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We have applied the Lie-Trotter operator splitting
method to model the dynamics of both the sum and difference of two correlated
constant elasticity of variance (CEV) stochastic variables. Within the
Lie-Trotter splitting approximation, both the sum and difference are shown to
follow a shifted CEV stochastic process, and approximate probability distributions
are determined in closed form. Illustrative numerical examples are presented to
demonstrate the validity and accuracy of these approximate distributions. These
approximate probability distributions can be used to valuate two-asset options,
e.g. spread options and basket
options, where the CEV variables represent the forward prices of the underlying
assets. Moreover, we believe that this new approach can be extended to study
the algebraic sum of N CEV variables
with potential applications in pricing multi-asset options.
In this paper, by means of the Lie-Trotter operator splitting method, we have presented a new unified approach not only to rigorously derive Kirk’s approximation but also to obtain a generalisation for multi-asset spread options in a straightforward manner. The derived price formula for the multi-asset spread option bears a great resemblance to Kirk’s approximation in the two-asset case. More importantly, our approach is able to provide a new perspective on Kirk’s approximation and the generalization; that is, they are simply equivalent to the Lie-Trotter operator splitting approximation to the Black-Scholes equation.
A novel prophylactic regimen is demanded for preventing bladder cancer recurrence, because of the high side-effect tolls of conventional adjuvant Bacillus Calmette-Guérin (BCG) immunotherapy, in addition to its only moderate efficacy. In vitro and animal studies have demonstrated the anti-cancer properties of a medicinal mushroom called Ganoderma lucidum (GL). In this study, a pre-malignant human uroepithelial cells (HUC-PC) model was utilized to compare the effectiveness between ethanol extract of GL (GLe) and BCG on interleukin-6 (IL-6) secretion and lactate dehydrogenase (LDH) cytotoxicity. Additionally, parameters relevant to the BCG efficacy and safety, including free soluble fibronectin (FN) and cell-surface glycosaminoglycans (GAGs) levels were tested, following the exposure of GLe to the cells. GLe at 100 μg/ml and BCG at 4.8 × 107 CFU were shown to induce equivalent levels of IL-6, suggesting the potential synergism, while the tested concentrations of GLe were non-cytotoxic. During the initial four hours of GLe exposure, the free FN concentrations in harvested media were significantly reduced that might facilitate the binding of BCG for uroepithelial internalization to enhance BCG efficacy. Furthermore, the cell membrane-bound GAGs levels of HUC-PC cells were significant increased in response to GLe to suggest cellular protection from BCG infection. In summary, current findings suggest the potential additive synergism of GLe with the BCG efficacy, as well as its protective effects, and thus reducing the BCG toxicity.