Continuous Semimartingales Research Articles

Robust replication in H-self-similar Gaussian market models under uncertainty

Abstract We consider the robust hedging problem in the framework of model uncertainty, where the log-returns of the stock price are Gaussian and H-self-similar with H∈(1/2,1). These assumptions lead to two natural but mutually exclusive hypotheses, both being self-contained to fix the probabilistic model for the stock price. Namely, the investor may assume that either the market is efficient, that is the stock price process is a continuous semimartingale, or that the centred log-returns have stationary distributions. We show that to be able to super-hedge a European contingent claim with a convex payoff robustly, the investor must assume that the markets are efficient. If it turns out that the stationarity hypothesis is true, then the investor can actually super-hedge the option and thereby receive some net profit.

On the combinatorics of iterated stochastic integrals

This paper derives several identities for the iterated integrals of a general semimartingale. They involve powers, brackets, exponential and the stochastic exponential. Their form and derivations are combinatorial. The formulae simplify for continuous or finite-variation semimartingales, especially for counting processes. The results are motivated by chaotic representation of martingales, and a simple such application is given.

On Covariation Estimation for Multivariate Continuous Itô Semimartingales with Noise in Non-Synchronous Observation Schemes

This paper presents a Hayashi-Yoshida type estimator for the covariation matrix of continuous Ito semimartingales observed with noise. The coordinates of the multivariate process are assumed to be observed at highly frequent nonsynchronous points. The estimator of the covariation matrix is designed via a certain combination of the local averages and the Hayashi-Yoshida estimator. Our method does not require any synchronization of the observation scheme (as e.g. previous tick method or refreshing time method) and it is robust to some dependence structure of the noise process. We show the associated central limit theorem for the proposed estimator and provide a feasible asymptotic result. Our proofs are based on a blocking technique and a stable convergence theorem for semimartingales. Finally, we show simulation results for the proposed estimator to illustrate its finite sample properties.

Nonsynchronous covariation process and limit theorems

An asymptotic distribution theory of the nonsynchronous covariation process for continuous semimartingales is presented. Two continuous semimartingales are sampled at stopping times in a nonsynchronous manner. Those sampling times possibly depend on the history of the stochastic processes and themselves. The nonsynchronous covariation process converges to the usual quadratic covariation of the semimartingales as the maximum size of the sampling intervals tends to zero. We deal with the case where the limiting variation process of the normalized approximation error is random and prove the convergence to mixed normality, or convergence to a conditional Gaussian martingale. A class of consistent estimators for the asymptotic variation process based on kernels is proposed, which will be useful for statistical applications to high-frequency data analysis in finance. As an illustrative example, a Poisson sampling scheme with random change point is discussed.

Backward stochastic PDEs related to the utility maximization problem

Abstract We study utility maximization problem for general utility functions using the dynamic programming approach. An incomplete financial market model is considered, where the dynamics of asset prices is described by an -valued continuous semimartingale. Under some regularity assumptions, we derive the backward stochastic partial differential equation related directly to the primal problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward stochastic differential equation. The cases of power, exponential and logarithmic utilities are considered as examples.

COMPUTATION OF VOLATILITY IN STOCHASTIC VOLATILITY MODELS WITH HIGH FREQUENCY DATA

We consider general stochastic volatility models driven by continuous Brownian semimartingales, we show that the volatility of the variance and the leverage component (covariance between the asset price and the variance) can be reconstructed pathwise by exploiting Fourier analysis from the observation of the asset price. Specifying parametrically the asset price model we show that the method allows us to compute the parameters of the model. We provide a Monte Carlo experiment to recover the volatility and correlation parameters of the Heston model.

Variance and Volatility Swaps: Bubbles and Fundamental Prices

The martingale theory of price bubbles defines an asset bubble to exist when the asset’s price process is a strict local martingale, that is, a local martingale that is not a martingale. Using this definition of a price bubble, for continuous semimartingales, we characterize the conditions under which variance and volatility swaps inherit the underlying asset’s price bubble. We show that the existence of variance swap market prices implies that there are no swap price bubbles. Furthermore, we also show that under some mild additional assumptions the discretely sampled variance swap’s market price can be well approximated by the expectation of the asset’s quadratic variation under the local martingale measure, providing a theoretical justification of a standard market practice.

Decomposition of Order Statistics of Semimartingales Using Local Times

Given a finite collection of continuous semimartingales, a semimartingale decomposition of the corresponding ranked (order-statistics) processes was derived recently in [1]. In this paper, we obtain a more general result for semimartingales (not necessarily continuous) using a simpler approach. Furthermore, we also give a generalization of Ouknine [7, 8] and Yan's [11] formula for local times of ranked processes.

Realized volatility with stochastic sampling

A central limit theorem for the realized volatility of a one-dimensional continuous semimartingale based on a general stochastic sampling scheme is proved. The asymptotic distribution depends on the sampling scheme, which is written explicitly in terms of the asymptotic skewness and kurtosis of returns. Conditions for the central limit theorem to hold are examined for several concrete examples of schemes. Lower bounds for mean squared error and for asymptotic conditional variance are given, which are attained by using a specific sampling scheme.

Mean Variance Hedging in a General Jump Model

We consider the mean-variance hedging of a contingent claim H when the discounted price process S is an -valued quasi-left continuous semimartingale with bounded jumps. We relate the variance-optimal martingale measure (VOMM) to a backward semimartingale equation (BSE) and show that the VOMM is equivalent to the original measure P if and only if the BSE has a solution. For a general contingent claim, we derive an explicit solution of the optimal strategy and the optimal cost of the mean-variance hedging by means of another BSE and an appropriate predictable process δ

Local Time Rough Path for Lévy Processes

In this paper, we will prove that the local time of a Levy process is a rough path of roughness $p$ a.s. for any $2 < p < 3$ under some condition for the Levy measure. This is a new class of rough path processes. Then for any function $g$ of finite $q$-variation ($1\leq q <3$), we establish the integral $\int _{-\infty}^{\infty}g(x)dL_t^x$ as a Young integral when $1\leq q<2$ and a Lyons' rough path integral when $2\leq q<3$. We therefore apply these path integrals to extend the Tanaka-Meyer formula for a continuous function $f$ if $f^\prime_{-}$ exists and is of finite $q$-variation when $1\leq q<3$, for both continuous semi-martingales and a class of Levy processes.

On the Microstructural Hedging Error

We consider the issue of hedging a European derivative security in the presence of microstructure noise. In a market where the efficient price of the asset is driven by a stochastic volatility process, we assume an agent wants to use a (possibly misspecified) local volatility-type replication strategy. Focusing on microstructure noise effects, our goal is to evaluate the error between the theoretical, but practically unfeasible, strategy and its market adapted versions. The microstructural hedging error is in particular due to transaction price discreteness and endogenous trading times. Thus, we consider a transaction price model that accommodates such inherent properties of ultrahigh frequency data with the assumption of a continuous semimartingale efficient price. In this framework, we study two hedging strategies derived from the local volatility-type hedging strategy: (i) the hedging portfolio is rebalanced every time that the transaction price moves; (ii) the hedging portfolio is rebalanced only once the transaction price has varied by more than a selected value. To assess these strategies, we use an asymptotic approach where the number of rebalancing transactions goes to infinity. For the first strategy, we show that, because of microstructure noise effects, the hedging error does not vanish. However, an optimal strategy of the second type enables us to reduce it significantly.

On Brownian motion as a prior for nonparametric regression

Abstract In this paper we consider the use of Brownian motion as a prior in a nonparametric, univariate regression setting. Using change of measure theory for continuous semimartingales we derive an explicit stochastic differential equation characterization for the posterior. In combination with stochastic calculus tools this dynamical characterization of the posterior allows us to derive new asymptotic properties of the posterior.

Exponential utility maximization under partial information

We consider the exponential utility maximization problem under partial information. The underlying asset price process follows a continuous semimartingale and strategies have to be constructed when only part of the information in the market is available. We show that this problem is equivalent to a new exponential optimization problem which is formulated in terms of observable processes. We prove that the value process of the reduced problem is the unique solution of a backward stochastic differential equation (BSDE) which characterizes the optimal strategy. We examine two particular cases of diffusion market models for which an explicit solution has been provided. Finally, we study the issue of sufficiency of partial information.

Hedging variance options on continuous semimartingales

We find robust model-free hedges and price bounds for options on the realized variance of [the returns on] an underlying price process. Assuming only that the underlying process is a positive continuous semimartingale, we superreplicate and subreplicate variance options and forward-starting variance options, by dynamically trading the underlying asset and statically holding European options. We thereby derive upper and lower bounds on values of variance options, in terms of Europeans.

Mean-Variance Hedging in Large Financial Markets

We consider a mean-variance hedging (MVH) problem for an arbitrage-free large financial market, that is, a financial market with countably many risky assets modelled by a sequence of continuous semimartingales. By using the stochastic integration theory for a sequence of semimartingales developed in De Donno and Pratelli [6], we extend the results about change of numéraire and MVH of Gourieroux et al. [12] to this setting. We also consider, for all n ≥ 1, the market formed by the first n risky assets and study the solutions to the corresponding n-dimensional MVH problem and their behaviour when n tends to infinity.

Illiquid financial market models and absence of arbitrage

We study financial market models with different liquidity effects. In the first part of this paper, we extend the short-term price impact model introduced by Rogers and Singh (2007) to a general semimartingale setup. We show the convergence of the discrete-time into the continuous-time modeling framework when trading times approach each other. In the second part, arbitrage opportunities in illiquid economies are considered, in particular a modification of the feedback effect model of Bank and Baum (2004). We demonstrate that a large trader cannot create wealth at no risk within this framework. Here we have to assume that the price process is described by a continuous semimartingale.

Limit theorems for bipower variation of semimartingales

This paper presents limit theorems for certain functionals of semimartingales observed at high frequency. In particular, we extend results from Jacod (2008) [5] 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.

A generalized occupation time formula for continuous semimartingales

We show that for a wide class of functions F we have lim \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim }\limits_{\varepsilon \downarrow 0} \frac{1}{\varepsilon }\int\limits_0^t {\{ F(s,X_s ) - F(s,X_s - \varepsilon )\} d\left\langle {X,X} \right\rangle _s = - } \int\limits_0^t {\int\limits_\mathbb{R} {F(s,x)dL_s^x } }$$ \end{document} where Xt is a continuous semimartingale, ( Ltx , x ∈ ℝ, t ≧ 0) its local time process and (〈 X, X 〉 t , t ≧ 0) its quadratic variation process.

A Fourier transform method for nonparametric estimation of multivariate volatility

We provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semi-martingales, which is based on Fourier analysis. The co-volatility is reconstructed as a stochastic function of time by establishing a connection between the Fourier transform of the prices process and the Fourier transform of the co-volatility process. A nonparametric estimator is derived given a discrete unevenly spaced and asynchronously sampled observations of the asset price processes. The asymptotic properties of the random estimator are studied: namely, consistency in probability uniformly in time and convergence in law to a mixture of Gaussian distributions.