State-dependent Jumps Research Articles

First-exit Problems for Integrated Diffusion Processes with State-dependent Jumps

First-exit Problems for Integrated Diffusion Processes with State-dependent Jumps

Approximating the First Passage Time Density of Diffusion Processes with State-Dependent Jumps

We study the problem of the first passage time through a constant boundary for a jump diffusion process whose infinitesimal generator is a nonlocal Jacobi operator. Due to the lack of analytical results, we address the problem using a discretization scheme for simulating the trajectories of jump diffusion processes with state-dependent jumps in both frequency and amplitude. We obtain numerical approximations on their first passage time probability density functions and results for the qualitative behavior of other statistics of this random variable. Finally, we provide two examples of application of the method for different choices of the distribution involved in the mechanism of generation of the jumps.

EXISTENCE AND UNIQUENESS OF DISCONTINUOUS PERIODIC ORBITS IN SECOND ORDER DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT IMPULSES

<abstract> In this paper, we are concerned with the existence and uniqueness of discontinuous periodic orbits for a class of second order impulsive differential equations with state-dependent jumps. we apply geometric method to estimate the time mapping of the equation, and then by using Poincaré-Bohl fixed point theorem to obtain some existence criteria under assumptions that the nonlinear term satisfies linear growth conditions. And, the uniqueness of the discontinuous periodic orbit is further proved. Finally, several specific impulsive functions are presented in examples to illustrate the obtained results. </abstract>

Unbiased Simulation Estimators for Multivariate Jump-Diffusions

We develop and analyze a class of unbiased Monte Carlo estimators for multivariate jump-diffusion processes with state-dependent drift, volatility, jump intensity and jump size. A change of measure argument is used to extend existing unbiased estimators for the inter-arrival diffusion to include state-dependent jumps. Under standard regularity conditions on the coefficient and target functions, we prove the unbiasedness and finite variance properties of the resulting jump-diffusion estimators. Numerical experiments illustrate the efficiency of our estimators.

First-passage problems for diffusion processes with state-dependent jumps

Let X(t) be a time-homogeneous jump-diffusion process. We assume that the jump size depends on the value of X(t). We obtain analytical results for the moments of T(x) and of where T(x) is the first time that the process leaves the interval (a, b). We also compute These results have applications in financial mathematics.

The ruin problem for a Wiener process with state-dependent jumps

Abstract Let X(t) be a jump-diffusion process whose continuous part is a Wiener process, and let T (x) be the first time it leaves the interval (0,b), where x = X(0). The jumps are negative and their sizes depend on the value of X(t). Moreover there can be a jump from X(t) to 0. We transform the integro-differential equation satisfied by the probability p(x) := P[X(T (x)) = 0] into an ordinary differential equation and we solve this equation explicitly in particular cases. We are also interested in the moment-generating function of T (x).

A recursive pricing method for autocallables under multivariate subordination

In this paper we develop a new class of models for pricing autocallables based on multivariate subordinate Orstein Uhlenbeck (OU) processes. Starting from d independent OU processes and an independent d-dimensional Levy subordinator, we construct a new process by time changing each of the OU processes with a coordinate of the Levy subordinator. The prices of underlying assets are then modeled as an exponential function of the subordinate processes. The new models introduce state-dependent jumps in the asset prices and the dependence among jumps is governed by the Levy measure of the d-dimensional subordinator. By employing the eigenfunction expansion technique, we are able to derive the analytical formulas for the worst-of autocallable prices. We also numerically implement a specific model and test its sensitivity to some of the key parameters of the model.

State-dependent jump processes: Itô-Stratonovich interpretations, potential, and transient solutions

The abrupt changes that are ubiquitous in physical and natural systems are often well characterized by shot noise with a state dependent recurrence frequency and jump amplitude. For such state dependent behavior, we derive the transition probability for both the It\^o and Stratonovich jump interpretations, and subsequently use the transition probability to pose a master equation for the jump process. For exponentially distributed inputs, we present a novel class of transient solutions, as well as a generic steady state solution in terms of a potential function and the Pope-Ching formula. These new results allow us to describe state dependent jumps in a double well potential for steady state particle dynamics, as well as transient salinity dynamics forced by state dependent jumps. Both examples showcase a stochastic description that is more general than the limiting case of Brownian motion to which the jump process defaults in the limit of infinitely frequent and small jumps. Accordingly, our analysis may be used to explore a continuum of stochastic behavior from infrequent, large jumps to frequent, small jumps approaching a diffusion process.

Pure jump models for pricing and hedging VIX derivatives

Recent non-parametric statistical analysis of high-frequency VIX data (Todorov and Tauchen, 2011) reveals that VIX dynamics is a pure jump semimartingale with infinite jump activity and infinite variation. To our best knowledge, existing models in the literature for pricing and hedging VIX derivatives do not have these features. This paper fills this gap by developing a novel class of parsimonious pure jump models with such features for VIX based on the additive time change technique proposed in Li et al. (2016a, 2016b). We time change the 3/2 diffusion by a class of additive subordinators with infinite activity, yielding pure jump Markov semimartingales with infinite activity and infinite variation. These processes have time and state dependent jumps that are mean reverting and are able to capture stylized features of VIX. Our models take the initial term structure of VIX futures as input and are analytically tractable for pricing VIX futures and European options via eigenfunction expansions. Through calibration exercises, we show that our model is able to achieve excellent fit for the VIX implied volatility surface which typically exhibits very steep skews. Comparison to two other models in terms of calibration reveals that our model performs better both in-sample and out-of-sample. We explain the ability of our model to fit the volatility surface by evaluating the matching of moments implied from market VIX option prices. To hedge VIX options, we develop a dynamic strategy which minimizes instantaneous jump risk at each rebalancing time while controlling transaction cost. Its effectiveness is demonstrated through a simulation study on hedging Bermudan style VIX options.

Model reduction for hybrid systems with state-dependent jumps

Abstract: In this paper we present a model reduction technique based on moment matching for a class of hybrid systems with state-dependent jumps. The problem of characterizing the steady-state for this class of systems is studied and a result which allows to describe the steady-state response of hybrid systems through the use of a hybrid mapping is given. A family of parameterized hybrid reduced order models which achieve moment matching. The special case of periodic input signals is analyzed and conditions for applying the technique are given for this class. A numerical simulation illustrates the results.

MULTIVARIATE SUBORDINATION OF MARKOV PROCESSES WITH FINANCIAL APPLICATIONS

This paper develops the procedure of multivariate subordination for a collection of independent Markov processes with killing. Starting from d independent Markov processes with killing and an independent d‐dimensional time change , we construct a new process by time, changing each of the Markov processes with a coordinate . When is a d‐dimensional Lévy subordinator, the time changed process is a time‐homogeneous Markov process with state‐dependent jumps and killing in the product of the state spaces of . The dependence among jumps of its components is governed by the d‐dimensional Lévy measure of the subordinator. When is a d‐dimensional additive subordinator, Y is a time‐inhomogeneous Markov process. When with forming a multivariate Markov process, is a Markov process, where each plays a role of stochastic volatility of . This construction provides a rich modeling architecture for building multivariate models in finance with time‐ and state‐dependent jumps, stochastic volatility, and killing (default). The semigroup theory provides powerful analytical and computational tools for securities pricing in this framework. To illustrate, the paper considers applications to multiname unified credit‐equity models and correlated commodity models.

Constructing Markov Processes with Dependent Jumps by Multivariate Subordination: Applications to Multi-Name Credit-Equity Modeling

This paper develops the procedure of multivariate subordination for a collection of independent Markov processes with killing. Starting from d independent Markov processes Xi with killing and an independent d-dimensional time change T, we construct a new process by time changing each of the Markov processes Xi with a coordinate Ti. When T is a d-dimensional Levy subordinator, the time changed process Yi:=Xi(Ti(t)) is a time-homogeneous Markov process with state-dependent jumps and killing in the product of the state spaces of Xi. The dependence among jumps of its components is governed by the d-dimensional Levy measure of the subordinator. When T is a d-dimensional additive subordinator, Y is a time-inhomogeneous Markov process. When Ti= ∫0tVsi ds with Vi forming a multi-variate Markov process, (Yi,Vi) is a Markov process, where each Vi plays a role of stochastic volatility of Yi. This construction provides a rich modeling architecture for building multivariate models in finance with time-dependent and state-dependent jumps, stochastic volatility, and killing (default). The semigroup theory provides powerful analytical and computational tools for securities pricing in this framework. To illustrate, the paper considers applications to multi-name unified credit-equity models and correlated commodity models.

Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models

Let denote the implied volatility at maturity t for a strike , where and is the current value of the underlying. We show that has a uniform (in x) limit as maturity t tends to infinity, given by the formula , for x in some compact neighbourhood of zero in the class of affine stochastic volatility models. Function is the convex dual of the limiting cumulant-generating function h of the scaled log-spot process. We express h in terms of the functional characteristics of the underlying model. The proof of the limiting formula rests on the large deviation behaviour of the scaled log-spot process as time tends to infinity. We apply our results to obtain the limiting smile for several classes of stochastic volatility models with jumps used in applications (e.g. Heston with state-independent jumps, Bates with state-dependent jumps and Barndorff-Nielsen–Shephard model).

Large Deviations and Stochastic Volatility with Jumps: Asymptotic Implied Volatility for Affine Models

We show that the implied volatility has a uniform (in log moneyness x) limit as the maturity tends to infinity, given by an explicit closed-form formula, for x in some compact neighborhood of zero in the class of affine stochastic volatility models. This expression is function of the convex dual of the limiting cumulant generating function h of the scaled log-spot process. We express h in terms of the functional characteristics of the underlying model. The proof of the limiting formula rests on the large deviation behavior of the scaled log-spot process as time tends to infinity. We apply our results to obtain the limiting smile for several classes of stochastic volatility models with jumps used in applications (e.g. Heston with state-independent jumps, Bates with state-dependent jumps and Barndorff-Nielsen-Shephard model).

TIME-CHANGED MARKOV PROCESSES IN UNIFIED CREDIT-EQUITY MODELING

This paper develops a novel class of hybrid credit-equity models with state-dependent jumps, local-stochastic volatility, and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time-changed Markov diffusion process with state-dependent local volatility and killing rate (default intensity). When the time change is a Lévy subordinator, the stock price process exhibits jumps with state-dependent Lévy measure. When the time change is a time integral of an activity rate process, the stock price process has local-stochastic volatility and default intensity. When the time change process is a Lévy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state-dependent jumps, local-stochastic volatility, and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the time-changed process is expressed in closed form as a single integral in the complex plane. If the payoff is square integrable, the complex integral is further reduced to a spectral expansion. To illustrate our general framework, we time change the jump-to-default extended constant elasticity of variance model of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local-stochastic volatility, and default intensity. These models can be used to jointly price equity and credit derivatives.

State Dependent Jumps in Stock Market Returns

State Dependent Jumps in Stock Market Returns

Time Changed Markov Processes in Unified Credit-Equity Modeling

This paper develops a novel class of hybrid credit-equity models with state-dependent jumps, local-stochastic volatility, and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time-changed Markov diffusion process with state-dependent local volatility and killing rate (default intensity). When the time change is a Levy subordinator, the stock price process exhibits jumps with state-dependent Levy measure. When the time change is a time integral of an activity rate process, the stock price process has local-stochastic volatility and default intensity. When the time change process is a Levy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state-dependent jumps, local-stochastic volatility, and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the time-changed process is expressed in closed form as a single integral in the complex plane. If the payoff is square integrable, the complex integral is further reduced to a spectral expansion. To illustrate our general framework, we time change the jump-to-default extended constant elasticity of variance model of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local-stochastic volatility, and default intensity. These models can be used to jointly price equity and credit derivatives.

On the Application of KAM Theory to Discontinuous Dynamical Systems

So far the application of Kolmogorov–Arnold–Moser (KAM) theory has been restricted to smooth dynamical systems. Since there are many situations which can be modeled only by differential equations containing discontinuous terms such as state-dependent jumps (e.g., in control theory or nonlinear oscillators), it is shown by a series of transformations how KAM theory can be used to analyze the dynamical behaviour of such discontinuous systems as well. The analysis is carried out for the examplex+x+asgn(x)=p(t)withp∈C6being periodic. It is known that all solutions are unbounded for smalla>0. We prove that all solutions are bounded fora>0 sufficiently large, and that there are infinitely many periodic and quasiperiodic solutions in this case.

Pricing currency options in target zone regime: Some consequences of state-dependent realignment

In this paper the pricing of currency options in a target zone model is discussed. The movement of the currency index is a product of the movement within the target zone and jumps in the target zone itself in the form of re- or devaluations. The probability of a re- or devaluation is modelled as dependent on the position of the exchange rate within the target zone. The traditional assumption of no dependence between the position of the exchange rate within the target zone and the probability of a re- or devaluation is compared with a model allowing state-dependent jumps. It is shown that the assumption of state-independent jump probabilities may lead to highly unrealistic outcomes, outcomes that will not arise if the probability of re- or devaluations is allowed to depend on the location of the exchange rate within the target zone. A comparison is made with the prices produced by the widely used Garman and Kohlhagen [ Journal of International Money and Finance (1983), Vol. 2, pp. 231–237] option pricing model.