Samim Ghamami

Dynamic Scheduling of a Two-Server Parallel Server System with Complete Resource Pooling and Reneging in Heavy Traffic: Asymptotic Optimality of a Two-Threshold Policy

We consider a dynamic control problem for a parallel server system commonly known as the N-system. An N-system is a two-server parallel server system with two job classes, one server that can serve both classes, and one server that can only serve one class. We assume that jobs within each class arrive according to a renewal process. The random service time of a job has a general distribution that may depend on both the jobs class and the server providing the service. Each job independently reneges, or abandons the queue without receiving service, if service does not begin within an exponentially distributed amount of time. The objective is to minimize the expected infinite horizon discounted cost of holding jobs in the system and having customers abandon, by dynamically scheduling waiting jobs to available servers. It is not possible to solve this control problem exactly, and so, we consider an asymptotic regime in which the system satisfies both a heavy traffic and a resource pooling condition. Then, we solve the limiting Brownian control problem, and interpret its solution as a policy in the original N-system. We label the servers and job classes so that server 1 can only serve class 1 and server 2 can serve both classes. The policy we propose has two thresholds. There is one threshold on the total number of jobs in the system, and one threshold on the number of class 1 jobs in the system. These thresholds are used to determine which job class server 2 should serve. We show that this proposed policy is asymptotically optimal within a specified class of admissible policies in the heavy traffic limit, and has the same limiting cost as the Brownian control problem solution.

Stochastic Intensity Models of Wrong Way Risk: Wrong Way CVA Need Not Exceed Independent CVA

A financial institution’s counterparty credit exposures may be correlated with the credit quality of a counterparty; wrong way risk refers to the case where this correlation is negative. Hull and White [9] are the first to model wrong way risk in Credit Value Adjustment (CVA) calculations by expressing the counterparty’s default intensity in terms of the financial institution’s credit exposure to the counterparty. We derive a formula for CVA for a class of models that includes the formulation of Hull and White [9], and we show that wrong way risk does not affect the credit quality of the counterparty. We provide numerical examples based on the Hull and White [9] formulation to estimate CVA for forward contracts and European options. These examples demonstrate that independent CVA can exceed wrong way CVA. This is inconsistent with the scalar multiples of independent CVA that have been adopted by regulators as a proxy for wrong way CVA.

Static Models of Central Counterparty Risk

Following the 2009 G-20 clearing mandate, international standard setting bodies (SSBs) have outlined a set of principles for central counterparty (CCP) risk management. They have also devised formulaic CCP risk capital requirements on clearing members for their central counterparty exposures. There is still no consensus among CCP regulators and bank regulators on how central counterparty risk should be measured coherently in practice. A conceptually sound and logically consistent definition of the CCP risk capital in the absence of a unifying CCP risk measurement framework is challenging. Incoherent CCP risk capital requirements may create an obscure environment disincentivizing the central clearing of over the counter (OTC) derivatives transactions. Based on novel applications of well-known mathematical models in finance, this paper introduces a risk measurement framework that coherently specifies all layers of the default waterfall resources of typical derivatives CCPs. The proposed framework gives the first risk sensitive definition of the CCP risk capital based on which less risk sensitive non-model-based methods can be evaluated.

Efficient Monte Carlo Barrier Option Pricing When the Underlying Security Price Follows a Jump-Diffusion Process

We present efficient simulation procedures for pricing barrier options when the underlying security price follows a geometric Brownian motion with jumps. Metwally and Atiya [2002] developed a simulation approach for pricing knock-out options in the same setting, but no variance reduction was introduced. We improve upon Metwally and Atiyas method by innovative applications of well-known variance reduction techniques. We also show how to use simulation to price knock-in options. Numerical examples show that our proposed Monte Carlo procedures lead to substantial variance reduction as well as a reduction in computing time.

Efficient Monte Carlo Counterparty Credit Risk Pricing and Measurement

Counterparty credit risk (CCR), a key driver of the 2007-08 credit crisis, has become one of the main focuses of the major global and U.S. regulatory standards. Financial institutions invest large amounts of resources employing Monte Carlo simulation to measure and price their counterparty credit risk. We develop efficient Monte Carlo CCR estimation frameworks by focusing on the most widely used and regulatory-driven CCR measures: expected positive exposure (EPE), credit value adjustment (CVA), and effective expected positive exposure (EEPE). Our numerical examples illustrate that our proposed efficient Monte Carlo estimators outperform the existing crude estimators of these CCR measures substantially in terms of mean square error (MSE). We also demonstrate that the two widely used sampling methods, the so-called Path Dependent Simulation (PDS) and Direct Jump to Simulation date (DJS), are not equivalent in that they lead to Monte Carlo CCR estimators which are drastically different in terms of their MSE.

Submodular Risk Allocation

We analyze the optimal allocation of trades to portfolios when the cost associated with an allocation is proportional to each portfolio’s risk. Our investigation is motivated by changes in the over-the-counter derivatives markets, under which some contracts may be traded bilaterally or through central counterparties, splitting a set of trades into two or more portfolios. A derivatives dealer faces risk-based collateral and capital costs for each portfolio, and it seeks to minimize total margin requirements through its allocation of trades to portfolios. When margin requirements are submodular, the problem becomes a submodular intersection problem. Its dual provides per-trade margin attributions, and assigning trades to portfolios based on the lowest attributed costs yields an optimal allocation. As part of this investigation, we derive conditions under which standard deviation and other risk measures are submodular functions of sets of trades. We compare systemwide optimality with individually optimal allocations in a market with multiple dealers.

Efficient monte carlo CVA estimation

This paper presents an overview of the efficient Monte Carlo counterparty credit risk (CCR) estimation framework recently developed by Ghamami and Zhang (2014). We focus on the estimation of credit value adjustment (CVA), one of the most widely used and regulatory-driven counterparty credit risk measures. Our proposed efficient CVA estimators are developed based on novel applications of well-known mean square error (MSE) reduction techniques in the simulation literature. Our numerical examples illustrate that the efficient estimators outperform the existing crude estimators of CVA substantially in terms of MSE.

Improving the normalized importance sampling estimator

The normalized importance sampling estimator allows the target density f to be known only up to a multiplicative constant. We indicate how it can be derived by a delta method-based approximation of a Rao-Blackwellized acceptance rejection estimator. Using additional terms in the delta method then results on a new estimator that also only requires f to be known only up to a multiplicative constant. Numerical examples indicate that the new estimator usually outperforms the normalized importance sampling estimator in terms of mean square error.