Application of Operator Splitting Methods in Finance

Financial derivatives pricing aims to find the fair value of a financial contract on an underlying asset. Here we consider option pricing in the partial differential equations framework. The contemporary models lead to one-dimensional or multidimensional parabolic problems of the convection-diffusion type and generalizations thereof. An overview of various operator splitting methods is presented for the efficient numerical solution of these problems. Splitting schemes of the Alternating Direction Implicit (ADI) type are discussed for multidimensional problems, e.g. given by stochastic volatility (SV) models. For jump models Implicit-Explicit (IMEX) methods are considered which efficiently treat the nonlocal jump operator. For American options an easy-to-implement operator splitting method is described for the resulting linear complementarity problems. Numerical experiments are presented to illustrate the actual stability and convergence of the splitting schemes. Here European and American put options are considered under four asset price models: the classical Black-Scholes model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV model with jumps

Stability of ADI schemes for multidimensional diffusion equations with mixed derivative terms

In this paper the unconditional stability of four well-known ADI schemes is analyzed in the application to time-dependent multidimensional diffusion equations with mixed derivative terms. Necessary and sufficient conditions on the parameter theta of each scheme are obtained that take into account the actual size of the mixed derivative coefficients. Our results generalize results obtained previously by Craig & Sneyd (1988) and In 't Hout & Welfert (2009). Numerical experiments are presented illustrating our main theorems

Texas Brazos River Flow and Mussel Growth Reconstructions Using Stable Oxygen, Hydrogen, and Carbon Isotopes and Trace Elements

The interaction between drought and river regulation is monitored to better understand river flow mixing, evaporation, and surface-groundwater exchange in changing regional climates and in increasingly regulated waterways. I compared Brazos River stable isotope (δ^18O and δD) and electrical conductivity values with reservoir, creek, and aquifer samples in the Brazos watershed, the largest watershed in Texas. Shells from two common species of Brazos River mussel, Amblema plicata and Cyrtonaias tampicoensis, were serially-sampled in the inner and outer shell layers for δ^18O, δ^13C, and trace elements to examine the isotopic and chemical signatures of the 2011-2014 drought. Predicted aragonite δ^18O for the 2012-13 study interval has an irregular pattern that complicates development of growth chronologies in modern shells. To circumvent this problem, clumped isotope (Δ47) temperature measurements were used for interpreting segments of shell growth chronologies. To characterize the influence that biological and environmental variables have on shell chemistry, one specimen from each of the above two mussel species were studied using paired isotope-trace element analyses and cathodoluminescence. The Brazos River Alluvium Aquifer and the Lake Whitney reservoir, both on the main river channel, represent water source endmembers of dilute runoff water and evaporated saline water, respectively. The difference between river and precipitation Δ^18O, or Δ^18ORIV-PPT, a measurement of degree of evaporation, ranged from 0.9‰ for a small creek, to 2.7‰ for the Brazos River, to at least 3.7‰ in Lake Whitney. Δ^18O values and trends were similar in coeval shell transects, indicating that δ^18O is a valid chronometer when calibrated, although all shell had winter growth cessations. Δ^13C trends were similar between shells, suggesting strong environmental control influenced by upstream dam releases. The shell isotope chronologies can be used to reconstruct variation in river discharge, flow source, and salinity. Shell δ^13C, Sr/Ca, and Mn/Ca generally covaried in the shell regions sampled, and shell δ^13C is thought to be controlled by upstream dam releases based on previous work. Relationships between Sr/Ca and temperature are consistent with temperature-paced metabolic control on shell Sr/Ca as in other studies

Stability of central finite difference schemes for the Heston PDE

This paper deals with stability in the numerical solution of the prominent Heston partial differential equation from mathematical finance. We study the well-known central second-order finite difference discretization, which leads to large semi-discrete systems with non-normal matrices A. By employing the logarithmic spectral norm we prove practical, rigorous stability bounds. Our theoretical stability results are illustrated by ample numerical experiments