Simulation-based Value-at-Risk for nonlinear portfolios

Abstract

Value-at-risk (VaR) has been playing the role of a standard risk measure since its introduction. In practice, the delta-normal approach is usually adopted to approximate the VaR of portfolios with option positions. Its effectiveness, however, substantially diminishes when the portfolios concerned involve a high dimension of derivative positions with nonlinear payoffs; lack of closed form pricing solution for these potentially highly correlated, American-style derivatives further complicates the problem. This paper proposes a generic simulation-based algorithm for VaR estimation that can be easily applied to any existing procedures. Our proposal leverages cross-sectional information and applies variable selection techniques to simplify the existing simulation framework. Asymptotic properties of the new approach demonstrate faster convergence due to the additional model selection component introduced. We have also performed sets of numerical results that verify the effectiveness of our approach in comparison with some existing strategies.

Leave‐one‐out least squares Monte Carlo algorithm for pricing Bermudan options

“…By using the LOOLSM method, practitioners can reliably obtain the low-biased price-even with higher-order regression basis functions. The LOOLSM algorithm can also be applied along with other regression methods proposed to improve least squares regression (Belomestny, 2011;Chen et al, 2019;Fabozzi et al, 2017;Ibáñez & Velasco, 2018;Ludkovski, 2018;Tompaidis & Yang, 2014).…”

“…Many studies have aimed to improve the LSM method using advanced regression methods, such as ridge regression (Tompaidis & Yang, 2014), least absolute shrinkage and selection operator (LASSO) (Chen et al, 2019;Tompaidis & Yang, 2014), weighted least squares regression (Fabozzi et al, 2017;Ibáñez & Velasco, 2018), and nonparametric kernel regression (Belomestny, 2011;Ludkovski, 2018). The LOOLSM method can be flexibly extended to these alternatives to the LSM method because they are essentially linear projections via the hat matrix.…”

“…Ridge regression and LASSO are linear regressions with L 2 and L 1 -regularization, respectively (Hastie et al, 2009, § 3.4). These methods outperform the LSM method in small simulation paths (Tompaidis & Yang, 2014) and provide stable estimates of the value-at-risk (Chen et al, 2019). The hat matrix of ridge regression is…”

“…Ridge regression and LASSO are linear regressions with $${L}^{2}$$ and $${L}^{1}$$‐regularization, respectively (Hastie et al, 2009, § 3.4). These methods outperform the LSM method in small simulation paths (Tompaidis & Yang, 2014) and provide stable estimates of the value‐at‐risk (Chen et al, 2019). The hat matrix of ridge regression is $$${{\boldsymbol{H}}}^{[i]}={\boldsymbol{X}}{({{\boldsymbol{X}}}^{\top }{\boldsymbol{X}}+\lambda {{\boldsymbol{I}}}_{M})}^{-1}{{\boldsymbol{X}}}^{\top },$$$ where $$\lambda $$ is the regularization strength.…”

Pathwise grid valuation of fixed-income portfolios with applications to risk management

“… Chen et al (2019) points out that nonlinearity reduces the effectiveness of approaches such as delta-normal and argues that the lack of a closed-form pricing solution for derivatives securities complicates the problem of measuring risk. To solve this problem, they propose a generic simulation-based algorithm to calculate the VaR for nonlinear portfolios.…”