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In this paper, we price the vulnerable option under the assumption that both underlying and counterparty asset price follow correlated jump diffusion processes. By the two-dimensional Itô-Doeblin formula, we initially derive a partial integro-differential equation (PIDE) satisfied by the price of the vulnerable option with flexible jumps. Then, the double Mellin transform converts the PIDE into an ordinary differential equation (ODE), which possesses an explicit solution. Further, the inverse Mellin transform of the explicit solution to the ODE turns out to be a rigorous solution to the PIDE for vulnerable option. Finally, by carrying out some numerical experiments, we demonstrate that the PIDE's solution produces price with a convincing degree of precision. [ABSTRACT FROM AUTHOR]

Next-generation sequencing (NGS) has been applied to monitor minimal residual disease (MRD) in multiple myeloma (MM). Standardized DNA input and sequencing depth is essential for achieving a uniform sensitivity in NGS-based MRD study. Herein, the sensitivity of 10−5 was verified by a standardized experimental design based on triplicate measurements of 1 μg DNA input and 1 million sequencing reads using the LymphoTrack-MiSeq platform. MRD level was defined as the mean MRD burden of the triplicates. Two spike-in controls at concentrations of 0.001% tumor plasma cells (PC) for verifying the sensitivity of 10−5 and 0.01% (or 0.005%) for MRD normalization were systematically analyzed. The spike-in control of 0.001% MRD was consistently detected in all samples, confirming a sensitivity of 10−5. Moreover, this standardized NGS approach yielded MRD measurements concordant with serological response and comparable to allele-specific oligonucleotide (ASO) real-time quantitative (RQ)-PCR. Moreover, NGS showed an improved sensitivity and provided quantification of MRD for cases assigned "positive but not quantifiable" (PNQ) by ASO RQ-PCR, even without the use of patient-specific probes/primers. Issues regarding the specificity of myeloma-specific sequences as MRD target, optimal input for spike-in normalization, and interpretation of MRD from triplicates are discussed. Herein, the standardized LymphoTrack-MiSeq-based method is verified to carry a sensitivity of 10−5, hence an effective tool for MRD monitoring in MM. As only a small number of samples are tested here, further study with a larger number of patients is warranted. [ABSTRACT FROM AUTHOR]