourier transform. Numerical results demonstrate the advantages of the method.

Pricing Variance and Volatility Swaps in a Stochastic Volatility Model with Regime Switching: Discrete Observations Case

This study presents a set of closed-form exact solutions for pricing discretely sampled variance swaps and volatility swaps, based on the Heston stochastic volatility model with regime switching. In comparison with all the previous studies in the literature, this research, which obtains closed-form exact solutions for variance and volatility swaps with discrete sampling times, serves several purposes. It verifies the degree of validity of Elliott et al.’s [Appl. Math. Finance, 2007, 14(1), 41–62] continuous-sampling-time approximation for variance and volatility swaps of relatively short sampling periods. It examines the effect of ignoring regime switching on pricing variance and volatility swaps. It contributes to bridging the gap between Zhu and Lian’s [Math. Finance, 2011, 21(2), 233–256] approach and Elliott et al.’s framework. Finally, it presents a semi-Monte-Carlo simulation for the pricing of other important realized variance based derivatives.

Bias in the estimation of the mean reversion parameter in continuous time models

It is well known that for continuous time models with a linear drift standard estimation methods yield biased estimators for the mean reversion parameter both in finite discrete samples and in large in-fill samples. In this paper, we obtain two expressions to approximate the bias of the least squares/maximum likelihood estimator of the mean reversion parameter in the Ornstein–Uhlenbeck process with a known long run mean when discretely sampled data are available. The first expression mimics the bias formula of Marriott and Pope (1954) for the discrete time model. Simulations show that this expression does not work satisfactorily when the speed of mean reversion is slow. Slow mean reversion corresponds to the near unit root situation and is empirically realistic for financial time series. An improvement is made in the second expression where a nonlinear correction term is included into the bias formula. It is shown that the nonlinear term is important in the near unit root situation. Simulations indicate that the second expression captures the magnitude, the curvature and the non-monotonicity of the actual bias better than the first expression.

Variance swaps on time-changed Lévy processes

We prove that a multiple of a log contract prices a variance swap, under arbitrary exponential Levy dynamics, stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the driving Levy process, subject to integrability conditions. We solve for the multiplier, which depends only on the Levy process, not on the clock. In the case of an arbitrary continuous underlying returns process, the multiplier is 2, which recovers the standard no-jump variance swap pricing formula. In the presence of negatively skewed jump risk, however, we prove that the multiplier exceeds 2, which agrees with calibrations of time-changed Levy processes to equity options data. Moreover, we show that discrete sampling increases variance swap values, under an independence condition; so if the commonly quoted multiple 2 undervalues the continuously sampled variance, then it undervalues even more the discretely sampled variance. Our valuations admit enforcement, in some cases, by hedging strategies which perfectly replicate variance swaps by holding log contracts and trading the underlying.

Valuation of Options on Discretely Sampled Variance

The values of options on realized variance are significantly impacted by the discrete sampling of realized variance and may be substantially higher than the values of options on continuously sampled variance (or, quadratic variation). Under arbitrary stochastic volatility dynamics, we analyze the discretization effect and obtain a simple analytical correction term to be applied to the value of options on continuously sampled variance. Our final result is remarkably compact and allows for a straightforward implementation in many of the standard stochastic volatility models proposed in the literature.

Spacetime could be simultaneously continuous and discrete, in the same way that information can be

There are competing schools of thought about the question of whether spacetime is fundamentally continuous or discrete. Here, we consider the possibility that spacetime could be simultaneously continuous and discrete, in the same mathematical way that information can be simultaneously continuous and discrete. The equivalence of continuous information and discrete information, which is of key importance in signal processing, is established by the Shannon sampling theory: for any band-limited signal, it suffices to record discrete samples to be able to perfectly reconstruct it everywhere, if the samples are taken at a rate of at least twice the band limit. It is known that physical fields on generic curved spaces obey a sampling theorem if they possess an ultraviolet cutoff. Most recently, methods of spectral geometry have been employed to show that also the very shape of a curved space (i.e. of a Riemannian manifold) can be discretely sampled and then reconstructed up to the cutoff scale. Here, we develop these results further and also consider the generalization to curved spacetimes, i.e. to Lorentzian manifolds.

Bayesian Inference for Discretely Sampled Markov Processes with Closed-Form Likelihood Expansions

This article proposes a new Bayesian Markov chain Monte Carlo (MCMC) methodology for estimation of a wide class of multidimensional jump-diffusion models. Our approach is based on the closed-form (CF) likelihood approximations of Aït-Sahalia (2002, 2008). The CF likelihood approximation does not integrate to 1; it is very close to 1 when in the center of the distribution but can differ markedly from 1 when far in the tails. We propose an MCMC algorithm that addresses the problems that arise when the CF approximation is applied in a Bayesian context. The efficacy of our approach is demonstrated in a simulation study of the Cox–Ingersoll–Ross and Heston models and is applied to two well-known datasets.