José E. Figueroa-López

Risk bounds for the non-parametric estimation of Lévy processes

Estimation methods for the Levy density of a Levy process are developed under mild qualitative assumptions. A classical model selection ap- proach made up of two steps is studied. The first step consists in the selection of a good estimator, from an approximating (finite-dimensional) linear model S for the true Levy density. The second is a data-driven selection of a linear model S, among a given collection {Sm}m∈M, that approximately realizes the best trade-off between the error of estimation within S and the error incurred when approximating the true Levy density by the linear model S. Using recent concentration inequalities for functionals of Poisson integrals, a bound for the risk of estimation is obtained. As a byproduct, oracle inequalities and long- run asymptotics for spline estimators are derived. Even though the resulting underlying statistics are based on continuous time observations of the process, approximations based on high-frequency discrete-data can be easily devised.

Dynamic Portfolio Optimization with a Defaultable Security and Regime Switching

We consider a portfolio optimization problem in a defaultable market with finitely-many economical regimes, where the investor can dynamically allocate her wealth among a defaultable bond, a stock, and a money market account. The market coefficients are assumed to depend on the market regime in place, which is modeled by a finite state continuous time Markov process. We rigorously deduce the dynamics of the defaultable bond price process in terms of a Markov modulated stochastic differential equation. Then, by separating the utility maximization problem into the pre-default and post-default scenarios, we deduce two coupled Hamilton-Jacobi-Bellman equations for the post and pre-default optimal value functions and show a novel verification theorem for their solutions. We obtain explicit optimal investment strategies and value functions for an investor with logarithmic utility. We finish with an economic analysis in the case of a market with two regimes and homogenous transition rates, and show the impact of the default intensities and loss rates on the optimal strategies and value functions.

Nonparametric estimation of time-changed Lévy models under high-frequency data

Let {Z t } t≥0 be a Lévy process with Lévy measure ν, and let τ(t)=∫0 t r(u) d u, where {r(t)} t≥0 is a positive ergodic diffusion independent from Z. Based upon discrete observations of the time-changed Lévy process X t ≔Z τt during a time interval [0,T], we study the asymptotic properties of certain estimators of the parameters β(φ)≔∫φ(x)ν(d x), which in turn are well known to be the building blocks of several nonparametric methods such as sieve-based estimation and kernel estimation. Under uniform boundedness of the second moments of r and conditions on φ necessary for the standard short-term ergodic property lim t→ 0 E φ(Z t )/t = β(φ) to hold, consistency and asymptotic normality of the proposed estimators are ensured when the time horizon T increases in such a way that the sampling frequency is high enough relative to T.

Small-time expansions of the distributions, densities, and option prices of stochastic volatility models with Lévy jumps

We consider a stochastic volatility model with Levy jumps for a log-return process Z=(Zt)t≥0 of the form Z=U+X, where U=(Ut)t≥0 is a classical stochastic volatility process and X=(Xt)t≥0 is an independent Levy process with absolutely continuous Levy measure ν. Small-time expansions, of arbitrary polynomial order, in time-t, are obtained for the tails P(Zt≥z), z>0, and for the call-option prices E(ez+Zt−1)+, z≠0, assuming smoothness conditions on the density of ν away from the origin and a small-time large deviation principle on U. Our approach allows for a unified treatment of general payoff functions of the form φ(x)1x≥z for smooth functions φ and z>0. As a consequence of our tail expansions, the polynomial expansions in t of the transition densities ft are also obtained under mild conditions.

Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility

In Figueroa-López et al. (Math. Finance, 2013), a second order approximation for at-the-money option prices is derived for a large class of exponential Lévy models, with or without a Brownian component. The purpose of the present article is twofold. First, we relax the regularity conditions imposed on the Lévy density to the weakest possible conditions for such an expansion to be well defined. Second, we show that the formulas extend both to the case of “close-to-the-money” strikes and to the case where the continuous Brownian component is replaced by an independent stochastic volatility process with leverage.

Jump-Diffusion Models Driven by Lévy Processes

During the past and this decade, a new generation of continuous-time financial models has been intensively investigated in a quest to incorporate the so-called stylized empirical features of asset priceslike fat-tails, high kurtosis, volatility clustering, and leverage. Modeling driven by “memoryless homogeneous” jump processes (Levy processes) constitutes one of the most viable directions in this enterprise. The basic principle is to replace the underlying Brownian motion of the Black-Scholes model with a type of jump-diffusion process. In this chapter, the basic results and tools behind jump-diffusion models driven by Levy processes are covered, providing an accessible overview, coupled with their financial applications and relevance. The material is drawn upon recent monographs (cf. Cont and Tankov (2004). Financial modelling with Jump Processes. Chapman & Hall.; Sato (1999). Levy Processes and Infinitely Divisible Distributions. Cambridge University Press.) and papers in the field.

Pricing and Semimartingale Representations of Vulnerable Contingent Claims in Regime‐Switching Markets

Using a suitable change of probability measure, we obtain a novel Poisson series representation for the arbitrage- free price process of vulnerable contingent claims in a regime-switching market driven by an underlying continuous- time Markov process. As a result of this representation, along with a short-time asymptotic expansion of the claims price process, we develop an efficient method for pricing claims whose payoffs may depend on the full path of the underlying Markov chain. The proposed approach is applied to price not only simple European claims such as defaultable bonds, but also a new type of path-dependent claims that we term self-decomposable, as well as the important class of vulnerable call and put options on a stock. We provide a detailed error analysis and illustrate the accuracy and computational complexity of our method on several market traded instruments, such as defaultable bond prices, barrier options, and vulnerable call options. Using again our Poisson series representation, we show differentiability in time of the pre-default price function of European vulnerable claims, which enables us to rigorously deduce Feynman-Kac representations for the pre-default pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk-neutral and objective probability measures.

Estimation of NIG and VG Models for High Frequency Financial Data

Numerous empirical studies have shown that certain exponential Levy models are able to fit the empirical distribution of daily financial returns quite well. By contrast, very few papers have considered intraday data in spite of their growing importance. In this paper, we fill this gap by studying the ability of the Normal Inverse Gaussian (NIG) and the Variance Gamma (VG) models to fit the statistical features of intraday data at different sampling frequencies. We propose to assess the suitability of the model by analyzing the signature plots of the point estimates at different sampling frequencies. Using high frequency transaction data from the U.S. equity market, we find the estimator of the volatility parameter to be quite stable at a wide range of intraday frequencies, in sharp contrast to the estimator of the kurtosis parameter, which is more sensitive to market microstructure effects. As a secondary contribution, we also assess the performance of the two most favored parametric estimation methods, the Method of Moments Estimators (MME) and the Maximum Likelihood Estimators (MLE), when dealing with high frequency observations. By Monte Carlo simulations, we show that neither high frequency sampling nor maxi- mum likelihood estimation significantly reduces the estimation error of the volatility parameter of the model. On the contrary, the estimation error of the parameter controlling the kurtosis of log returns can be significantly reduced by using MLE and high-frequency sampling. Both of these results appear to be new in the literature on statistical analysis of high frequency data.

Short-Time Expansions for Call Options on Leveraged ETFs Under Exponential Lévy Models with Local Volatility

In this article, we consider the small-time asymptotics of options on a \emph{Leveraged Exchange-Traded Fund} (LETF) when the underlying Exchange Traded Fund (ETF) exhibits both local volatility and jumps of either finite or infinite activity. Our main results are closed-form expressions for the leading order terms of off-the-money European call and put LETF option prices, near expiration, with explicit error bounds. We show that the price of an out-of-the-money European call on a LETF with positive (negative) leverage is asymptotically equivalent, in short-time, to the price of an out-of-the-money European call (put) on the underlying ETF, but with modified spot and strike prices. Similar relationships hold for other off-the-money European options. In particular, our results suggest a method to hedge off-the-money LETF options near expiration using options on the underlying ETF. Finally, a second order expansion for the corresponding implied volatility is also derived and illustrated numerically.

Nonparametric Regression with Rescaled Time Series Errors

We consider a heteroscedastic nonparametric regression model with an autoregressive error process of finite known order p. The heteroscedasticity is incorporated using a scaling function defined at uniformly spaced design points on an interval [0,1]. We provide an innovative nonparametric estimator of the variance function and establish its consistency and asymptotic normality. We also propose a semiparametric estimator for the vector of autoregressive error process coefficients that is consistent and asymptotically normal for a sample size T. Explicit asymptotic variance covariance matrix is obtained as well. Finally, the finite sample performance of the proposed method is tested in simulations.