When Moving‐Average Models Meet High‐Frequency Data: Uniform Inference on Volatility

We conduct inference on volatility with noisy high‐frequency data. We assume the observed transaction price follows a continuous‐time Itô‐semimartingale, contaminated by a discrete‐time moving‐average noise process associated with the arrival of trades. We estimate volatility, defined as the quadratic variation of the semimartingale, by maximizing the likelihood of a misspecified moving‐average model, with its order selected based on an information criterion. Our inference is uniformly valid over a large class of noise processes whose magnitude and dependence structure vary with sample size. We show that the convergence rate of our estimator dominates n 1/4 as noise vanishes, and is determined by the selected order of noise dependence when noise is sufficiently small. Our implementation guarantees positive estimates in finite samples.

Estimating fast mean-reverting jumps in electricity market models

Based on empirical evidence of fast mean-reverting spikes, electricity spot prices are often modeled X + Zβ as the sum of a continuous Itô semimartingale X and a mean-reverting compound Poisson process Ztβ=∫0t ∫ℝxe−β(t−s)p̲(ds,dt) where p̲(ds,dt) is Poisson random measure with intensity λds ⊗dt. In a first part, we investigate the estimation of (λ, β) from discrete observations and establish asymptotic efficiency in various asymptotic settings. In a second part, we discuss the use of our inference results for correcting the value of forward contracts on electricity markets in presence of spikes. We implement our method on real data in the French, German and Australian market over 2015 and 2016 and show in particular the effect of spike modelling on the valuation of certain strip options. In particular, we show that some out-of-the-money options have a significant value if we incorporate spikes in our modelling, while having a value close to 0 otherwise.

The Euler Scheme for a Stochastic Differential Equation Driven by Pure Jump Semimartingales

In this paper we propose the asymptotic error distributions of the Euler scheme for a stochastic differential equation driven by Itô semimartingales. Jacod (2004) studied this problem for stochastic differential equations driven by pure jump Lévy processes and obtained quite sharp results. We extend his results to a more general pure jump Itô semimartingale.

The Euler Scheme for a Stochastic Differential Equation Driven by Pure Jump Semimartingales

In this paper we propose the asymptotic error distributions of the Euler scheme for a stochastic differential equation driven by Itô semimartingales. Jacod (2004) studied this problem for stochastic differential equations driven by pure jump Lévy processes and obtained quite sharp results. We extend his results to a more general pure jump Itô semimartingale.

Testing for Jumps

This chapter is devoted to the most basic question about jumps: are they present at all? As seen in Chapter 5, this question can be answered unambiguously when the full path of the underlying process X is observed over the time interval of interest [0, T]. However, we suppose that X is discretely observed along a regular scheme with lag Δ‎₀, so no jump can actually be exactly observed, since observing a large discrete increment may be suggestive that a jump took place, but provides no certitude. We wish to derive testing procedures which are at least consistent. This can only be done under some structural hypotheses on X, and the property of being an Itô semimartingale is a suitable one.

Is Brownian Motion Really Necessary?

The mathematical treatment of models relying on pure jump processes is quite different from the treatment of models where a Brownian motion is present. For instance, risk management procedures, derivative pricing and portfolio optimization are all significantly altered, so there is interest from the mathematical finance side in finding out which model is more likely to have generated the data. This chapter provides explicit testing procedures to decide whether the Brownian motion is necessary to model the observed path, or whether the process is entirely driven by its jumps. The structural assumption is the same as in the previous two chapters, with the underlying process X being a one-dimensional Itô semimartingale, since in the multi-dimensional case we can again perform the test on each component separately.