4 research outputs found
Sub-sampling and other considerations for efficient risk estimation in large portfolios
Computing risk measures of a financial portfolio comprising thousands of
options is a challenging problem because (a) it involves a nested expectation
requiring multiple evaluations of the loss of the financial portfolio for
different risk scenarios and (b) evaluating the loss of the portfolio is
expensive and the cost increases with its size. In this work, we look at
applying Multilevel Monte Carlo (MLMC) with adaptive inner sampling to this
problem and discuss several practical considerations. In particular, we discuss
a sub-sampling strategy that results in a method whose computational complexity
does not increase with the size of the portfolio. We also discuss several
control variates that significantly improve the efficiency of MLMC in our
setting
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