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In this paper, we theoretically and empirically study the intrahorizon value at risk (iVaR) in a general jump-diffusion setting. We propose a new class of models of asset returns, the displaced mixed exponential model, which can arbitrarily closely approximate finite and infinite activity Lévy processes. We then derive analytical results for the iVaR and disentangle, in a theoretically consistent way, the jump and diffusion contributions to the intrahorizon risk. We estimate historical and option-implied value at risk and iVaR for several popular jump models using the Standard & Poor's (S&P) 100 Index and American options. Empirically disentangling the contribution of the jumps from the contribution of the diffusion, we conclude that jumps account for about 90% of the iVaR on average. Our back-testing results indicate that the option-implied estimates are much more responsive to market changes than their historical counterparts, which perform poorly. This paper was accepted by Gustavo Manso, finance. [ABSTRACT FROM AUTHOR]

We consider a Lévy process that is reflected at 0 and at K > 0. The reflected process is obtained by adding the difference between the local time at 0 and the local time at K to the sum of the feeding Lévy process and an initial condition. We define the loss rate to be the expectation of the local time at K at time 1 under stationary conditions. The main result of the paper is the identification of the loss rate in terms of the stationary measure of the reflected process and the characteristic triplet of the Lévy process. We also derive asymptotics of the loss rate as K → ∞ when the drift of the feeding process is negative and the Lévy measure is light tailed. Finally, we extend the results for Lévy processes to hold for Markov-modulated Lévy processes. [ABSTRACT FROM AUTHOR]

Statement of Scope and Mission Reflected hyperexponential jump diffusion processes (RHEPs) have broad applications in operations research, financial engineering, insurance, economics, and queueing. Nonetheless, the extant literature lacks efficiently computable formulae for quantities related to first passage problems of RHEPs. This paper provides a new approach to derive such expressions. As a by-product, we obtain a formula for the Laplace transform of the RHEP transition density. Our approach relies on solving specific OIDEs numerically, applying martingale methods, conducting expansions, and performing matrix analysis. Numerical results show that, for cases where rounding error matters, our approach can provide effective approximation that is substantially more efficient than using "multiprecision computing." The extant literature on first passage problems of reflected hyperexponential jump diffusion processes (RHEPs) lacks efficiently computable formulae for the Laplace transform of the joint distribution of the RHEP and its first passage time, cumulative distribution function of the overshoot, expected cumulative value of the discounted increments of the local time up to the first passage time, expected cumulative discounted value of the RHEP up to the first passage time, and expectation of the first passage time. We combine numerical solutions to ordinary integro-differential equations and martingale methods in a novel manner to derive such expressions. For some of these quantities, our approach can deal with the subtle case in which both the RHEP's overall drift and the discount rate equal zero. As a by-product, we obtain a formula for the Laplace transform of the RHEP transition density. We illustrate the numerical performance of our methodology through a few examples. We observe that, when the RHEP's overall drift and the discount rate are very close to zero, rounding errors can make the evaluation of some of our formulae unreliable. In these situations our exact expression for the case in which the RHEP's overall drift and discount rate are both zero can be an effective approximation for the quantities in question that is substantially more efficient than reliably calculating them using their exact expressions and "multiprecision computing." Our research has applications in financial engineering, insurance, economics, and queueing. [ABSTRACT FROM AUTHOR]

A superposition of a large number of infinite source Poisson inputs or that of a large number of ON-OFF inputs with heavy tails can look like either a fractional Brownian motion or a stable Lévy motion, depending on the magnification at which we are looking at the input process (Mikosch et al. 2002). In this paper, we investigate what happens to a queue driven by such inputs. Under such conditions, we study the output of a single fluid server and the behavior of a fluid queueing network. For the network we obtain random field limits describing the activity at different stations. In general, both kinds of stations arise in the same network: the stations of the first kind experience loads driven by a fractional Brownian motion, while the stations of the second kind experience loads driven by a stable Lévy motion. [ABSTRACT FROM AUTHOR]

This note presents two counterexamples to illustrate that neither implication of Theorem 4 in Levy (1992) is correct. [ABSTRACT FROM AUTHOR]

Abstract. We present a two-armed bandit model of decision making under uncertainty where the expected return to investing in the "risky arm" increases when choosing that arm and decreases when choosing the "safe" arm. These dynamics are natural in applications such as human capital development, job search, and occupational choice. Using new insights from stochastic control, along with a monotonicity condition on the payoff dynamics, we show that optimal strategies in our model are stopping rules that can be characterized by an index which formally coincides with Gittins' index. Our result implies the indexability of a new class of restless bandit models. [ABSTRACT FROM AUTHOR]

We propose an accurate method for pricing arithmetic Asian options on the discrete or continuous average in a general model setting by means of a lower bound approximation. In particular, we derive analytical expressions for the lower bound in the Fourier domain. This is then recovered by a single univariate inversion and sharpened using an optimization technique. In addition, we derive an upper bound to the error from the lower bound price approximation. Our proposed method can be applied to computing the prices and price sensitivities of Asian options with fixed or floating strike price, discrete or continuous averaging, under a wide range of stochastic dynamic models, including exponential Lévy models, stochastic volatility models, and the constant elasticity of variance diffusion. Our extensive numerical experiments highlight the notable performance and robustness of our optimized lower bound for different test cases. [ABSTRACT FROM AUTHOR]

We consider tempered stable Lévy subordinators and develop a bridge sampling method. An approximate conditional probability density function (PDF) given the terminal values is derived with stable index less than one, using the double saddlepoint approximation. We then propose an acceptance-rejection algorithm based on the existing gamma bridge and the inverse Gaussian bridge as proposal densities. Its performance is comparable to existing sequential sampling methods such as Devroye (2009) [Devroye L (2009) Random variate generation for exponentially and ploynomially tilted stable distributions. ACM Trans. Modeling Comput. Simulation 19(4):18:1-20.] and Hofert (2011) [Hofert M (2011) Sampling exponentially tilted stable distributions. ACM Trans. Modeling Comput. Simulation 22(1):3:1-11.] when generating a fixed number of observations. As applications, we consider option pricing problems in Lévy models. First, we demonstrate the effectiveness of bridge sampling when combined with adaptive sampling under finite-variance CGMY processes. Second, further efficiency gain is achieved in terms of variance reduction via stratified sampling. [ABSTRACT FROM AUTHOR]

Bandit problems model the trade-off between exploration and exploitation in various decision problems. We study two-armed bandit problems in continuous time, where the risky arm can have two types: High or Low; both types yield stochastic payoffs generated by a Lévy process. We show that the optimal strategy is a cut-off strategy and we provide an explicit expression for the cut-off and for the optimal payoff. [ABSTRACT FROM AUTHOR]

We study stochastic tree fluid networks driven by a multidimensional Lévy process. We are interested in (the joint distribution of) the steady-state content in each of the buffers, the busy periods, and the idle periods. To investigate these fluid networks, we relate the above three quantities to fluctuations of the input Lévy process by solving a multidimensional Skorokhod reflection problem. This leads to the analysis of the distribution of the componentwise maximums, the corresponding epochs at which they are attained, and the beginning of the first last-passage excursion. Using the notion of splitting times, we are able to find their Laplace transforms. It turns out that, if the components of the Lévy process are "ordered," the Laplace transform has a so-called quasi-product form. The theory is illustrated by working out special cases, such as tandem networks and priority queues. [ABSTRACT FROM AUTHOR]

We study the multidimensional reflection map on the spaces D([0, T], R[supk]) and D([0, ∞), R[supk]) of right-continuous R[supk] -valued functions on [0,T] or [0, ∞) with left limits, endowed with variants of the Skorohod (1956) M[sub1] topology. The reflection map was used with the continuous mapping theorem by Harrison and Reiman (1981) and Reiman (1984) to establish heavy-traffic limit theorems with reflected Brownian motion limit processes for vector-valued queue length, waiting time, and workload stochastic processes in single-class open queueing networks. Since Brownian motion and reflected Brownian motion have continuous sample paths, the topology of uniform convergence over bounded intervals could be used for those results. Variants of the M[sub1] topologies are needed to obtain alternative discontinuous limits approached gradually by the converging processes, as occurs in stochastic fluid networks with bursty exogenous input processes, e.g., with on-off sources having heavy-tailed on periods or off periods (having in finite variance). We show that the reflection map is continuous at limits without simultaneous jumps of opposite sign in the coordinate functions, provided that the product M[sub1] topology is used. As a consequence, the reflection map is continuous with the product M[sub1] topology at all functions that have discontinuities in only one coordinate at a time. That continuity property also holds for more general reflection maps and is sufficient to support limit theorems for stochastic processes in most applications. We apply the continuity of the reflection map to obtain limits for buffer-content stochastic processes in stochastic fluid networks. [ABSTRACT FROM AUTHOR]