This paper presents a novel method to price discretely monitored single-and doublebarrier options in Lévy process-based models. The method involves a sequential evaluation of Hilbert transforms of the product of the Fourier transform of the value function at the previous barrier monitoring date and the characteristic function of the (Esscher transformed) Lévy process. A discrete approximation with exponentially decaying errors is developed based on the Whittaker cardinal series (Sinc expansion) in Hardy spaces of functions analytic in a strip. An efficient computational algorithm is developed based on the fast Hilbert transform that, in turn, relies on the FFT-based Toeplitz matrix-vector multiplication. Our method also provides a natural framework for credit risk applications, where the firm value follows an exponential Lévy process and default occurs at the first time the firm value is below the default barrier on one of a discrete set of monitoring dates.

Laser Raman spectroscopy and cathodoluminescence (CL) images show that zircon from Sulu-Dabie dolomitic marbles is characterized by distinctive domains of inherited (detrital), prograde, ultrahighpressure (UHP) and retrograde metamorphic growths. The inherited zircon domains are darkluminescent in CL images and contain mineral inclusions of Qtz + Cal + Ap. The prograde metamorphic domains are white-luminescent in CL images and preserve a quartz eclogite facies assemblage of Qtz + Dol + Grt + Omp + Phe + Ap, formed at 542-693°C and 1.8-2.1 GPa. In contrast, the UHP metamorphic domains are grey-luminescent in CL images, retain the UHP assemblage of Coe + Grt + Omp + Arg + Mgs + Ap, and record UHP conditions of 739-866°C and >5.5 GPa. The outermost retrograde rims have dark-luminescent CL images, and contain low-P minerals such as calcite, related to the regional amphibolite facies retrogression. Laser ablation ICP-MS trace-element data show striking difference between the inherited cores of mostly magmatic origin and zircon domains grown in response to prograde, UHP and retrograde metamorphism. SHRIMP U-Pb dating on these zoned zircon identified four discrete 206 Pb/ 238 U age groups: 1823-503 Ma is recorded in the inherited (detrital) zircon derived from various Proterozoic protoliths, the prograde domains record the quartz eclogite facies metamorphism at 254-239 Ma, the UHP growth domains occurred at 238-230 Ma, and the late amphibolite facies retrogressive overprint in the outermost rims was restricted to 218-206 Ma. Thus, Proterozoic continental materials of the Yangtze craton were subducted to 55-60 km depth during the Early Triassic and recrystallized at quartz eclogite facies conditions. Then these metamorphic rocks were further subducted to depths of 165-175 km in the Middle Triassic and experienced UHP metamorphism, and finally these UHP metamorphic rocks were exhumed to midcrustal levels (about 30 km) in the Late Triassic and overprinted by regional amphibolite facies metamorphism. The subduction and exhumation rates deduced from the SHRIMP data and metamorphic P-T conditions are 9-10 km Myr )1 and 6.4 km Myr )1 , respectively, and these rapid subduction-exhumation rates may explain the obtained P-T-t path. Such a fast exhumation suggests that Sulu-Dabie UHP rocks that returned towards crustal depths were driven by buoyant forces, caused as a consequence of slab breakoff at mantle depth.

We propose a new computational method for the valuation of options in jump-diffusion models. The option value function for European and barrier options satisfies a partial integrodifferential equation (PIDE). This PIDE is commonly integrated in time by implicit-explicit (IMEX) time discretization schemes, where the differential (diffusion) term is treated implicitly, while the integral (jump) term is treated explicitly. In particular, the popular IMEX Euler scheme is first-order accurate in time. Second-order accuracy in time can be achieved by using the IMEX midpoint scheme. In contrast to the above approaches, we propose a new high-order time discretization scheme for the PIDE based on the extrapolation approach to the solution of ODEs that also treats the diffusion term implicitly and the jump term explicitly. The scheme is simple to implement, can be added to any PIDE solver based on the IMEX Euler scheme, and is remarkably fast and accurate. We demonstrate our approach on the examples of Merton's and Kou's jump-diffusion models, the diffusion-extended variance gamma model, as well as the two-dimensional Duffie-Pan-Singleton model with correlated and contemporaneous jumps in the stock price and its volatility. By way of example, pricing a one-year double-barrier option in Kou's jump-diffusion model, our scheme attains accuracy of 10 −5 in 72 time steps (in 0.05 seconds). In contrast, it takes the first-order IMEX Euler scheme more than 1.3 million time steps (in 873 seconds) and the second-order IMEX midpoint scheme 768 time steps (in 0.49 seconds) to attain the same accuracy. Our scheme is also well suited for Bermudan options. Combining simplicity of implementation and remarkable gains in computational efficiency, we expect this method to be very attractive to financial engineering modelers.

In this paper, we consider an optimal portfolio deleveraging problem, where the objective is to meet specified debt/equity requirements at the minimal execution cost. Permanent and temporary price impact is taken into account. With no restrictions on the relative magnitudes of permanent and temporary price impact, the optimal deleveraging problem reduces to a nonconvex quadratic program with quadratic and box constraints. Analytical results on the optimal deleveraging strategy are obtained. They provide guidance on how we liquidate a portfolio according to endogenous and exogenous factors. A Lagrangian method is proposed to solve the nonconvex quadratic program numerically. By studying the breakpoints of the Lagrangian problem, we obtain conditions under which the Lagrangian method returns an optimal solution of the deleveraging problem. When the Lagrangian algorithm returns a suboptimal approximation, we present upper bounds on the loss in equity caused by using such an approximation.Subject classifications: optimal deleveraging; permanent and temporary price impact; nonconvex quadratic program; Lagrangian method; breakpoint. Area of review: Optimization.

[1] Predominant stretching structures in the Greater Himalayan Crystalline Complex (GHC) trend perpendicular to the belt and are linked to the southward exhumation or emplacement of the GHC between the South Tibet Detachment (STD) and the Main Central Thrust. However, our field investigations in southern Tibet reveal the widespread presence of gently dipping shear zones with a penetrative orogen-parallel stretching lineation, which separates the Tethyan Himalayan Sequence and the underlying GHC. The shear zones are well preserved in the upper part of the GHC, south to and structurally lower than the STD. Field criteria, microstructures, and quartz fabrics indicate top-to-the-east shearing in the Yadong shear zone (eastern GHC), coexistence of top-to-the-east and topto-the-west shearing in the Nyalam shear zone (central GHC), but top-to-the-west shearing in the Pulan shear zone (western GHC). Characteristic microstructures and slip systems of quartz in the high-grade GHC rocks resulted from the lateral flow under upper amphibolite (up to 650-700 C) to greenschist facies conditions. U-Pb ages of metamorphic zircon rims by sensitive high-resolution ion microprobe (SHRIMP) and laser ablation multi-collector inductively coupled plasma mass spectrometry ( Ar cooling ages of biotite and muscovite suggest cessation of ductile sharing at 13-11 Ma on the Yadong shear zone, which is coeval with the activation of the STD. Combined with previous studies, we propose that initiation of orogen-parallel extension marks the transition from burial/crustal thickening to exhumation of the GHC. Due to lateral crustal thickness gradients in a thickened crust, orogen-parallel gravitational collapse occurred within the convergent Himalayan orogen in the late Oligocene-Miocene. This tectonic denudation triggered and enhanced partial melting and ductile extrusion of the GHC in the Miocene.