Showing papers on "Semimartingale published in 2019"

Adapted Wasserstein Distances and Stability in Mathematical Finance

Abstract: Assume that an agent models a financial asset through a measure Q with the goal to price / hedge some derivative or optimize some expected utility. Even if the model Q is chosen in the most skilful and sophisticated way, she is left with the possibility that Q does not provide an "exact" description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of Q? If we measure proximity with the usual Wasserstein distance (say), the answer is NO. Models which are similar w.r.t. Wasserstein distance may provide dramatically different information on which to base a hedging strategy. Remarkably, this can be overcome by considering a suitable "adapted" version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested distance as pioneered by Pflug and Pichler \cite{Pf09,PfPi12,PfPi14}. It allows us to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time. Notably, these abstract results are sharp already for Brownian motion and European call options.

Canonical RDEs and general semimartingales as rough paths

Abstract: In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of Lepingle’s BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased a uniformly-controlled-variations (UCV) condition (Kurtz–Protter, Jakubowski–Memin–Pages). A number of examples illustrate the scope of our results.

Estimating the Spot Covariation of Asset Prices—Statistical Theory and Empirical Evidence

Abstract: We propose a new estimator for the spot covariance matrix of a multi-dimensional continuous semimartingale log asset price process, which is subject to noise and nonsynchronous observations. The es...

Dynamic risk measure for BSVIE with jumps and semimartingale issues

Abstract: Risk measure is a fundamental concept in finance and in the insurance industry. It is used to adjust life insurance rates. In this article, we will study dynamic risk measures by means of backward ...

Efficient estimation of integrated volatility functionals via multiscale jackknife

Abstract: We propose semiparametrically efficient estimators for general integrated volatility functionals of multivariate semimartingale processes. A plug-in method that uses nonparametric estimates of spot volatilities is known to induce high-order biases that need to be corrected to obey a central limit theorem. Such bias terms arise from boundary effects, the diffusive and jump movements of stochastic volatility and the sampling error from the nonparametric spot volatility estimation. We propose a novel jackknife method for bias correction. The jackknife estimator is simply formed as a linear combination of a few uncorrected estimators associated with different local window sizes used in the estimation of spot volatility. We show theoretically that our estimator is asymptotically mixed Gaussian, semiparametrically efficient, and more robust to the choice of local windows. To facilitate the practical use, we introduce a simulation-based estimator of the asymptotic variance, so that our inference is derivative-free, and hence is convenient to implement.

Implications of Kunita-It\^o-Wentzell formula for $k$-forms in stochastic fluid dynamics

Abstract: We extend the Ito-Wentzell formula for the evolution of a time-dependent stochastic field along a semimartingale to $k$-form-valued stochastic processes. The result is the Kunita-Ito-Wentzell (KIW) formula for $k$-forms. We also establish a correspondence between the KIW formula for $k$-forms derived here and a certain class of stochastic fluid dynamics models which preserve the geometric structure of deterministic ideal fluid dynamics. This geometric structure includes Eulerian and Lagrangian variational principles, Lie--Poisson Hamiltonian formulations and natural analogues of the Kelvin circulation theorem, all derived in the stochastic setting.

Nonparametric implied Lévy densities

Abstract: This paper develops a nonparametric estimator for the Levy density of an asset price, following an Ito semimartingale, implied by short-maturity options. The asymptotic setup is one in which the time to maturity of the available options decreases, the mesh of the available strike grid shrinks and the strike range expands. The estimation is based on aggregating the observed option data into nonparametric estimates of the conditional characteristic function of the return distribution, the derivatives of which allow to infer the Fourier transform of a known transform of the Levy density in a way which is robust to the level of the unknown diffusive volatility of the asset price. The Levy density estimate is then constructed via Fourier inversion. We derive an asymptotic bound for the integrated squared error of the estimator in the general case as well as its probability limit in the special Levy case. We further show rate optimality of our Levy density estimator in a minimax sense. An empirical application to market index options reveals relative stability of the left tail decay during high and low volatility periods.

Sensitivity analysis of the utility maximisation problem with respect to model perturbations

Abstract: We consider the expected utility maximisation problem and its response to small changes in the market price of risk in a continuous semimartingale setting. Assuming that the preferences of a rational economic agent are modelled by a general utility function, we obtain a second-order expansion of the value function, a first-order approximation of the terminal wealth, and we construct trading strategies that match the indirect utility function up to the second order. The method, which is presented in an abstract version, relies on a simultaneous expansion with respect to both the state variable and the parameter, and convex duality in the direction of the state variable only (as there is no convexity with respect to the parameter). If a risk-tolerance wealth process exists, using it as numeraire and under an appropriate change of measure, we reduce the approximation problem to a Kunita–Watanabe decomposition.

Affine processes beyond stochastic continuity

Abstract: In this paper, we study time-inhomogeneous affine processes beyond the common assumption of stochastic continuity. In this setting, times of jumps can be both inaccessible and predictable. To this end, we develop a general theory of finite dimensional affine semimartingales under very weak assumptions. We show that the corresponding semimartingale characteristics have affine form and that the conditional characteristic function can be represented with solutions to measure differential equations of Riccati type. We prove existence of affine Markov processes and affine semimartingales under mild conditions and elaborate on examples and applications including affine processes in discrete time.

Compactness criterion for semimartingale laws and semimartingale optimal transport

Abstract: We provide a compactness criterion for the set of laws $\mathfrak{P}^{ac}_{sem}(\Theta)$ on the Skorokhod space for which the canonical process $X$ is a semimartingale having absolutely continuous characteristics with differential characteristics taking values in some given set $\Theta$ of Levy triplets. Whereas boundedness of $\Theta$ implies tightness of $\mathfrak{P}^{ac}_{sem}(\Theta)$, closedness fails in general, even when choosing $\Theta$ to be additionally closed and convex, as a sequence of purely discontinuous martingales may converge to a diffusion. To that end, we provide a necessary and sufficient condition that prevents the purely discontinuous martingale part in the canonical representation of $X$ to create a diffusion part in the limit. As a result, we obtain a sufficient criterion for $\mathfrak{P}^{ac}_{sem}(\Theta)$ to be compact, which turns out to be also a necessary one if the geometry of $\Theta$ is similar to a box on the product space. As an application, we consider a semimartingale optimal transport problem, where the transport plans are elements of $\mathfrak{P}^{ac}_{sem}(\Theta)$. We prove the existence of an optimal transport law $\widehat{\mathbb{P}}$ and obtain a duality result extending the classical Kantorovich duality to this setup.

A central limit theorem for the realised covariation of a bivariate Brownian semistationary process

Abstract: This article presents a weak law of large numbers and a central limit theorem for the scaled realised covariation of a bivariate Brownian semistationary process. The novelty of our results lies in the fact that we derive the suitable asymptotic theory both in a multivariate setting and outside the classical semimartingale framework. The proofs rely heavily on recent developments in Malliavin calculus.

Mixed-normal limit theorems for multiple Skorohod integrals in high-dimensions, with application to realized covariance

Abstract: This paper develops mixed-normal approximations for probabilities that vectors of multiple Skorohod integrals belong to random convex polytopes when the dimensions of the vectors possibly diverge to infinity. We apply the developed theory to establish the asymptotic mixed normality of the realized covariance matrix of a high-dimensional continuous semimartingale observed at a high-frequency, where the dimension can be much larger than the sample size. We also present an application of this result to testing the residual sparsity of a high-dimensional continuous-time factor model.

Option pricing under fast‐varying long‐memory stochastic volatility

Abstract: Recent empirical studies suggest that the volatility of an underlying price process may have correlations that decay slowly under certain market conditions. In this paper, the volatility is modeled as a stationary process with long‐range correlation properties in order to capture such a situation, and we consider European option pricing. This means that the volatility process is neither a Markov process nor a martingale. However, by exploiting the fact that the price process is still a semimartingale and accordingly using the martingale method, we can obtain an analytical expression for the option price in the regime where the volatility process is fast mean reverting. The volatility process is modeled as a smooth and bounded function of a fractional Ornstein–Uhlenbeck process. We give the expression for the implied volatility, which has a fractional term structure.

A local gaussian bootstrap method for realized volatility and realized beta

Abstract: This article introduces a local Gaussian bootstrap method useful for the estimation of the asymptotic distribution of high-frequency data-based statistics such as functions of realized multivariate volatility measures as well as their asymptotic variances. The new approach consists of dividing the original data into nonoverlapping blocks of M consecutive returns sampled at frequency h (where h−1 denotes the sample size) and then generating the bootstrap observations at each frequency within a block by drawing them randomly from a mean zero Gaussian distribution with a variance given by the realized variance computed over the corresponding block.Our main contributions are as follows. First, we show that the local Gaussian bootstrap is first-order consistent when used to estimate the distributions of realized volatility and realized betas under assumptions on the log-price process which follows a continuous Brownian semimartingale process. Second, we show that the local Gaussian bootstrap matches accurately the first four cumulants of realized volatility up to o(h), implying that this method provides third-order refinements. This is in contrast with the wild bootstrap of Goncalves and Meddahi (2009, Econometrica 77(1), 283–306), which is only second-order correct. Third, we show that the local Gaussian bootstrap is able to provide second-order refinements for the realized beta, which is also an improvement of the existing bootstrap results in Dovonon, Goncalves, and Meddahi (2013, Journal of Econometrics 172, 49–65) (where the pairs bootstrap was shown not to be second-order correct under general stochastic volatility). In addition, we highlight the connection between the local Gaussian bootstrap and the local Gaussianity approximation of continuous semimartingales established by Mykland and Zhang (2009, Econometrica 77, 1403–1455) and show the suitability of this bootstrap method to deal with the new class of estimators introduced in that article. Lastly, we provide Monte Carlo simulations and use empirical data to compare the finite sample accuracy of our new bootstrap confidence intervals for integrated volatility with the existing results.

Central limit theorems for discretized occupation time functionals

Abstract: The approximation of integral type functionals is studied for discrete observations of a continuous Ito semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved for $L^2$-Sobolev functions with fractional smoothness. An explicit $L^2$-lower bound shows that already lower order quadrature rules, such as the trapezoidal rule and the classical Riemann estimator, are rate optimal, but only the trapezoidal rule is efficient, achieving the minimal asymptotic variance.

Local times for continuous paths of arbitrary regularity.

Abstract: We study a continuous pathwise local time of order p for continuous functions with finite p-th variation along a sequence of time partitions, for even integers p >= 2. With this notion, we establish a Tanaka-type change of variable formula, as well as Tanaka-Meyer formulae. We also derive some identities involving this high-order pathwise local time, each of which generalizes a corresponding identity from semimartingale theory. We then use collision local times between multiple functions of arbitrary regularity, to study the dynamics of ranked continuous functions of arbitrary regularity. We present also another definition of pathwise local time which is more natural for fractional Brownian Motions, and give a connection with the previous notion of local time.

Probability-free models in option pricing: statistically indistinguishable dynamics and historical vs implied volatility

Abstract: We investigate whether it is possible to formulate option pricing and hedging models without using probability. We present a model that is consistent with two notions of volatility: a historical volatility consistent with statistical analysis, and an implied volatility consistent with options priced with the model. The latter will be also the quadratic variation of the model, a pathwise property. This first result, originally presented in Brigo and Mercurio (1998, 2000), is then connected with the recent work of Armstrong et al (2018, 2021), where using rough paths theory it is shown that implied volatility is associated with a purely pathwise lift of the stock dynamics involving no probability and no semimartingale theory in particular, leading to option models without probability. Finally, an intermediate result by Bender et al. (2008) is recalled. Using semimartingale theory, Bender et al. showed that one could obtain option prices based only on the semimartingale quadratic variation of the model, a pathwise property, and highlighted the difference between historical and implied volatility. All three works confirm the idea that while historical volatility is a statistical quantity, implied volatility is a pathwise one. This leads to a 20 years mini-anniversary of pathwise pricing through 1998, 2008 and 2018, which is rather fitting for a talk presented at the conference for the 45 years of the Black, Scholes and Merton option pricing paradigm.

Stochastic port--Hamiltonian systems

Abstract: In the present work we formally extend the theory of port--Hamiltonian systems to include random perturbations. In particular, suitably choosing the space of flows and effort variables we will show how several elements coming from possibly different physical domains can be interconnected in order to describe a dynamics perturbed by general semimartingale. In this sense the noise does not enter the system solely as an external random perturbation but each port is a semimartingale in itself. We will show how the present treatment, extend pseudo-Poisson an pre--symplectic geometric mechanics. At last, we will show that a power preserving interconnection of stochastic port--Hamiltonian system defines again a stochastic port--Hamiltonian system.

Semimartingale theory of monotone mean--variance portfolio allocation

Abstract: We study dynamic optimal portfolio allocation for monotone mean--variance preferences in a general semimartingale model. Armed with new results in this area we revisit the work of Cui, Li, Wang and Zhu (2012, MAFI) and fully characterize the circumstances under which one can set aside a non-negative cash flow while simultaneously improving the mean--variance efficiency of the left-over wealth. The paper analyzes, for the first time, the monotone hull of the Sharpe ratio and highlights its relevance to the problem at hand.

Option pricing by using a mixed fractional Brownian motion with jumps

Abstract: Option pricing is conventionally based on a Brownian motion (Bm). The Bm is a semimartingale process with stationary and independent increments. However, there are several stock returns that have a long memory or have high autocorrelation for long lags. A fractional Brownian motion (fBm) is one of the models that can solve this problem, but a model option with fBm is not arbitrage-free. A mixed fractional Brownian motion (mfBm) is a linear combination of a Bm and an independent fBm which can overcome the arbitrage problem. A jump process in time series is another problem found in stock price modeling. This paper deals with the problem of options pricing by using mfBm with jumps. Based on quasi-conditional expectation and Fourier transform method, we obtain a pricing formula for a stock option.

Comparison of path-dependent functionals of semimartingales

Abstract: Based on an extension of the martingale comparison method some comparison results for path-dependent functions of semimartingales are established. The proof makes essential use of the functional Ito calculus. A main tool is an extension of the Kolmogorov backwards equation to path-dependent functions. The paper also derives criteria for the regularity conditions of the comparison theorems and discusses applications as to the comparison of Asian options for semimartingale models.

Quadratic variation of a càdlàg semimartingale as a.s. limit of the normalized truncated variations

Abstract: For a real cadlag path x, we define sequence of semi-explicit quantities, which do not depend on any partitions and such that whenever x is a path of a cadlag semimartingale then these quantities t...

Stochastic integration with respect to arbitrary collections of continuous semimartingales and applications to Mathematical Finance

Abstract: Stochastic integrals are defined with respect to a collection $P = (P_i; \, i \in I)$ of continuous semimartingales, imposing no assumptions on the index set $I$ and the subspace of $\mathbb{R}^I$ where $P$ takes values. The integrals are constructed though finite-dimensional approximation, identifying the appropriate local geometry that allows extension to infinite dimensions. For local martingale integrators, the resulting space $\mathsf{S} (P)$ of stochastic integrals has an operational characterisation via a corresponding set of integrands $\mathsf{R} (C)$, constructed with only reference the covariation structure $C$ of $P$. This bijection between $\mathsf{R} (C)$ and the (closed in the semimartingale topology) set $\mathsf{S} (P)$ extends to families of continuous semimartingale integrators for which the drift process of $P$ belongs to $\mathsf{R} (C)$. In the context of infinite-asset models in Mathematical Finance, the latter structural condition is equivalent to a certain natural form of market viability. The enriched class of wealth processes via extended stochastic integrals leads to exact analogues of optional decomposition and hedging duality as the finite-asset case. A corresponding characterisation of market completeness in this setting is provided.

Semimartingales on Duals of Nuclear Spaces

Abstract: This work is devoted to the study of semimartingales on the dual of a general nuclear space. We start by establishing conditions for a cylindrical semimartingale in the strong dual $\Phi'$ of a nuclear space $\Phi$ to have a $\Phi'$-valued semimartingale version whose paths are right-continuous with left limits. Results of similar nature but for more specific classes of cylindrical semimartingales and examples are also provided. Later, we will show that under some general conditions every semimartingale taking values in the dual of a nuclear space has a canonical representation. The concept of predictable characteristics is introduced and is used to establish necessary and sufficient conditions for a $\Phi'$-valued semimartingale to be a $\Phi'$-valued Levy process.

Pure-jump semimartingales

Abstract: A new integral with respect to an integer-valued random measure is introduced. In contrast to the finite variation integral ubiquitous in semimartingale theory (Jacod and Shiryaev, 2003, II.1.5), the new integral is closed under stochastic integration, composition, and smooth transformations. The new integral gives rise to a previously unstudied class of pure-jump processes: the sigma-locally finite variation pure-jump processes. As an application, it is shown that every semimartingale $X$ has a unique decomposition $$X = X_0 + X^{\mathrm{qc}}+X^{\mathrm{dp}},$$ where $X^{\mathrm{qc}}$ is quasi-left-continuous and $X^{\mathrm{dp}}$ is a sigma-locally finite variation pure-jump process that jumps only at predictable times, both starting at zero. The decomposition mirrors the classical result for local martingales (Yoeurp, 1976, Theoreme~1.4) and gives a rigorous meaning to the notions of continuous-time and discrete-time components of a semimartingale. Against this backdrop, the paper investigates a wider class of processes that are equal to the sum of their jumps in the semimartingale topology and constructs a taxonomic hierarchy of pure-jump semimartingales.