Infinite Variation Research Articles

Truncated Realized Covariance When Prices Have Infinite Variation Jumps

The speed of convergence of the truncated realized covariance to the integrated covariation between the Brownian parts of two semimartingales is heavily influenced by the presence of infinite activity jumps with infinite variation. Namely, the two small jumps processes play a crucial role through their degree of dependence, other than through their jump activity indices. This theoretical result is established when the marginal small jumps constitute alpha-stable processes and the semimartingales are observed discretely on a finite time horizon. The estimator in many (but not all) cases is less efficient than when the model only has finite variation jumps.The result of this paper is relevant in financial economics, since by the truncated realized covariance it is possible to separately estimate the common jumps among two assets, which has important implications in risk management and contagion modeling.

Infinite networks and variation of conductance functions in discrete Laplacians

For a given infinite connected graph G = (V, E) and an arbitrary but fixed conductance function c, we study an associated graph Laplacian Δc; it is a generalized difference operator where the differences are measured across the edges E in G; and the conductance function c represents the corresponding coefficients. The graph Laplacian (a key tool in the study of infinite networks) acts in an energy Hilbert space ℋE computed from c. Using a certain Parseval frame, we study the spectral theoretic properties of graph Laplacians. In fact, for fixed c, there are two versions of the graph Laplacian, one defined naturally in the l2 space of V and the other in ℋE. The first is automatically selfadjoint, but the second involves a Krein extension. We prove that, as sets, the two spectra are the same, aside from the point 0. The point zero may be in the spectrum of the second, but not the first. We further study the fine structure of the respective spectra as the conductance function varies, showing now how the spectrum changes subject to variations in the function c. Specifically, we study an order on the spectra of the family of operators Δc, and we compare it to the ordering of pairs of conductance functions. We show how point-wise estimates for two conductance functions translate into spectral comparisons for the two corresponding graph Laplacians, involving a certain similarity: We prove that point-wise ordering of two conductance functions c on E induces a certain similarity of the corresponding (Krein extensions computed from the) two graph Laplacians Δc. The spectra are typically continuous, and precise notions of fine-structure of spectrum must be defined in terms of equivalence classes of positive Borel measures (on the real line). Our detailed comparison of spectra is analyzed this way.

Testing for pure-jump processes for high-frequency data

Pure-jump processes have been increasingly popular in modeling high-frequency financial data, partially due to their versatility and flexibility. In the meantime, several statistical tests have been proposed in the literature to check the validity of using pure-jump models. However, these tests suffer from several drawbacks, such as requiring rather stringent conditions and having slow rates of convergence. In this paper, we propose a different test to check whether the underlying process of high-frequency data can be modeled by a pure-jump process. The new test is based on the realized characteristic function, and enjoys a much faster convergence rate of order $O(n^{1/2})$ (where $n$ is the sample size) versus the usual $o(n^{1/4})$ available for existing tests; it is applicable much more generally than previous tests; for example, it is robust to jumps of infinite variation and flexible modeling of the diffusion component. Simulation studies justify our findings and the test is also applied to some real high-frequency financial data.

The distribution of the supremum for spectrally asymmetric Lévy processes

In this article we derive formulas for the probability $\mathbb{P}(\sup_{t\leq T} X(t)>u)$, $T>0$ and $\mathbb{P}(\sup_{t u)$ where $X$ is a spectrally positive Levy process with infinite variation. The formulas are generalizations of the well-known Takacs formulas for stochastic processes with non-negative and interchangeable increments. Moreover, we find the joint distribution of $\inf_{t\leq T} Y(t)$ and $Y(T)$ where $Y$ is a spectrally negative Levy process.

On the Convergence Rate of Randomized Quasi--Monte Carlo for Discontinuous Functions

This paper studies the convergence rate of randomized quasi--Monte Carlo (RQMC) for discontinuous functions, which are often of infinite variation in the sense of Hardy and Krause. It was previously known that the root mean square error (RMSE) of RQMC is only $o(n^{-1/2})$ for discontinuous functions. For certain discontinuous functions in $d$ dimensions, we prove that the RMSE of RQMC is $O(n^{-1/2-1/(4d-2)+\epsilon})$ for any $\epsilon>0$ and arbitrary $n$. If some discontinuity boundaries are parallel to some coordinate axes, the rate can be improved to $O(n^{-1/2-1/(4d_u-2)+\epsilon})$, where $d_u$ denotes the so-called irregular dimension, that is, the number of axes which are not parallel to the discontinuity boundaries. Moreover, this paper shows that the RMSE is $O(n^{-1/2-1/(2d)})$ for certain indicator functions.

Viscosity methods giving uniqueness for martingale problems

Let $E$ be a complete, separable metric space and $A$ be an operator on $C_b(E)$. We give an abstract definition of viscosity sub/supersolution of the resolvent equation $\lambda u-Au=h$ and show that, if the comparison principle holds, then the martingale problem for $A$ has a unique solution. Our proofs work also under two alternative definitions of viscosity sub/supersolution which might be useful, in particular, in infinite dimensional spaces, for instance to study measure-valued processes. We prove the analogous result for stochastic processes that must satisfy boundary conditions, modeled as solutions of constrained martingale problems. In the case of reflecting diffusions in $D\subset {\bf R}^d$, our assumptions allow $ D$ to be nonsmooth and the direction of reflection to be degenerate. Two examples are presented: A diffusion with degenerate oblique direction of reflection and a class of jump diffusion processes with infinite variation jump component and possibly degenerate diffusion matrix.

Ghost calibration and the pricing of barrier options and CDS in spectrally one-sided Lévy models: the parabolic Laplace inversion method

Recently, the advantages of conformal deformations of the contours of integration in pricing formulas were demonstrated in the context of wide classes of Lévy models and the Heston model. In the present paper, we construct efficient conformal deformations of the contours of integration in the pricing formulas for barrier options and CDS in the setting of spectrally one-sided Lévy models, taking advantage of Rogers’s trick [J. Appl. Prob., 2000, 37, 1173–1180] that greatly simplifies calculation of the Wiener-Hopf factors. We extend the trick to wide classes of Lévy processes of infinite variation with zero diffusion component. In the resulting formulas (both in the finite variation and the infinite variation cases), we make quasi-parabolic deformations as in Boyarchenko and Levendorskiĭ [Int. J. Theor. Appl. Finance, 2013, 16 (3), 1350011], which greatly increase the rate of convergence of the integrals. We demonstrate that in many cases the proposed method is more accurate than the standard realization of Laplace inversion. We also exhibit examples in which the standard realization is so unstable that it cannot be used for any choice of the error control parameters. This may lead to a ghost calibration: a situation where a parameter set of a model is declared to be a ‘good fit’ to the data only because the errors of calibration and of the numerical method used for pricing (almost) cancel each other out.

Ghost Calibration and Pricing Barrier Options and CDS in Spectrally One-Sided L'evy Models: The Parabolic Laplace Inversion Method

Recently, the advantages of conformal deformations of the contours of integration in pricing formulas were demonstrated in the context of wide classes of Levy models and the Heston model. In the present paper, we construct efficient conformal deformations of the contours of integration in the pricing formulas for barrier options and CDS in the setting of spectrally one-sided Levy models, taking advantage of Rogers’s trick [J. Appl. Prob., 2000, 37, 1173–1180] that greatly simplifies calculation of the Wiener-Hopf factors. We extend the trick to wide classes of Levy processes of infinite variation with zero diffusion component. In the resulting formulas (both in the finite variation and the infinite variation cases), we make quasi-parabolic deformations as in Boyarchenko and Levendorskiĭ [Int. J. Theor. Appl. Finance, 2013, 16 (3), 1350011], which greatly increase the rate of convergence of the integrals. We demonstrate that in many cases the proposed method is more accurate than the standard realization o...

Efficient estimation of integrated volatility in presence of infinite variation jumps

We propose new nonparametric estimators of the integrated volatility of an It\^{o} semimartingale observed at discrete times on a fixed time interval with mesh of the observation grid shrinking to zero. The proposed estimators achieve the optimal rate and variance of estimating integrated volatility even in the presence of infinite variation jumps when the latter are stochastic integrals with respect to locally "stable" L\'{e}vy processes, that is, processes whose L\'{e}vy measure around zero behaves like that of a stable process. On a first step, we estimate locally volatility from the empirical characteristic function of the increments of the process over blocks of shrinking length and then we sum these estimates to form initial estimators of the integrated volatility. The estimators contain bias when jumps of infinite variation are present, and on a second step we estimate and remove this bias by using integrated volatility estimators formed from the empirical characteristic function of the high-frequency increments for different values of its argument. The second step debiased estimators achieve efficiency and we derive a feasible central limit theorem for them.

Lévy Processes with Marked Jumps I: Limit Theorems

Consider a sequence \(({\tilde{Z}}_n,{\tilde{Z}}_n^{{m}})\) of bivariate Lévy processes, such that \({\tilde{Z}}_n\) is a spectrally positive Lévy process with finite variation, and \({\tilde{Z}}_n^{{m}}\) is the counting process of marks in \(\{0,1\}\) carried by the jumps of \({\tilde{Z}}_n\). The study of these processes is justified by their interpretation as contour processes of a sequence of splitting trees (Lambert in Ann Probab 38(1):348–395, 2010) with mutations at birth. Indeed, this paper is the first part of a work (Delaporte in Lévy processes with marked jumps II: application to a population model with mutations at birth) aiming to establish an invariance principle for the genealogies of such populations enriched with their mutational histories. To this aim, we define a bivariate subordinator that we call the marked ladder height process of \(({\tilde{Z}}_n,{\tilde{Z}}_n^{{m}})\), as a generalization of the classical ladder height process to our Lévy processes with marked jumps. Assuming that the sequence \(({\tilde{Z}}_n)\) converges towards a Lévy process \(Z\) with infinite variation, we first prove the convergence in distribution, with two possible regimes for the marks, of the marked ladder height process of \(({\tilde{Z}}_n,{\tilde{Z}}_n^{{m}})\). Then, we prove the joint convergence in law of \({\tilde{Z}}_n\) with its local time at the supremum and its marked ladder height process. The proof of this latter result is an adaptation of Chaumont and Doney (Ann Probab 38(4):1368–1389, 2010) to the finite variation case.

Monte Carlo Simulation of the CGMY Process and Option Pricing

AbstractWe present a joint Monte Carlo‐Fourier transform sampling scheme for pricing derivative products under a Carr–Geman–Madan–Yor (CGMY) model (Carr et al. [Journal of Business, 75, 305–332, 2002]) exhibiting jumps of infinite activity and finite or infinite variation. The approach relies on numerical transform inversion with computable error estimates, which allow generating the unknown cumulative distribution function of the CGMY process increments at the desired accuracy level. We use this to generate samples and simulate the entire trajectory of the process without need of truncating the process small jumps. We illustrate the computational efficiency of the proposed method by comparing it to the existing methods in the literature on pricing a wide range of option contracts, including path‐dependent univariate and multivariate products. © 2014 Wiley Periodicals, Inc. Jrl Fut Mark 34:1095–1121, 2014

The Kingman tree length process has infinite quadratic variation

In the case of neutral populations of fixed sizes in equilibrium whose genealogies are described by the Kingman N-coalescent back from time t consider the associated processes of total tree length as t increases. We show that the (càdlàg) process to which the sequence of compensated tree length processes converges as N tends to infinity is a process of infinite quadratic variation; therefore this process cannot be a semimartingale. This answers a question posed in Pfaffelhuber et al. (2011).

$L_1$-distance for additive processes with time-homogeneous Lévy measures

We give an explicit bound for the $L_1$-distance between two additive processes of local characteristics $(f_j(\cdot),\sigma^2(\cdot),\nu_j)$, $j = 1,2$. The cases $\sigma =0$ and $\sigma(\cdot) > 0$ are both treated. We allow $\nu_1$ and $\nu_2$ to be time-homogeneous Levy measures, possibly with infinite variation. Some examples of possible applications are discussed.

Pricing Barrier Options and Credit Default Swaps (CDS) in Spectrally One-Sided Levy Models: The Parabolic Laplace Inversion Method

Recently, the advantages of conformal deformations of the contours of integration in pricing formulas were demonstrated in the context of wide classes of Levy models and the Heston model. In the present paper we construct efficient conformal deformations of the contours of integration in the pricing formulas for barrier options and CDS in the setting of spectrally one-sided Levy models, taking advantage of Rogers's trick (J. Appl. Prob. 2000) that greatly simplifies calculation of the Wiener-Hopf factors. We extend the trick to wide classes of Levy processes of infinite variation with zero diffusion component. In the resulting formulas (both in the finite variation and the infinite variation cases), we make quasi-parabolic deformations as in S. Boyarchenko and Levendorskii (IJTAF 2013), which greatly increase the rate of convergence of the integrals. We demonstrate that the proposed method is more accurate than the standard realization of Laplace inversion in many cases. We also exhibit examples in which the standard realization is so unstable that it cannot be used for any choice of the error control parameters.

Arbitrage Pricing Theory Under Transaction Costs and Application to the Tobin Tax

This paper studies arbitrage pricing theory in financial markets with transaction costs. We extend the existing theory to include the more realistic possibility that the price at which the investors trade is dependent on the traded volume. The investors in the market always buy at the ask and sell at the bid price. Transaction costs are composed of two terms, one is able to capture the implicit transaction costs, and the second the price impact. Moreover, a new definition of a self-financing portfolio is obtained. The self-financing condition suggests that continuous trading is possible, but is restricted to predictable trading strategies having caglad (right-continuous with left limits) and caglad (left-continuous with right limits) paths of bounded quadratic variation and of finitely many jumps. That is, caglad and caglad predictable trading strategies of infinite variation, with finitely many jumps and of finite quadratic variation are allowed in our setting. Restricting ourselves to caglad predictable trading strategies, we show that the existence of an equivalent probability measure is equivalent to the absence of arbitrage opportunities, so that the first fundamental theorem of asset pricing (FFTAP) holds. It is also shown that the use of continuous and bounded variation trading strategies can improve the efficiency of hedging in a market with transaction costs. To better understand how to apply the theory proposed we provide an example with linear and non-linear transaction costs.

A NEW LOOK AT SHORT‐TERM IMPLIED VOLATILITY IN ASSET PRICE MODELS WITH JUMPS

We analyze the behavior of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalization of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal–Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short‐end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant model‐independent slope. This result gives a theoretical justification for the preference of the infinite variation Lévy models over the finite variation ones in the calibration based on short‐maturity option prices.

Volatility activity: Specification and estimation

The paper examines volatility activity and its asymmetry and undertakes further specification analysis of volatility models based on it. We develop new nonparametric statistics using high-frequency option-based VIX data to test for asymmetry in volatility jumps. We also develop methods for estimating and evaluating, using price data alone, a general encompassing model for volatility dynamics where volatility activity is unrestricted. The nonparametric application to VIX data, along with model estimation for S&P index returns, suggests that volatility moves are best captured by an infinite variation pure-jump martingale with a symmetric jump compensator around zero. The latter provides a parsimonious generalization of the jump-diffusions commonly used for volatility modeling.

Monte Carlo Option Pricing Coupled with Fourier Transform for the CGMY Process

We present a joint Monte Carlo-Fourier transform sampling scheme for pricing derivative products under a Carr-Geman-Madan-Yor (CGMY) model (Carr et al. [Journal of Business, 75, 305-332, 2002]) exhibiting jumps of infinite activity and finite or infinite variation. The approach relies on numerical transform inversion with computable error estimates, which allow generating the unknown cumulative distribution function of the CGMY process increments at the desired accuracy level. We use this to generate samples and simulate the entire trajectory of the process without need of truncating the process small jumps. We illustrate the computational efficiency of the proposed method by comparing to the existing methods in the literature on pricing a wide range of option contracts, including path-dependent univariate and multivariate products.

Explicit formula for the supremum distribution of a spectrally negative stable process

In this article we get simple formulas for $E\sup_{s\leq t}X(s)$ where $X$ is a spectrally positive or negative Levy process with infinite variation. As a consequence we derive a generalization of the well-known formula for the supremum distribution of Wiener process that is we obtain $P(\sup_{s\leq t}Z_{\alpha}(s)\geq u)=\alpha\,P(Z_{\alpha}(t)\geq u)$ for $u\geq 0$ where $Z_{\alpha}$ is a spectrally negative $\alpha$-stable Levy process with $1<\alpha\leq 2$ which also stems from Kendall's identity for the first crossing time. Our proof uses a formula for the supremum distribution of a spectrally positive Levy process which follows easily from the elementary Seal's formula.

On the Impulse Control of Jump Diffusions

Regularity of the impulse control problem for a nondegenerate $n$-dimensional jump diffusion with infinite activity and finite variation jumps was recently examined in [M. H. A. Davis, X. Guo, and G. Wu, SIAM J. Control Optim., 48 (2010), pp. 5276--5293]. Here we extend the analysis to include infinite activity and infinite variation jumps. More specifically, we show that the value function $u$ of the impulse control problem satisfies $u\in W^{2,p}_{\mathrm{loc}}(\mathbb{R}^{n})$.