Compound Poisson Jumps Research Articles

Prediction of Gaussian Volterra processes with compound Poisson jumps

We consider a Gaussian Volterra process with compound Poisson jumps and derive its prediction law.

Affine term structure models: A time‐change approach with perfect fit to market curves

AbstractWe address the so‐called calibration problem, which consists of fitting in a tractable way a given model to a specified term structure such as yield, prepayment or default probability curves. Time‐homogeneous affine jump diffusions (HAJD) are tractable processes but have limited flexibility; they fail to perfectly replicate actual market curves. Applying a deterministic shift to the latter is a simple but efficient solution that is widely used by both academics and practitioners. However, the shift approach may not be appropriate when positivity is required, a common constraint when dealing with credit spreads or default intensities. In this paper, we address this problem by adopting a time‐change technique. Specific attention is paid to the Cox–Ingersoll–Ross model with compound Poisson jumps (JCIR), which remains standard for modeling intensities. Our time‐changed JCIR (TC‐JCIR) is compared to the shifted JCIR (JCIR++) in various credit applications such as credit default swap (CDS), credit default swaption, and credit valuation adjustment (CVA) under wrong‐way risk (WWR). The TC‐JCIR model is able to generate much larger implied volatilities and covariance effects than JCIR++ under positivity constraints and represents an appealing alternative to the latter.

Estimating diffusion with compound Poisson jumps based on self-normalized residuals

We consider parametric estimation of the continuous part of a class of ergodic diffusions with jumps based on high-frequency samples. Various papers previously proposed threshold based methods, which enable us to distinguish whether observed increments have jumps or not at each small-time interval, hence to estimate the unknown parameters separately. However, a data-adapted and quantitative choice of the threshold parameter is known to be a subtle and sensitive problem. In this paper, we present a simple alternative based on the Jarque–Bera normality test for the Euler residuals. Different from the threshold based method, the proposed method does not require any sensitive fine tuning, hence is of practical value. It is shown that under suitable conditions the proposed estimator is asymptotically equivalent to an estimator constructed by the unobserved fluctuation of the continuous part of the solution process, hence is asymptotically efficient. Some numerical experiments are conducted to observe finite-sample performance of the proposed method.

Ergodic control of diffusions with compound Poisson jumps under a general structural hypothesis

We study the ergodic control problem for a class of controlled jump diffusions driven by a compound Poisson process. This extends the results of Arapostathis et al. (2019) to running costs that are not near-monotone. This generality is needed in applications such as optimal scheduling of large-scale parallel server networks.We provide a full characterizations of optimality via the Hamilton–Jacobi–Bellman (HJB) equation, for which we additionally exhibit regularity of solutions under mild hypotheses. In addition, we show that optimal stationary Markov controls are a.s. pathwise optimal. Lastly, we show that one can fix a stable control outside a compact set and obtain near-optimal solutions by solving the HJB on a sufficiently large bounded domain. This is useful for constructing asymptotically optimal scheduling policies for multiclass parallel server networks.

Affine Term-Structure Models: A Time-Changed Approach with Perfect Fit to Market Curves

We address the so-called calibration problem which consists of fitting in a tractable way a given model to a specified term structure like, e.g., yield, prepayment or default probability curves. Time-homogeneous jump-diffusions like Vasicek or Cox-Ingersoll-Ross (possibly coupled with compound Poisson jumps, JCIR), are tractable processes but have limited flexibility; they fail to replicate actual market curves. The deterministic shift extension of the latter (Hull-White or JCIR++) is a simple but yet efficient solution that is widely used by both academics and practitioners. However, the shift approach is often not appropriate when positivity is required, which is a common constraint when dealing with credit spreads or default intensities. In this paper, we tackle this problem by adopting a time change approach. On the top of providing an elegant solution to the calibration problem under positivity constraint, our model features additional interesting properties in terms of implied volatilities. It is compared to the shift extension on various credit risk applications such as credit default swap, credit default swaption and credit valuation adjustment under wrong-way risk. The time change approach is able to generate much larger volatility and covariance effects under the positivity constraint. Our model offers an appealing alternative to the shift in such cases.

Option pricing in time-changed Lévy models with compound Poisson jumps

The problem of European-style option pricing in time-changed Lévy models in the presence of compound Poisson jumps is considered. These jumps relate to sudden large drops in stock prices induced by political or economical hits. As the time-changed Lévy models, the variance-gamma and the normal-inverse Gaussian models are discussed. Exact formulas are given for the price of digital asset-or-nothing call option on extra asset in foreign currency. The prices of simpler options can be derived as corollaries of our results and examples are presented. Various types of dependencies between stock prices are mentioned.

Cliquet Option Pricing in a Jump-Diffusion LLvy Model

We investigate the pricing of cliquet options in a jump-diffusion model. The considered option is of monthly sum cap style while the underlying stock price model is driven by a drifted Levy process entailing a Brownian diffusion component as well as compound Poisson jumps. We also derive representations for the density and distribution function of the emerging Levy process. In this setting, we infer semi-analytic expressions for the cliquet option price by two different approaches. The first one involves the probability distribution function of the driving Levy process whereas the second draws upon Fourier transform techniques. With view on sensitivity analysis and hedging purposes, we eventually deduce representations for several greeks while putting emphasis on the vega.

On magnitude, asymptotics and duration of drawdowns for Lévy models

This paper considers magnitude, asymptotics and duration of drawdowns for some L\'{e}vy processes. First, we revisit some existing results on the magnitude of drawdowns for spectrally negative L\'{e}vy processes using an approximation approach. For any spectrally negative L\'{e}vy process whose scale functions are well-behaved at $0+$, we then study the asymptotics of drawdown quantities when the threshold of drawdown magnitude approaches zero. We also show that such asymptotics is robust to perturbations of additional positive compound Poisson jumps. Finally, thanks to the asymptotic results and some recent works on the running maximum of L\'{e}vy processes, we derive the law of duration of drawdowns for a large class of L\'{e}vy processes (with a general spectrally negative part plus a positive compound Poisson structure). The duration of drawdowns is also known as the "Time to Recover" (TTR) the historical maximum, which is a widely used performance measure in the fund management industry. We find that the law of duration of drawdowns qualitatively depends on the path type of the spectrally negative component of the underlying L\'{e}vy process.

Option pricing for stochastic volatility model with infinite activity Lévy jumps

The purpose of this paper is to apply the stochastic volatility model driven by infinite activity Lévy processes to option pricing which displays infinite activity jumps behaviors and time varying volatility that is consistent with the phenomenon observed in underlying asset dynamics. We specially pay attention to three typical Lévy processes that replace the compound Poisson jumps in Bates model, aiming to capture the leptokurtic feature in asset returns and volatility clustering effect in returns variance. By utilizing the analytical characteristic function and fast Fourier transform technique, the closed form formula of option pricing can be derived. The intelligent global optimization search algorithm called Differential Evolution is introduced into the above highly dimensional models for parameters calibration so as to improve the calibration quality of fitted option models. Finally, we perform empirical researches using both time series data and options data on financial markets to illustrate the effectiveness and superiority of the proposed method.

Ambiguous Jump-Diffusions and Optimal Stopping

An ambiguity averse decision-maker contemplates investment of a fixed size capital into a project with a stochastic profit stream under the Knightian uncertainty. Multiple priors are modeled as a "cloud" of diffusion processes with embedded compound Poisson jumps. The "cloud" contains the Brownian motion (BM) as a process with zero density of jumps. The decision-maker has recursive multiple priors utility as in Epstein and Schneider (2003) and chooses the optimal investment timing. We demonstrate that if the expected present value (EPV) of the project is the same for each jump-diffusion prior at the moment of investment, then the BM is the worst prior in the waiting region. The same conclusion holds for some parameter values even when the BM gives the highest EPV of the project. For other parameter values, it is possible that the local dynamics of the worst case prior is given by a jump-diffusion in a vicinity of the investment threshold and by the BM in a vicinity of negative infinity. Explicit formulas for the value functions and investment thresholds are derived.

Quantile regression estimation for discretely observed SDE models with compound Poisson jumps

This paper proposes a quantile regression estimator for the diffusion parameter in diffusion processes with compound Poisson jumps. The method is based on discretely sampled observations at high frequency. We verify its consistency and exhibit its robustness to jumps through a simulation study.

MCMC ESTIMATION OF LÉVY JUMP MODELS USING STOCK AND OPTION PRICES

We examine the performances of several popular Levy jump models and some of the most sophisticated affine jump-diffusion models in capturing the joint dynamics of stock and option prices. We develop efficient Markov chain Monte Carlo methods for estimating parameters and latent volatility/jump variables of the Levy jump models using stock and option prices. We show that models with infinite-activity Levy jumps in returns significantly outperform affine jump-diffusion models with compound Poisson jumps in returns and volatility in capturing both the physical and risk-neutral dynamics of the S&P 500 index. We also find that the variance gamma model of Madan, Carr, and Chang with stochastic volatility has the best performance among all the models we consider.

Moderate deviations for estimators of quadratic variational process of diffusion with compound Poisson jumps

We consider the stochastic differential equation driven by Lévy processes. Using discrete observations, moderate deviations for the threshold estimator of the quadratic variational process are studied. Moreover, we also obtain the functional moderate deviations.

Approximate self-weighted LAD estimation of discretely observed ergodic Ornstein-Uhlenbeck processes

We consider drift estimation of a discretely observed Ornstein-Uhlenbeck process driven by a possibly heavy-tailed symmetric Lévy process with positive activity index β. Under an infill and large-time sampling design, we first establish an asymptotic normality of a self-weighted least absolute deviation estimator with the rate of convergence being $\sqrt{n}h_{n}^{1-1/\beta}$, where n denotes sample size and hn>0 the sampling mesh satisfying that hn→0 and nhn→∞. This implies that the rate of convergence is determined by the most active part of the driving Lévy process; the presence of a driving Wiener part leads to $\sqrt{nh_{n}}$, which is familiar in the context of asymptotically efficient estimation of diffusions with compound Poisson jumps, while a pure-jump driving Lévy process leads to a faster one. Also discussed is how to construct corresponding asymptotic confidence regions without full specification of the driving Lévy process. Second, by means of a polynomial type large deviation inequality we derive convergence of moments of our estimator under additional conditions.

Return Dynamics With Lévy Jumps: Evidence from Stock and Option Prices

We examine the performances of L evy jump models and a ne jump-di usion models in capturing the joint dynamics of stock and option prices. We discuss the change of measure for in nite-activity L evy jumps and develop e cient Markov chain Monte Carlo methods for estimating model parameters and latent volatility and jump variables using stock and option prices. Using daily returns and option prices of the SP E-mail: htli@umich.edu; phone: (734)-764-6409. 2Wells is from the Department of Biological Statistics and Computational Biology and the Department of Social Statistics, Cornell University, Ithaca, NY 14853; E-mail: mtw1@cornell.edu; phone: (607)-255-4388; he gratefully acknowledges the support of NSF Grant DMS 02-04252. 3Yu is from the Department of Statistics, Iowa State University, Ames, IA 50011; E-mail: cindyyu@iastate.edu; phone: (515)-294-3319. We would like to thank Torben Andersen and Luca Benzoni for their help in the early stage of this study. We thank Yacine A t-Sahalia (the editor), Antje Berndt, Peter Carr, Bjorn Eraker, Bob Jarrow, Nour Meddahi, Ray Renken, Sidney Resnick, Ernst Schaumburg, Neil Shephard, George Tauchen, Liuren Wu, an anonymous referee, and seminar participants at Cornell University, the 2004 CIREQ/CIRANO nancial econometrics conference, the 2004 International Chinese Statistical Association Applied Statistics Symposium, and the 2004 Institute of Mathematical Statistics Annual Meeting/6th Bernoulli World Congress for helpful comments. We are responsible for any remaining errors. Return Dynamics with L evy Jumps: Evidence from Stock and Option Prices ABSTRACT We examine the performances of L evy jump models and a ne jump-di usion models in capturing the joint dynamics of stock and option prices. We discuss the change of measure for in nite-activity L evy jumps and develop e cient Markov chain Monte Carlo methods for estimating model parameters and latent volatility and jump variables using stock and option prices. Using daily returns and option prices of the S&P 500 index, we show that models with in nite-activity L evy jumps in returns signi cantly outperform a ne jump-di usion models with compound Poisson jumps in returns and volatility in capturing both the physical and the risk-neutral dynamics of the S&P 500 index. Modeling the dynamics of stock returns is a key issue in modern asset pricing. A realistic model of return dynamics is essential for option pricing, portfolio analysis, and risk management. One of the most popular continuous-time models for return dynamics in the current literature is the a ne jump-di usion (hereafter AJD) models of Du e, Pan, and Singleton (2000) (hereafter DPS). In AJD models, stock returns are driven by a ne di usions and compound Poisson processes. AJD models capture important stylized behaviors of index returns and are highly tractable. They allow closed-form pricing formulae for a wide range of equity and xed-income derivatives. Despite the successes of AJD models, Brownian motion and compound Poisson process (the two main building blocks of AJD models) are only two special cases of L evy processes, which are continuous-time stochastic processes with stationary and independent increments. L evy processes are much more exible than Brownian motion and compound Poisson process for modeling purposes. For example, L evy processes allow non-normal increments (compared to normal increments of Brownian motion) and much richer jump structures than compound Poisson process. Moreover, Carr and Wu (2004) show that L evy processes are as tractable as AJD models for pricing purposes: Closed-form pricing formulae are available for a wide range of derivative securities under L evy processes. These appealing features of L evy processes have spurred a fast-growing literature that models return dynamics using L evy processes in recent years.1 This new development, however, has raised some challenging theoretical and empirical issues in the current literature. While existing studies of L evy processes have mainly focused on either the physical or the risk-neutral return dynamics, a key remaining question is whether L evy processes have signi cant empirical advantages over AJD models in modeling the joint return dynamics.2 This is an important question because the ultimate See Wu (2006) for an excellent review of the current literature on L evy processes. See A t-Sahalia (2004) and A t-Sahalia and Jacod (2004) on some fundamental issues on statistical inferences of L evy processes. Madan, Carr, and Chang (1998), Carr, Geman, Madan, and Yor (2002), Huang and Wu (2003), and Carr and Wu