Jump Size Distribution Research Articles

Confined sandpile in two dimensions: Percolation and singular diffusion.

We investigate the properties of a two-state sandpile model subjected to a confining potential in two dimensions. From the microdynamical description, we derive a diffusion equation, and find a stationary solution for the case of a parabolic confining potential. By studying the systems at different confining conditions, we observe two scale-invariant regimes. At a given confining potential strength, the cluster size distribution takes the form of a power law. This regime corresponds to the situation in which the density at the center of the system approaches the critical percolation threshold. The analysis of the fractal dimension of the largest cluster frontier provides evidence that this regime is reminiscent of gradient percolation. By increasing further the confining potential, most of the particles coalesce in a giant cluster, and we observe a regime where the jump size distribution takes the form of a power law. The onset of this second regime is signaled by a maximum in the fluctuation of energy.

Option Pricing for a Jump-Diffusion Model with General Discrete Jump-Size Distributions

We obtain a closed-form solution for pricing European options under a general jump-diffusion model that can incorporate arbitrary discrete jump-size distributions, including nonparametric distributions such as an empirical distribution. The flexibility in the jump-size distribution allows the model to better capture leptokurtic features found in real-world data. The model uses a discrete-time framework and leads to a pricing formula that is provably convergent to the continuous-time price as the discretization is increased. The solution is easy to implement with fast convergence properties. Numerical results illustrate the efficiency and accuracy of the proposed model and highlight its robustness and flexibility. This paper was accepted by Noah Gans, stochastic models and simulation.

Sample path large deviations for Lévy processes and random walks with Weibull increments

We study sample path large deviations for Lévy processes and random walks with heavy-tailed jump-size distributions that are of Weibull type. The sharpness and applicability of these results are illustrated by a counterexample proving the nonexistence of a full LDP in the $J_{1}$ topology, and by an application to a first passage problem.

Hedging With Credibility When Assets Can Jump

Hedging options in non-Gaussian models is a well-known and difficult task, yet remaining important for practitioners. Hence, solutions through the quadratic hedging methods have been suggested, see the recent contributions of Riesner (2006), Vandaele and Vanmaele (2008) and that in Cont and Tankov (2004). Although their suggested ratios to invest in the underlying asset for an optimal replication are different from each to other, they, however, share a common structure which makes their implementation non obvious. This structure originates in the integral part of the partial integro-differential equation and stems from the expectation of option prices taken over the Lévy measure. Although non-straightforward numerical integrations can be used to implement this quantity, they have to be modified and adapted to suit the choice of the random jump size distributions, resulting in a cumbersome task. Hence, implementation efficiency has still to be addressed. Using a locally risk-minimizing hedging strategy together with an elegant result of Hille and Phillips (1957), this paper shows how to efficiently compute the expectation of the option prices taken over the Lévy measure of any Lévy processes, be they of the finite or of the infinite activity. Hence, all the optimal ratios suggested by the aforementioned authors can be evaluated by adding a minor factor to a fast Fourier transform pricing formula and thereby gaining its computation efficiency.

A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps

After the recent financial crisis, the market for volatility derivatives has expanded rapidly to meet the demand from investors, risk managers and speculators seeking diversification of the volatility risk. In this paper, we develop a novel and efficient transform-based method to price swaps and options related to discretely-sampled realized variance under a general class of stochastic volatility models with jumps. We utilize frame duality and density projection method combined with a novel continuous-time Markov chain (CTMC) weak approximation scheme of the underlying variance process. Contracts considered include discrete variance swaps, discrete variance options, and discrete volatility options. Models considered include several popular stochastic volatility models with a general jump size distribution: Heston, Scott, Hull–White, Stein–Stein, α-Hypergeometric, 3/2 and 4/2 models. Our framework encompasses and extends the current literature on discretely sampled volatility derivatives, and provides highly efficient and accurate valuation methods. Numerical experiments confirm our findings.

Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps

In this paper, we develop a novel and efficient transform-based method to price equity-linked annuities (ELAs), including equity-indexed annuities (EIAs) and cliquet-style payoff structures popular in the insurance market under a general class of stochastic volatility models with jumps. We utilize frame duality and density projection combined with a continuous-time Markov chain (CTMC) weak approximation scheme and spectral filtering. Contracts considered include EIAs with return guarantees of a cliquet style. Models considered include exponential Lévy processes, regime-switching Lévy processes, and stochastic volatility models with a general jump size distribution including Heston, Scott’s, Hull–White, Schöbel–Zhu, and the 3/2 models. We also consider some recently proposed stochastic volatility models in the literature such as the α-Hypergeometric model, and the 4/2 model. Our framework encompasses and extends the current literature on EIAs with highly efficient and accurate valuation methods. Numerical experiments confirm our findings.

The Jump Size Distribution of the Commodity Spot Price and Its Effect on Futures and Option Prices

In this paper, we analyze the role of the jump size distribution in the US natural gas prices when valuing natural gas futures traded at New York Mercantile Exchange (NYMEX) and we observe that a jump-diffusion model always provides lower errors than a diffusion model. Moreover, we also show that although the Normal distribution offers lower errors for short maturities, the Exponential distribution is quite accurate for long maturities. We also price natural gas options and we see that, in general, the model with the Normal jump size distribution underprices these options with respect to the Exponential distribution. Finally, we obtain the futures risk premia in both cases and we observe that for long maturities the term structure of the risk premia is negative. Moreover, the Exponential distribution provides the highest premia in absolute value.

Tail Risk and Robust Portfolio Decisions

This paper formulates a portfolio choice problem in a multiasset incomplete market characterized by ambiguous jumps and arbitrary tail assumptions. We derive the optimal portfolio in closed form through a decomposition approach. We show that, due to fear of tail incidents, an investor diminishes portfolio diversification, and even more so under heavy-tailed jumps that intensify misspecification concerns. We then implement our model in international equity markets to quantify the impact of tail risk on portfolio selection, through comparisons between a normal and a slowly decaying jump size distribution. We find that, without jump ambiguity, constant relative risk aversion (CRRA) investors increase their jump exposures merely slightly and suffer negligible wealth losses from underestimating tail risk, given that the first two moments of the jump size distributions are preserved regardless of the tail properties. In stark contrast, sizable portfolio rebalancing and subsequent wealth losses are encountered in the presence of jump ambiguity. This paper was accepted by David Simchi-Levi, finance.

Option Pricing Based on Alternative Jump Size Distributions

It is well known that volatility smirks and heavy-tailed asset return distributions are two violations of the Black-Scholes model. This paper investigates the role of jump size distribution played in explaining these two abnormalities. We consider a jump-diffusion model with Laplace jump size distribution, in comparison to the conventional normal distribution. In addition, our analysis is built upon a pure exchange economy, in which the representative agenti¯s risk preference shows a fanning characteristic. We find that, when a fanning effect is present, Laplace model produces a more remarkable leptokurtic pattern of the risk-neutral distribution implied by options, as well as generating more pronounced volatility smirks than the normal model.

Estimating Jump-Diffusions Using Closed-Form Likelihood Expansions -- Online Supplementary Material

This supplementary material for Estimating Jump-Diffusions Using Closed-form Likelihood Expansions contains (1) closed-form expansion formulas for the examples (Section 1), (2) proofs of the results in the appendices (Section 2), (3) expectation (D.4) under correlated normal distribution of jump size (Section 3), (4) expectation (D.4) under the jump-size distribution of the CEV-SVCJ model (Section 4), (5) validity of the density expansion (Section 5), (6) approximate maximum-likelihood estimation via the density expansion (Section 6).

Optimal Exercise Strategies for Operational Risk Insurance via Multiple Stopping Times

In this paper we demonstrate how to develop analytic closed form solutions to optimal multiple stopping time problems arising in the setting in which the value function acts on a compound process that is modified by the actions taken at the stopping times. This class of problem is particularly relevant in insurance and risk management settings and we demonstrate this on an important application domain based on insurance strategies in Operational Risk management for financial institutions. In this area of risk management the most prevalent class of loss process models is the Loss Distribution Approach (LDA) framework which involves modelling annual losses via a compound process. Given an LDA model framework, we consider Operational Risk insurance products that mitigate the risk for such loss processes and may reduce capital requirements. In particular, we consider insurance products that grant the policy holder the right to insure k of its annual Operational losses in a horizon of T years. We consider two insurance product structures and two general model settings, the first are families of relevant LDA loss models that we can obtain closed form optimal stopping rules for under each generic insurance mitigation structure and then secondly classes of LDA models for which we can develop closed form approximations of the optimal stopping rules. In particular, for losses following a compound Poisson process with jump size given by an Inverse-Gaussian distribution and two generic types of insurance mitigation, we are able to derive analytic expressions for the loss process modified by the insurance application, as well as closed form solutions for the optimal multiple stopping rules in discrete time (annually). When the combination of insurance mitigation and jump size distribution does not lead to tractable stopping rules we develop a principled class of closed form approximations to the optimal decision rule. These approximations are developed based on a class of orthogonal Askey polynomial series basis expansion representations of the annual loss compound process distribution and functions of this annual loss.

Pricing turbo warrants under mixed-exponential jump diffusion model

Turbo warrant is a special type of barrier options in which the rebate is calculated as another exotic option. In this paper, using Laplace transforms we obtain the valuation of turbo warrant under the mixed-exponential jump diffusion model, which is able to approximate any jump size distribution. The numerical Laplace inversion examples verify that the analytical solutions are accurate. The results of simulation confirm the argument that jump risk should not be ignored in the valuation of turbo warrants.

Disorder-induced power-law response of a superconducting vortex on a plane

We report drive-response experiments on individual superconducting vortices on a plane, a realization for a 1+1-dimensional directed polymer in random media. For this we use magnetic force microscopy (MFM) to image and manipulate individual vortices trapped on a twin boundary in YBCO near optimal doping. We find that when we drag a vortex with the magnetic tip it moves in a series of jumps. As theory suggests the jump-size distribution does not depend on the applied force and is consistent with power-law behavior. The measured power is much larger than widely accepted theoretical calculations.

Real-World Forward Rate Dynamics With Affine Realizations

We investigate the existence of affine realizations for Lévy driven interest rate term structure models under the real-world probability measure, which so far has only been studied under an assumed risk-neutral probability measure. For models driven by Wiener processes, all results obtained under the risk-neutral approach concerning the existence of affine realizations are transferred to the general case. A similar result holds true for models driven by compound Poisson processes with finite jump size distributions. However, in the presence of jumps with infinite activity we obtain severe restrictions on the structure of the market price of risk; typically, it must even be constant.

Filtering and Likelihood Estimation of Latent Factor Jump-Diffusions with an Application to Stochastic Volatility Models

In this article we use a partial integral-differential approach to construct and extend a non-linear filter to include jump components in the system state. We employ the enhanced filter to estimate the latent state of multivariate parametric jump-diffusions. The devised procedure is flexible and can be applied to non-affine diffusions as well as to state dependent jump intensities and jump size distributions. The particular design of the system state can also provide an estimate of the jump times and sizes. With the same approch by which the filter has been devised, we implement an approximate likelihood for the parameter estimation of models of the jump-diffusion class. In the development of the estimation function, we take particular care in designing a simplified algorithm for computing. The likelihood function is then characterised in the application to stochastic volatility models with jumps. In the empirical section we validate the proposed approach via Monte Carlo experiments. We deal with the volatility as an intrinsic latent factor, which is partially observable through the integrated variance, a new system state component that is introduced to increase the filtered information content, allowing a closer tracking of the latent volatility factor. Further, we analyse the structure of the measurement error, particularly in relation to the presence of jumps in the system. In connection to this, we detect and address an issue arising in the update equation, improving the system state estimate.

Hedging with Credibility When Assets Can Jump

Hedging options in non-Gaussian models is a well-known and difficult task, yet remaining important for risk practitioners from banks to insurance companies. Hence, solutions through the quadratic hedging methods have been recently suggested, see Cont and Tankov (2004), Riesner (2006) and Vandaele and Vanmaele (2008). Although their suggested ratios to invest in the underlying asset for an optimal replication are different from each to other, they, however, share a common structure which makes their implementation non obvious. This structure originates in the integral part of the partial integro-differential equation and stems from the expectation of option prices taken over the random jump sizes. Although non-straightforward numerical integrations can be used to implement this quantity, they have to be modified and adapted to suit the choice of the random jump size distributions, resulting in a cumbersome task. Hence, implementation efficiency has still to be addressed. Using a locally risk-minimizing hedging strategy together with an elegant result of Hille and Phillips (1957), this paper shows how to efficiently compute the expectation of the option prices taken over the random jump sizes of any Levy processes, be they of the finite or of the infinite activity. Hence, all the optimal ratios suggested by the aforementioned authors can be evaluated by adding a minor factor to a fast Fourier transform pricing formula and thereby gaining its computation efficiency.

Martingale Approach to Optimal Portfolio-Consumption Problems in Markov-Modulated Pure-Jump Models

We study optimal investment strategies that maximize expected utility from consumption and terminal wealth in a pure-jump asset price model with Markov-modulated (regime switching) jump-size distributions. We give sufficient conditions for existence of optimal policies and find closed-form expressions for the optimal value function for agents with logarithmic and fractional power (CRRA) utility in the case of two-state Markov chains. The main tools are convex duality techniques, stochastic calculus for pure-jump processes, and explicit formulae for the moments of telegraph processes with Markov-modulated random jumps.

The Role of the Risk-Neutral Jump Size Distribution in Single-Factor Interest Rate Models

We obtain a result that relates the risk-neutral jump size of interest rates with yield curve data. This function is unobservable; therefore, this result opens a way to estimate the jump size directly from data in the markets together with the risk-neutral drift and jump intensity estimations. Then, we investigate the finite sample performance of this approach with a test problem. Moreover, we analyze the effect of estimating the risk-neutral jump size instead of assuming that it is artificially absorbed by the jump intensity, as usual in the interest rate literature. Finally, an application to US Treasury Bill data is also illustrated.

Tail Asymptotics of the Stationary Distribution of a Two-Dimensional Reflecting Random Walk with Unbounded Upward Jumps

We consider a two-dimensional reflecting random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary of the quadrant, but may have unbounded jumps in the opposite direction, which are referred to as upward jumps. We are interested in the tail asymptotic behavior of its stationary distribution, provided it exists. Assuming that the upward jump size distributions have light tails, we find the rough tail asymptotics of the marginal stationary distributions in all directions. This generalizes the corresponding results for the skip-free reflecting random walk in Miyazawa (2009). We exemplify these results for a two-node queueing network with exogenous batch arrivals.

Tail Asymptotics of the Stationary Distribution of a Two-Dimensional Reflecting Random Walk with Unbounded Upward Jumps

We consider a two-dimensional reflecting random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary of the quadrant, but may have unbounded jumps in the opposite direction, which are referred to as upward jumps. We are interested in the tail asymptotic behavior of its stationary distribution, provided it exists. Assuming that the upward jump size distributions have light tails, we find the rough tail asymptotics of the marginal stationary distributions in all directions. This generalizes the corresponding results for the skip-free reflecting random walk in Miyazawa (2009). We exemplify these results for a two-node queueing network with exogenous batch arrivals.