Guangwu Liu

Simulating Sensitivities of Conditional Value at Risk

Conditional value at risk (CVaR) is both a coherent risk measure and a natural risk statistic. It is often used to measure the risk associated with large losses. In this paper, we study how to estimate the sensitivities of CVaR using Monte Carlo simulation. We first prove that the CVaR sensitivity can be written as a conditional expectation for general loss distributions. We then propose an estimator of the CVaR sensitivity and analyze its asymptotic properties. The numerical results show that the estimator works well. Furthermore, we demonstrate how to use the estimator to solve optimization problems with CVaR objective and/or constraints, and compare it to a popular linear programming-based algorithm.

Kernel Estimation of the Greeks for Options with Discontinuous Payoffs

The Greeks are the derivatives (also known as sensitivities) of the option prices with respect to market parameters. They play an important role in financial risk management. Among many Monte Carlo methods of estimating the Greeks, the classical pathwise method requires only the pathwise information that is directly observable from simulation and is generally easier to implement than many other methods. However, the classical pathwise method is generally not applicable to the Greeks of options with discontinuous payoffs and the second-order Greeks. In this paper, we generalize the classical pathwise method to allow discontinuity in the payoffs. We show how to apply the new pathwise method to the first-and second-order Greeks and propose kernel estimators that require little analytical efforts and are very easy to implement. The numerical results show that our estimators work well for practical problems.

Pathwise Estimation of Probability Sensitivities Through Terminating or Steady-State Simulations

A probability is the expectation of an indicator function. However, the standard pathwise sensitivity estimation approach, which interchanges the differentiation and expectation, cannot be directly applied because the indicator function is discontinuous. In this paper, we design a pathwise sensitivity estimator for probability functions based on a result of Hong [Hong, L. J. 2009. Estimating quantile sensitivities. Oper. Res.57(1) 118--130]. We show that the estimator is consistent and follows a central limit theorem for simulation outputs from both terminating and steady-state simulations, and the optimal rate of convergence of the estimator is n-2/5 where n is the sample size. We further demonstrate how to use importance sampling to accelerate the rate of convergence of the estimator to n-1/2, which is the typical rate of convergence for statistical estimation. We illustrate the performances of our estimators and compare them to other well-known estimators through several examples.

Monte Carlo Methods for Value-at-Risk and Conditional Value-at-Risk: A Review

Value-at-risk (VaR) and conditional value-at-risk (CVaR) are two widely used risk measures of large losses and are employed in the financial industry for risk management purposes. In practice, loss distributions typically do not have closed-form expressions, but they can often be simulated (i.e., random observations of the loss distribution may be obtained by running a computer program). Therefore, Monte Carlo methods that design simulation experiments and utilize simulated observations are often employed in estimation, sensitivity analysis, and optimization of VaRs and CVaRs. In this article, we review some of the recent developments in these methods, provide a unified framework to understand them, and discuss their applications in financial risk management.

Monte Carlo estimation of value-at-risk, conditional value-at-risk and their sensitivities

Value-at-risk and conditional value at risk are two widely used risk measures, employed in the financial industry for risk management purposes. This tutorial discusses Monte Carlo methods for estimating value-at-risk, conditional value-at-risk and their sensitivities. By relating the mathematical representation of value-at-risk to that of conditional value-at-risk, it provides a unified view of simulation methodologies for both risk measures and their sensitivities.

Simulating Risk Contributions of Credit Portfolios

The 2007-2009 financial turmoil highlighted the need for more active management of credit portfolios. After measuring portfolio credit risk, an important step toward active risk management is to measure risk contributions of individual obligors to the overall risk of the portfolio. In practice, value-at-risk is often used as a risk measure for credit portfolios, and it can be decomposed into a sum of the risk contributions of individual obligors. Estimation of these risk contributions is computationally challenging, mainly because they are expectations conditioned on a rare event. In this paper, we tackle this computational problem by developing a restricted importance sampling RIS method for a class of conditional-independence credit risk models, where defaults of obligors are conditionally independent given an appropriately chosen random vector. We propose fast estimators for risk contributions and their confidence intervals. Furthermore, we study the incorporation of traditional importance sampling methods into the RIS method to further improve its efficiency for the widely used Gaussian copula model. Numerical examples show that the proposed method works well.

Kernel Smoothing for Nested Estimation with Application to Portfolio Risk Measurement.

Nested estimation involves estimating an expectation of a function of a conditional expectation via simulation. This problem has of late received increasing attention amongst researchers due to its broad applicability particularly in portfolio risk measurement and in pricing complex derivatives. In this paper, we study a kernel smoothing approach. We analyze its asymptotic properties, and present efficient algorithms for practical implementation. While asymptotic results suggest that the kernel smoothing approach is preferable over nested simulation only for low-dimensional problems, we propose a decomposition technique for portfolio risk measurement, through which a high-dimensional problem may be decomposed into low-dimensional ones that allow an efficient use of the kernel smoothing approach. Numerical studies show that, with the decomposition technique, the kernel smoothing approach works well for a reasonably large portfolio with 200 risk factors. This suggests that the proposed methodology may serve...

Kernel estimation for quantile sensitivities

Quantiles, also known as value-at-risk in financial applications, are important measures of random performance. Quantile sensitivities provide information on how changes in the input parameters affect the output quantiles. In this paper, we study the estimation of quantile sensitivities using simulation. We propose a new estimator by employing kernel method and show its consistency and asymptotic normality for i.i.d. data. Numerical results show that our estimator works well for the test problems.

Iimportance sampling for risk contributions of credit portfolios

Value-at-Risk is often used as a risk measure of credit portfolios, and it can be decomposed into a sum of risk contributions associated with individual obligors. These risk contributions play an important role in risk management of credit portfolios. They can be used to measure risk-adjusted performances of subportfolios and to allocate risk capital. Mathematically, risk contributions can be represented as conditional expectations, which are conditioned on rare events. In this paper, we develop a restricted importance sampling (IS) method for simulating risk contributions, and devise estimators whose mean square errors converge in a rate of n−1. Furthermore, we combine our method with the IS method in the literature to improve the efficiency of the estimators. Numerical examples show that the proposed method works quite well.

Revisit of stochastic mesh method for pricing American options

We revisit the stochastic mesh method for pricing American options, from a conditioning viewpoint, rather than the importance sampling viewpoint of Broadie and Glasserman (1997). Starting from this new viewpoint, we derive the weights proposed by Broadie and Glasserman (1997) and show that their weights at each exercise date use only the information of the next exercise date (therefore, we call them forward-looking weights). We also derive new weights that exploit not only the information of the next exercise date but also the information of the last exercise date (therefore, we call them binocular weights). We show how to apply the binocular weights to the Black-Scholes model, more general diffusion models, and the variance-gamma model. We demonstrate the performance of the binocular weights and compare to the performance of the forward-looking weights through numerical experiments.