Financial econometrics

Abstract

We consider high-frequency observations from a one-dimensional time-homogeneous diffusion process Y. We assume that the diffusion coefficient  is continuously differentiable in y, but with a jump discontinuity at some level y, say . We first study sign-constrained kernel estimators of functions of the left and right limits of  at . These functions intricately depend on both limits. We propose a method to extricate these functions by searching for bandwidths where the kernel estimators are stable by iteration. We finally provide an estimator of the discontinuity jump size. We prove its convergence in probability and discuss its rate of convergence. A Monte Carlo study shows the finite sample properties of this estimator.

Abstract

We consider high-frequency observations from a one-dimensional time-homogeneous diffusion process Y. We assume that the diffusion coefficient  is continuously differentiable in y, but with a jump discontinuity at some level y, say . We first study sign-constrained kernel estimators of functions of the left and right limits of  at . These functions intricately depend on both limits. We propose a method to extricate these functions by searching for bandwidths where the kernel estimators are stable by iteration. We finally provide an estimator of the discontinuity jump size. We prove its convergence in probability and discuss its rate of convergence. A Monte Carlo study shows the finite sample properties of this estimator.

Abstract

At the ultra high frequency level, the notion of price of an asset is very ambiguous. Indeed, many different prices can be defined (last traded price, best bid price, mid price, etc.). Thus, in practice, market participants face the problem of choosing a price when implementing their strategies. In this work, we propose a notion of efficient price which seems relevant in practice. Furthermore, we provide a statistical methodology enabling to estimate this price from the order flow.

Abstract

We consider high frequency observations of a semi-martingale. From these data, we build simple test statistics allowing to distinguish between the two following situations: i) the data generating process is an Itō semi-martingale; ii) the data generating process is a Multifractal Random Walk. We also investigate the finite sample behavior of the test statistics on some simulated data.

Abstract

We consider two continuous-time Gaussian processes, one being partially correlated to a time-lagged version of the other. We first give the limiting spectral distribution for the covariance matrices of the increments of the processes when the span between two observations tends to zero. Then, we derive the limiting distribution of the eigenvalues of the sample covariance matrices. This result is obtained when the number of paths of the processes is asymptotically proportional to the number of observations for each single path. As an application, we use the second moment of this distribution together with auxiliary volatility and correlation estimates to construct an adaptive estimator of the time lag between the two processes. Finally, we provide an asymptotic theory for our estimation procedure.

Abstract

In this paper, we provide a model which accommodates the assumption of a continuous efficient price with the inherent properties of ultra-high-frequency transaction data (price discreteness, irregular temporal spacing, diurnal patterns...). Our approach consists in designing a stochastic mechanism for deriving the transaction prices from the latent efficient price. The main idea behind the model is that, if a transaction occurs at some value on the tick grid and leads to a price change, then the efficient price has been close enough to this value shortly before the transaction. We call uncertainty zones the bands around the mid-tick grid where the efficient price is too far from the tick grid to trigger a price change. In our setting, the width of these uncertainty zones quantifies the aversion to price changes of the market participants. Furthermore, this model enables us to derive approximated values of the efficient price at some random times, which is particularly useful for building statistical procedures. Convincing results are obtained through a simulation study and the use of the model over 10 representative stocks.

Abstract

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.

Abstract

Using a market-making inventory model, we analyze the impact of order preferencing on dealers’ quoting behavior by changing the degree of quote disclosure. We find that preferenced orders raise the inventory-holding costs of preferenced dealers, making them less able to post attractive quotes. In turn, competitors choose less aggressive prices, but still attract more likely public orders. Price competition is smoothed and expected market spreads widen. Promoting competition might be, however, enforced by (i) fine tuning through the degree of market transparency, (ii) favoring the entry of unpreferenced dealers, or (iii) requiring preferenced market-makers to have more funding capital.

Abstract

This paper considers practically appealing procedures for estimating intraday volatility measures of financial assets. The underlying microstructure model accommodates the inherent properties of ultra high-frequency data with the assumption of continuous efficient price processes. In this model, microstructure noise and trading times are endogenous but do not only depend on the prices. Using the (observed) last traded prices of the assets, we develop a new approach that enables to approximate the values of the efficient prices at some random times. Based on these approximated values, we build an estimator of the integrated volatility and give its asymptotic theory. We also give a consistent estimator of the integrated covariation when two assets (asynchronous by construction of the model) are observed.

Abstract

This paper develops a dynamic switching model, with a random walk anda stationary regime, where the time spent in the random walk regimeis endogeneously predetermined. More precisely, we assume that theprocess is recursively defined byYt = μ +Yt−1 +εt, with stochastic probabilityπrw(Yt−1),Yt = μ +εt, with stochastic probability1 −πrw(Yt−1),where (εt) is a strong white noise andπrw is a nondecreasing function.Then, the dynamics of the process(Yt), itsmarginal distribution, and the distribution of the time spent in theunit root regime depend on the pattern of random walk intensityπrw and on the noisedistribution F. Moreover, we study the linksbetween the endogeneous switching regime and the degree ofpersistence of the process(Yt).