Multilevel Richardson-Romberg and Importance Sampling in Derivative Pricing

In this paper, we propose and analyze a novel combination of multilevel Richardson-Romberg (ML2R) and importance sampling algorithm, with the aim of reducing the overall computational time, while achieving desired root-mean-squared error while pricing. We develop an idea to construct the Monte-Carlo estimator that deals with the parametric change of measure. We rely on the Robbins-Monro algorithm with projection, in order to approximate optimal change of measure parameter, for various levels of resolution in our multilevel algorithm. Furthermore, we propose incorporating discretization schemes with higher-order strong convergence, in order to simulate the underlying stochastic differential equations (SDEs) thereby achieving better accuracy. In order to do so, we study the Central Limit Theorem for the general multilevel algorithm. Further, we study the asymptotic behavior of our estimator, thereby proving the Strong Law of Large Numbers. Finally, we present numerical results to substantiate the efficacy of our developed algorithm

Loan portfolio management and Liquidity Risk: The impact of limited liability and haircut

In this article, we consider the problem of a bank's loan portfolio in the context of liquidity risk, while allowing for the limited liability protection enjoyed by the bank. Accordingly, we construct a novel loan portfolio model with limited liability, while maintaining a threshold level of haircut in the portfolio. For the constructed three-time step loan portfolio, at the initial time, the bank raises capital via debt and equity, investing the same in several classes of loans, while at the final time, the bank either meets its liabilities or becomes insolvent. At the intermediate time step, a fraction of the deposits are withdrawn, resulting in liquidation of some of the bank's assets. The liquidated portfolio is designed with the goal of minimizing the liquidation cost. Our theoretical results show that model with the haircut constraint leads to lesser liquidity risk, as compared to the scenario of no haircut constraint being imposed. Finally, we present numerical results to illustrate the theoretical results which were obtained

Analysis of Hepatitis C Viral Dynamics Using Latin Hypercube Sampling

We consider a mathematical model comprising of four coupled ordinary differential equations (ODEs) for studying the hepatitis C (HCV) viral dynamics. The model embodies the efficacies of a combination therapy of interferon and ribavirin. A condition for the stability of the uninfected and the infected steady states is presented. A large number of sample points for the model parameters (which were physiologically feasible) were generated using Latin hypercube sampling. Analysis of our simulated values indicated approximately 24% cases as having an uninfected steady state. Statistical tests like the chi-square-test and the Spearman's test were also done on the sample values. The results of these tests indicate a distinctly differently distribution of certain parameter values and not in case of others, vis-a-vis, the stability of the uninfected and the infected steady states