In this talk, we discuss two kinds of dynamic financial decision - dynamic hedging of contingent claim and dynamic asset allocation - in the presence of concerns on financial risk and the implementation of financial risk management. For the dynamic hedging decision, we focus on the asset pricing implications of financial risks, such as liquidity risk and model risk. We also discuss some challenges of deep dynamic hedging of contingent claim. Then we discuss the dynamic asset allocation under several important financial risk management measures and formulate the equilibrium (game) under financial risks.
Building out a Robust and Efficient Risk Management - Alan CheungLászló Árvai
Credit Derivatives are off-balance sheet financial statements that permit one party to transfer the risk of a reference asset, which it typically owns, to another one party (the guarantor) without actually selling the assets.
Given the current regulatory environment and the resulting changes going on in the industry today, the chief risk officer has become the most important person in the financial institution.
WolfPAC Solutions Group Director Michael Cohn interviewed chief risk officers at financial institutions across the country to find out how they became a CRO, what skills and experience they bring to the role, and what is expected of them now.
MODULE 3:
Credit Risks Credit Risk Management models - Introduction, Motivation, Funtionality of good credit. Risk Management models- Review of Markowitz’s Portfolio selection theory –Credit Risk Pricing Model – Capital and Rgulation. Risk management of Credit Derivatives.
Building out a Robust and Efficient Risk Management - Alan CheungLászló Árvai
Credit Derivatives are off-balance sheet financial statements that permit one party to transfer the risk of a reference asset, which it typically owns, to another one party (the guarantor) without actually selling the assets.
Given the current regulatory environment and the resulting changes going on in the industry today, the chief risk officer has become the most important person in the financial institution.
WolfPAC Solutions Group Director Michael Cohn interviewed chief risk officers at financial institutions across the country to find out how they became a CRO, what skills and experience they bring to the role, and what is expected of them now.
MODULE 3:
Credit Risks Credit Risk Management models - Introduction, Motivation, Funtionality of good credit. Risk Management models- Review of Markowitz’s Portfolio selection theory –Credit Risk Pricing Model – Capital and Rgulation. Risk management of Credit Derivatives.
A presentation on the proposed ERM risk evaluation standard by the US Actuarial Standards Board.
Présentation de la norme ERM du Actuarial Standards Board des USA
Kuala Lumpur - PMI Global Congress 2009 - Risk ManagementTorsten Koerting
Presentation on Risk Management Tools, like Risk Register, Risk Profile Presentation Options, How to facilitate a Risk Assessment and effective Processes for day to day application of Risk Management in your Project
How to Manage Increasing Data Compliance Issues in Community BanksColleen Beck-Domanico
During one of RMA’s Credit Risk Management Audio Conferences, H. Walter Young, chief liquidity risk officer, M&T Bank and chief data officer, CCAR, shared strategies and best practices for community banks facing increased data compliance and integrity issues, once deemed as “big bank issues."
This presentation was shared with the Risk Model Working Group (RMWG) committee led by PHMSA. The committee asked Dynamic Risk to present on designing risk models that include elements of both qualitative and quantitative, and overall insight into pipeline safety and reliability.
Value Engineering. Measuring and managing risks in the wind energy industryStavros Thomas
As wind turbines number increased around the world, the number of hazardous accidents is also rising, causing critics to question overall safety. A recent study from Anemorphosis Research Group reveals how wind power professionals manage risk, from current areas of concern to anticipated challenges.
Mercer Capital's Community Bank Stress Testing: What You Need to KnowMercer Capital
While there is no legal requirement for community banks to perform stress tests, recent regulatory commentary suggests that community banks should be developing and implementing some form of stress testing on at least an annual basis.
Whether you are considering performing the test in-house or with outside assistance, this webinar will be of interest to you. This webinar: covers the basics of community bank stress testing; reviews the economic scenarios published by the Federal Reserve; provides detail on the key steps to developing a sound community bank stress test; and discusses how to analyze and act upon the outputs of your stress tests.
ISOL 533 - Information Security and Risk Management R.docxchristiandean12115
ISOL 533 - Information Security and Risk Management Risk Management Plan
University of the Cumberlands
Executive Summary
<Review the Scenario on Page #2 of the publisher’s Project: Risk Management Plan. Summarize the information about the company provided in the scenario and place it into this section of the report. Remove these instructions and all other instructions below before submitting the document for grading.>
This Risk Management Plan covers the Risks, Threats and Weaknesses of the Health Network, Inc. (Health Network).Risks - Threats – Weaknesses within each domain
<Using the Threats listed on Page #3 of the publisher’s Project: Risk Management Plan and the 7 Domains diagram on Page #3 of this template, complete the table on Page #2 of this template (review your Lab #1 solution). Once you enter the Threats into the table, list one or more Weaknesses that might exist in a typical organization using research and your imagination) and then list the Risk to the company if the Threat exploits that Weakness. Then group these Risks-Threats-Weaknesses (R-T-W) by Domain and discuss them below in this section.>
User Domain: <list each User Domain R-T--W identified in the table>
Workstation Domain: <list each Workstation Domain R-T--W identified in the table>
LAN Domain: <list each User Domain R-T--W identified in the table>
WAN-to-LAN Domain: <list each Workstation Domain R-T--W identified in the table>
WAN Domain: <list each User Domain R-T--W identified in the table>
Remote Access Domain: <list each Workstation Domain R-T--W identified in the table>
System/Application Domain: <list each User Domain R-T--W identified in the table>Compliance Laws and Regulations
<List the laws and regulations that affect this industry.>
…
Your Organization
.
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
Enter details about the organization and it IT Infrastructure.
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Management University of the Cumberlands
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ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
organization changes to the
systems, applications and organizational data can undermine the organization's
violations of federal or state mandates and laws can
lead to major . potential to impact the
organization
organization
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
organization
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
central respoitory accessible via the
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ISOL 533 - InfoSecurity & Risk
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A presentation on the proposed ERM risk evaluation standard by the US Actuarial Standards Board.
Présentation de la norme ERM du Actuarial Standards Board des USA
Kuala Lumpur - PMI Global Congress 2009 - Risk ManagementTorsten Koerting
Presentation on Risk Management Tools, like Risk Register, Risk Profile Presentation Options, How to facilitate a Risk Assessment and effective Processes for day to day application of Risk Management in your Project
How to Manage Increasing Data Compliance Issues in Community BanksColleen Beck-Domanico
During one of RMA’s Credit Risk Management Audio Conferences, H. Walter Young, chief liquidity risk officer, M&T Bank and chief data officer, CCAR, shared strategies and best practices for community banks facing increased data compliance and integrity issues, once deemed as “big bank issues."
This presentation was shared with the Risk Model Working Group (RMWG) committee led by PHMSA. The committee asked Dynamic Risk to present on designing risk models that include elements of both qualitative and quantitative, and overall insight into pipeline safety and reliability.
Value Engineering. Measuring and managing risks in the wind energy industryStavros Thomas
As wind turbines number increased around the world, the number of hazardous accidents is also rising, causing critics to question overall safety. A recent study from Anemorphosis Research Group reveals how wind power professionals manage risk, from current areas of concern to anticipated challenges.
Mercer Capital's Community Bank Stress Testing: What You Need to KnowMercer Capital
While there is no legal requirement for community banks to perform stress tests, recent regulatory commentary suggests that community banks should be developing and implementing some form of stress testing on at least an annual basis.
Whether you are considering performing the test in-house or with outside assistance, this webinar will be of interest to you. This webinar: covers the basics of community bank stress testing; reviews the economic scenarios published by the Federal Reserve; provides detail on the key steps to developing a sound community bank stress test; and discusses how to analyze and act upon the outputs of your stress tests.
ISOL 533 - Information Security and Risk Management R.docxchristiandean12115
ISOL 533 - Information Security and Risk Management Risk Management Plan
University of the Cumberlands
Executive Summary
<Review the Scenario on Page #2 of the publisher’s Project: Risk Management Plan. Summarize the information about the company provided in the scenario and place it into this section of the report. Remove these instructions and all other instructions below before submitting the document for grading.>
This Risk Management Plan covers the Risks, Threats and Weaknesses of the Health Network, Inc. (Health Network).Risks - Threats – Weaknesses within each domain
<Using the Threats listed on Page #3 of the publisher’s Project: Risk Management Plan and the 7 Domains diagram on Page #3 of this template, complete the table on Page #2 of this template (review your Lab #1 solution). Once you enter the Threats into the table, list one or more Weaknesses that might exist in a typical organization using research and your imagination) and then list the Risk to the company if the Threat exploits that Weakness. Then group these Risks-Threats-Weaknesses (R-T-W) by Domain and discuss them below in this section.>
User Domain: <list each User Domain R-T--W identified in the table>
Workstation Domain: <list each Workstation Domain R-T--W identified in the table>
LAN Domain: <list each User Domain R-T--W identified in the table>
WAN-to-LAN Domain: <list each Workstation Domain R-T--W identified in the table>
WAN Domain: <list each User Domain R-T--W identified in the table>
Remote Access Domain: <list each Workstation Domain R-T--W identified in the table>
System/Application Domain: <list each User Domain R-T--W identified in the table>Compliance Laws and Regulations
<List the laws and regulations that affect this industry.>
…
Your Organization
.
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
Enter details about the organization and it IT Infrastructure.
•
•
•
organization
division
organization's
organizational
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
organization .
organization
organization d
organization'
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
organization changes to the
systems, applications and organizational data can undermine the organization's
violations of federal or state mandates and laws can
lead to major . potential to impact the
organization
organization
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
organization
ISOL 533 - InfoSecurity & Risk
Management University of the Cumberlands
central respoitory accessible via the
orporate
ISOL 533 - InfoSecurity & Risk
Management Uni.
Session 04_Risk Assessment Program for YSP_Risk Analysis IMuizz Anibire
Program Objectives
In light of industrialization trends across the globe, new hazards are constantly introduced in many workplaces. This program aims to provide Young Safety Professionals (YSPs) from diverse backgrounds with the requisite skill to address the health and safety hazards in the modern workplace.
Recently, the machine learning community has expressed strong interest in applying latent variable modeling strategies to causal inference problems with unobserved confounding. Here, I discuss one of the big debates that occurred over the past year, and how we can move forward. I will focus specifically on the failure of point identification in this setting, and discuss how this can be used to design flexible sensitivity analyses that cleanly separate identified and unidentified components of the causal model.
I will discuss paradigmatic statistical models of inference and learning from high dimensional data, such as sparse PCA and the perceptron neural network, in the sub-linear sparsity regime. In this limit the underlying hidden signal, i.e., the low-rank matrix in PCA or the neural network weights, has a number of non-zero components that scales sub-linearly with the total dimension of the vector. I will provide explicit low-dimensional variational formulas for the asymptotic mutual information between the signal and the data in suitable sparse limits. In the setting of support recovery these formulas imply sharp 0-1 phase transitions for the asymptotic minimum mean-square-error (or generalization error in the neural network setting). A similar phase transition was analyzed recently in the context of sparse high-dimensional linear regression by Reeves et al.
Many different measurement techniques are used to record neural activity in the brains of different organisms, including fMRI, EEG, MEG, lightsheet microscopy and direct recordings with electrodes. Each of these measurement modes have their advantages and disadvantages concerning the resolution of the data in space and time, the directness of measurement of the neural activity and which organisms they can be applied to. For some of these modes and for some organisms, significant amounts of data are now available in large standardized open-source datasets. I will report on our efforts to apply causal discovery algorithms to, among others, fMRI data from the Human Connectome Project, and to lightsheet microscopy data from zebrafish larvae. In particular, I will focus on the challenges we have faced both in terms of the nature of the data and the computational features of the discovery algorithms, as well as the modeling of experimental interventions.
Bayesian Additive Regression Trees (BART) has been shown to be an effective framework for modeling nonlinear regression functions, with strong predictive performance in a variety of contexts. The BART prior over a regression function is defined by independent prior distributions on tree structure and leaf or end-node parameters. In observational data settings, Bayesian Causal Forests (BCF) has successfully adapted BART for estimating heterogeneous treatment effects, particularly in cases where standard methods yield biased estimates due to strong confounding.
We introduce BART with Targeted Smoothing, an extension which induces smoothness over a single covariate by replacing independent Gaussian leaf priors with smooth functions. We then introduce a new version of the Bayesian Causal Forest prior, which incorporates targeted smoothing for modeling heterogeneous treatment effects which vary smoothly over a target covariate. We demonstrate the utility of this approach by applying our model to a timely women's health and policy problem: comparing two dosing regimens for an early medical abortion protocol, where the outcome of interest is the probability of a successful early medical abortion procedure at varying gestational ages, conditional on patient covariates. We discuss the benefits of this approach in other women’s health and obstetrics modeling problems where gestational age is a typical covariate.
Difference-in-differences is a widely used evaluation strategy that draws causal inference from observational panel data. Its causal identification relies on the assumption of parallel trends, which is scale-dependent and may be questionable in some applications. A common alternative is a regression model that adjusts for the lagged dependent variable, which rests on the assumption of ignorability conditional on past outcomes. In the context of linear models, Angrist and Pischke (2009) show that the difference-in-differences and lagged-dependent-variable regression estimates have a bracketing relationship. Namely, for a true positive effect, if ignorability is correct, then mistakenly assuming parallel trends will overestimate the effect; in contrast, if the parallel trends assumption is correct, then mistakenly assuming ignorability will underestimate the effect. We show that the same bracketing relationship holds in general nonparametric (model-free) settings. We also extend the result to semiparametric estimation based on inverse probability weighting.
We develop sensitivity analyses for weak nulls in matched observational studies while allowing unit-level treatment effects to vary. In contrast to randomized experiments and paired observational studies, we show for general matched designs that over a large class of test statistics, any valid sensitivity analysis for the weak null must be unnecessarily conservative if Fisher's sharp null of no treatment effect for any individual also holds. We present a sensitivity analysis valid for the weak null, and illustrate why it is conservative if the sharp null holds through connections to inverse probability weighted estimators. An alternative procedure is presented that is asymptotically sharp if treatment effects are constant, and is valid for the weak null under additional assumptions which may be deemed reasonable by practitioners. The methods may be applied to matched observational studies constructed using any optimal without-replacement matching algorithm, allowing practitioners to assess robustness to hidden bias while allowing for treatment effect heterogeneity.
The world of health care is full of policy interventions: a state expands eligibility rules for its Medicaid program, a medical society changes its recommendations for screening frequency, a hospital implements a new care coordination program. After a policy change, we often want to know, “Did it work?” This is a causal question; we want to know whether the policy CAUSED outcomes to change. One popular way of estimating causal effects of policy interventions is a difference-in-differences study. In this controlled pre-post design, we measure the change in outcomes of people who are exposed to the new policy, comparing average outcomes before and after the policy is implemented. We contrast that change to the change over the same time period in people who were not exposed to the new policy. The differential change in the treated group’s outcomes, compared to the change in the comparison group’s outcomes, may be interpreted as the causal effect of the policy. To do so, we must assume that the comparison group’s outcome change is a good proxy for the treated group’s (counterfactual) outcome change in the absence of the policy. This conceptual simplicity and wide applicability in policy settings makes difference-in-differences an appealing study design. However, the apparent simplicity belies a thicket of conceptual, causal, and statistical complexity. In this talk, I will introduce the fundamentals of difference-in-differences studies and discuss recent innovations including key assumptions and ways to assess their plausibility, estimation, inference, and robustness checks.
We present recent advances and statistical developments for evaluating Dynamic Treatment Regimes (DTR), which allow the treatment to be dynamically tailored according to evolving subject-level data. Identification of an optimal DTR is a key component for precision medicine and personalized health care. Specific topics covered in this talk include several recent projects with robust and flexible methods developed for the above research area. We will first introduce a dynamic statistical learning method, adaptive contrast weighted learning (ACWL), which combines doubly robust semiparametric regression estimators with flexible machine learning methods. We will further develop a tree-based reinforcement learning (T-RL) method, which builds an unsupervised decision tree that maintains the nature of batch-mode reinforcement learning. Unlike ACWL, T-RL handles the optimization problem with multiple treatment comparisons directly through a purity measure constructed with augmented inverse probability weighted estimators. T-RL is robust, efficient and easy to interpret for the identification of optimal DTRs. However, ACWL seems more robust against tree-type misspecification than T-RL when the true optimal DTR is non-tree-type. At the end of this talk, we will also present a new Stochastic-Tree Search method called ST-RL for evaluating optimal DTRs.
A fundamental feature of evaluating causal health effects of air quality regulations is that air pollution moves through space, rendering health outcomes at a particular population location dependent upon regulatory actions taken at multiple, possibly distant, pollution sources. Motivated by studies of the public-health impacts of power plant regulations in the U.S., this talk introduces the novel setting of bipartite causal inference with interference, which arises when 1) treatments are defined on observational units that are distinct from those at which outcomes are measured and 2) there is interference between units in the sense that outcomes for some units depend on the treatments assigned to many other units. Interference in this setting arises due to complex exposure patterns dictated by physical-chemical atmospheric processes of pollution transport, with intervention effects framed as propagating across a bipartite network of power plants and residential zip codes. New causal estimands are introduced for the bipartite setting, along with an estimation approach based on generalized propensity scores for treatments on a network. The new methods are deployed to estimate how emission-reduction technologies implemented at coal-fired power plants causally affect health outcomes among Medicare beneficiaries in the U.S.
Laine Thomas presented information about how causal inference is being used to determine the cost/benefit of the two most common surgical surgical treatments for women - hysterectomy and myomectomy.
We provide an overview of some recent developments in machine learning tools for dynamic treatment regime discovery in precision medicine. The first development is a new off-policy reinforcement learning tool for continual learning in mobile health to enable patients with type 1 diabetes to exercise safely. The second development is a new inverse reinforcement learning tools which enables use of observational data to learn how clinicians balance competing priorities for treating depression and mania in patients with bipolar disorder. Both practical and technical challenges are discussed.
The method of differences-in-differences (DID) is widely used to estimate causal effects. The primary advantage of DID is that it can account for time-invariant bias from unobserved confounders. However, the standard DID estimator will be biased if there is an interaction between history in the after period and the groups. That is, bias will be present if an event besides the treatment occurs at the same time and affects the treated group in a differential fashion. We present a method of bounds based on DID that accounts for an unmeasured confounder that has a differential effect in the post-treatment time period. These DID bracketing bounds are simple to implement and only require partitioning the controls into two separate groups. We also develop two key extensions for DID bracketing bounds. First, we develop a new falsification test to probe the key assumption that is necessary for the bounds estimator to provide consistent estimates of the treatment effect. Next, we develop a method of sensitivity analysis that adjusts the bounds for possible bias based on differences between the treated and control units from the pretreatment period. We apply these DID bracketing bounds and the new methods we develop to an application on the effect of voter identification laws on turnout. Specifically, we focus estimating whether the enactment of voter identification laws in Georgia and Indiana had an effect on voter turnout.
We study experimental design in large-scale stochastic systems with substantial uncertainty and structured cross-unit interference. We consider the problem of a platform that seeks to optimize supply-side payments p in a centralized marketplace where different suppliers interact via their effects on the overall supply-demand equilibrium, and propose a class of local experimentation schemes that can be used to optimize these payments without perturbing the overall market equilibrium. We show that, as the system size grows, our scheme can estimate the gradient of the platform’s utility with respect to p while perturbing the overall market equilibrium by only a vanishingly small amount. We can then use these gradient estimates to optimize p via any stochastic first-order optimization method. These results stem from the insight that, while the system involves a large number of interacting units, any interference can only be channeled through a small number of key statistics, and this structure allows us to accurately predict feedback effects that arise from global system changes using only information collected while remaining in equilibrium.
We discuss a general roadmap for generating causal inference based on observational studies used to general real world evidence. We review targeted minimum loss estimation (TMLE), which provides a general template for the construction of asymptotically efficient plug-in estimators of a target estimand for realistic (i.e, infinite dimensional) statistical models. TMLE is a two stage procedure that first involves using ensemble machine learning termed super-learning to estimate the relevant stochastic relations between the treatment, censoring, covariates and outcome of interest. The super-learner allows one to fully utilize all the advances in machine learning (in addition to more conventional parametric model based estimators) to build a single most powerful ensemble machine learning algorithm. We present Highly Adaptive Lasso as an important machine learning algorithm to include.
In the second step, the TMLE involves maximizing a parametric likelihood along a so-called least favorable parametric model through the super-learner fit of the relevant stochastic relations in the observed data. This second step bridges the state of the art in machine learning to estimators of target estimands for which statistical inference is available (i.e, confidence intervals, p-values etc). We also review recent advances in collaborative TMLE in which the fit of the treatment and censoring mechanism is tailored w.r.t. performance of TMLE. We also discuss asymptotically valid bootstrap based inference. Simulations and data analyses are provided as demonstrations.
We describe different approaches for specifying models and prior distributions for estimating heterogeneous treatment effects using Bayesian nonparametric models. We make an affirmative case for direct, informative (or partially informative) prior distributions on heterogeneous treatment effects, especially when treatment effect size and treatment effect variation is small relative to other sources of variability. We also consider how to provide scientifically meaningful summaries of complicated, high-dimensional posterior distributions over heterogeneous treatment effects with appropriate measures of uncertainty.
Climate change mitigation has traditionally been analyzed as some version of a public goods game (PGG) in which a group is most successful if everybody contributes, but players are best off individually by not contributing anything (i.e., “free-riding”)—thereby creating a social dilemma. Analysis of climate change using the PGG and its variants has helped explain why global cooperation on GHG reductions is so difficult, as nations have an incentive to free-ride on the reductions of others. Rather than inspire collective action, it seems that the lack of progress in addressing the climate crisis is driving the search for a “quick fix” technological solution that circumvents the need for cooperation.
This seminar discussed ways in which to produce professional academic writing, from academic papers to research proposals or technical writing in general.
Machine learning (including deep and reinforcement learning) and blockchain are two of the most noticeable technologies in recent years. The first one is the foundation of artificial intelligence and big data, and the second one has significantly disrupted the financial industry. Both technologies are data-driven, and thus there are rapidly growing interests in integrating them for more secure and efficient data sharing and analysis. In this paper, we review the research on combining blockchain and machine learning technologies and demonstrate that they can collaborate efficiently and effectively. In the end, we point out some future directions and expect more researches on deeper integration of the two promising technologies.
In this talk, we discuss QuTrack, a Blockchain-based approach to track experiment and model changes primarily for AI and ML models. In addition, we discuss how change analytics can be used for process improvement and to enhance the model development and deployment processes.
More from The Statistical and Applied Mathematical Sciences Institute (20)
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdf
GDRR Opening Workshop - Dynamic Financial Decisions under Financial Risks - Weidong Tian, August 6, 2019
1. Dynamic Decisions under Financial Risks
Weidong Tian
University of North Carolina at Charlotte
GDRR-SAMSI Workshop, August, 2019
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 1 / 27
2. Introduction
Financial risks are pervasive
Market risk (due to moves in market factors)
Credit risk and counterparty risk (due to the credit or default to its
counterparty)
Liquidity risk (due to the illiquidity on both macro and micro-level
environment)
Operational risk (the loss rusting from inadequate or failed process,
people, system or external events)
Model risk (the adverse consequence from decisions based on incorrect
or misused model outputs and reports)
Risk category including data reporting, fair lending, fintech, financial
crime, etc.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 2 / 27
3. Introduction
Financial risks are pervasive
Market risk (due to moves in market factors)
Credit risk and counterparty risk (due to the credit or default to its
counterparty)
Liquidity risk (due to the illiquidity on both macro and micro-level
environment)
Operational risk (the loss rusting from inadequate or failed process,
people, system or external events)
Model risk (the adverse consequence from decisions based on incorrect
or misused model outputs and reports)
Risk category including data reporting, fair lending, fintech, financial
crime, etc.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 2 / 27
4. Introduction
Financial risks are pervasive
Market risk (due to moves in market factors)
Credit risk and counterparty risk (due to the credit or default to its
counterparty)
Liquidity risk (due to the illiquidity on both macro and micro-level
environment)
Operational risk (the loss rusting from inadequate or failed process,
people, system or external events)
Model risk (the adverse consequence from decisions based on incorrect
or misused model outputs and reports)
Risk category including data reporting, fair lending, fintech, financial
crime, etc.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 2 / 27
5. Introduction
Financial risks are pervasive
Market risk (due to moves in market factors)
Credit risk and counterparty risk (due to the credit or default to its
counterparty)
Liquidity risk (due to the illiquidity on both macro and micro-level
environment)
Operational risk (the loss rusting from inadequate or failed process,
people, system or external events)
Model risk (the adverse consequence from decisions based on incorrect
or misused model outputs and reports)
Risk category including data reporting, fair lending, fintech, financial
crime, etc.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 2 / 27
6. Introduction
Financial risks are pervasive
Market risk (due to moves in market factors)
Credit risk and counterparty risk (due to the credit or default to its
counterparty)
Liquidity risk (due to the illiquidity on both macro and micro-level
environment)
Operational risk (the loss rusting from inadequate or failed process,
people, system or external events)
Model risk (the adverse consequence from decisions based on incorrect
or misused model outputs and reports)
Risk category including data reporting, fair lending, fintech, financial
crime, etc.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 2 / 27
7. Introduction
Financial risks are pervasive
Market risk (due to moves in market factors)
Credit risk and counterparty risk (due to the credit or default to its
counterparty)
Liquidity risk (due to the illiquidity on both macro and micro-level
environment)
Operational risk (the loss rusting from inadequate or failed process,
people, system or external events)
Model risk (the adverse consequence from decisions based on incorrect
or misused model outputs and reports)
Risk category including data reporting, fair lending, fintech, financial
crime, etc.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 2 / 27
8. Introduction
Market risk measure from Basel
Minimal capital requirement
Value ar risk requirement
Var(p) = min {m : P{V(0) − V(∆t) ≥ m} ≤ 1 − p} , p = 0.99
Expected shortfall
ES(p) = EP
[V(0) − V(∆t)|V(0) − V(∆t) ≥ Var(p)]
Risk management constraint (Stressed VaR, stress testing), and capital
charge
BCBS, “Minimal Capital Requirements for Market Risks" (standards),
January 2016.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 3 / 27
9. Introduction
Market risk measure from Basel
Minimal capital requirement
Value ar risk requirement
Var(p) = min {m : P{V(0) − V(∆t) ≥ m} ≤ 1 − p} , p = 0.99
Expected shortfall
ES(p) = EP
[V(0) − V(∆t)|V(0) − V(∆t) ≥ Var(p)]
Risk management constraint (Stressed VaR, stress testing), and capital
charge
BCBS, “Minimal Capital Requirements for Market Risks" (standards),
January 2016.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 3 / 27
10. Introduction
Market risk measure from Basel
Minimal capital requirement
Value ar risk requirement
Var(p) = min {m : P{V(0) − V(∆t) ≥ m} ≤ 1 − p} , p = 0.99
Expected shortfall
ES(p) = EP
[V(0) − V(∆t)|V(0) − V(∆t) ≥ Var(p)]
Risk management constraint (Stressed VaR, stress testing), and capital
charge
BCBS, “Minimal Capital Requirements for Market Risks" (standards),
January 2016.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 3 / 27
11. Introduction
Market risk measure from Basel
Minimal capital requirement
Value ar risk requirement
Var(p) = min {m : P{V(0) − V(∆t) ≥ m} ≤ 1 − p} , p = 0.99
Expected shortfall
ES(p) = EP
[V(0) − V(∆t)|V(0) − V(∆t) ≥ Var(p)]
Risk management constraint (Stressed VaR, stress testing), and capital
charge
BCBS, “Minimal Capital Requirements for Market Risks" (standards),
January 2016.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 3 / 27
12. Introduction
Market risk measure from Basel
Minimal capital requirement
Value ar risk requirement
Var(p) = min {m : P{V(0) − V(∆t) ≥ m} ≤ 1 − p} , p = 0.99
Expected shortfall
ES(p) = EP
[V(0) − V(∆t)|V(0) − V(∆t) ≥ Var(p)]
Risk management constraint (Stressed VaR, stress testing), and capital
charge
BCBS, “Minimal Capital Requirements for Market Risks" (standards),
January 2016.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 3 / 27
13. Introduction
Counterparty credit risk management from Basel
Credit risk value at risk to capture the credit downgrade or default
Regulator capital (advanced internal credit model) for counterparty risk:
the effective expected exposure
Additional cost or adjustments for credit risk (XVA)
BCBS, “Margin requirements for non-centrally cleared derivatives",
September 2013; “Review of the credit valuation adjustment risk
framework", July 2015.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 4 / 27
14. Introduction
Counterparty credit risk management from Basel
Credit risk value at risk to capture the credit downgrade or default
Regulator capital (advanced internal credit model) for counterparty risk:
the effective expected exposure
Additional cost or adjustments for credit risk (XVA)
BCBS, “Margin requirements for non-centrally cleared derivatives",
September 2013; “Review of the credit valuation adjustment risk
framework", July 2015.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 4 / 27
15. Introduction
Counterparty credit risk management from Basel
Credit risk value at risk to capture the credit downgrade or default
Regulator capital (advanced internal credit model) for counterparty risk:
the effective expected exposure
Additional cost or adjustments for credit risk (XVA)
BCBS, “Margin requirements for non-centrally cleared derivatives",
September 2013; “Review of the credit valuation adjustment risk
framework", July 2015.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 4 / 27
16. Introduction
Counterparty credit risk management from Basel
Credit risk value at risk to capture the credit downgrade or default
Regulator capital (advanced internal credit model) for counterparty risk:
the effective expected exposure
Additional cost or adjustments for credit risk (XVA)
BCBS, “Margin requirements for non-centrally cleared derivatives",
September 2013; “Review of the credit valuation adjustment risk
framework", July 2015.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 4 / 27
17. Introduction
Liquidity and Operational risk management from Basel
Liquidity coverage ratio
Operational risk capital y
Prob (Loss portfolio <= y) = 0.001
BCBS, “Liquidity Coverage Ratio and liquidity risk monitoring tools",
January 2013
BCBS, “Standardised Measurement Approach for operational risk",
March 2016
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 5 / 27
18. Introduction
Liquidity and Operational risk management from Basel
Liquidity coverage ratio
Operational risk capital y
Prob (Loss portfolio <= y) = 0.001
BCBS, “Liquidity Coverage Ratio and liquidity risk monitoring tools",
January 2013
BCBS, “Standardised Measurement Approach for operational risk",
March 2016
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 5 / 27
19. Introduction
Liquidity and Operational risk management from Basel
Liquidity coverage ratio
Operational risk capital y
Prob (Loss portfolio <= y) = 0.001
BCBS, “Liquidity Coverage Ratio and liquidity risk monitoring tools",
January 2013
BCBS, “Standardised Measurement Approach for operational risk",
March 2016
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 5 / 27
20. Introduction
Liquidity and Operational risk management from Basel
Liquidity coverage ratio
Operational risk capital y
Prob (Loss portfolio <= y) = 0.001
BCBS, “Liquidity Coverage Ratio and liquidity risk monitoring tools",
January 2013
BCBS, “Standardised Measurement Approach for operational risk",
March 2016
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 5 / 27
21. Introduction
Model risk management
Black-Scholes model and 1987 Black Monday
Gaussian copula model and 2007-2008 financial crisis
Model risk capital (inherent risk, residual risk, aggregate risk)
Bayesian model average approach
The worst-case scenario approach
Federal Reserve Supervisory Bulletin 2011-7; Office of the Comptroller
of the Currency (OCC) 2011-12
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 6 / 27
22. Introduction
Model risk management
Black-Scholes model and 1987 Black Monday
Gaussian copula model and 2007-2008 financial crisis
Model risk capital (inherent risk, residual risk, aggregate risk)
Bayesian model average approach
The worst-case scenario approach
Federal Reserve Supervisory Bulletin 2011-7; Office of the Comptroller
of the Currency (OCC) 2011-12
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 6 / 27
23. Introduction
Model risk management
Black-Scholes model and 1987 Black Monday
Gaussian copula model and 2007-2008 financial crisis
Model risk capital (inherent risk, residual risk, aggregate risk)
Bayesian model average approach
The worst-case scenario approach
Federal Reserve Supervisory Bulletin 2011-7; Office of the Comptroller
of the Currency (OCC) 2011-12
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 6 / 27
24. Introduction
Model risk management
Black-Scholes model and 1987 Black Monday
Gaussian copula model and 2007-2008 financial crisis
Model risk capital (inherent risk, residual risk, aggregate risk)
Bayesian model average approach
The worst-case scenario approach
Federal Reserve Supervisory Bulletin 2011-7; Office of the Comptroller
of the Currency (OCC) 2011-12
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 6 / 27
25. Introduction
Model risk management
Black-Scholes model and 1987 Black Monday
Gaussian copula model and 2007-2008 financial crisis
Model risk capital (inherent risk, residual risk, aggregate risk)
Bayesian model average approach
The worst-case scenario approach
Federal Reserve Supervisory Bulletin 2011-7; Office of the Comptroller
of the Currency (OCC) 2011-12
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 6 / 27
26. Introduction
Model risk management
Black-Scholes model and 1987 Black Monday
Gaussian copula model and 2007-2008 financial crisis
Model risk capital (inherent risk, residual risk, aggregate risk)
Bayesian model average approach
The worst-case scenario approach
Federal Reserve Supervisory Bulletin 2011-7; Office of the Comptroller
of the Currency (OCC) 2011-12
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 6 / 27
27. Introduction
Outlines
We discuss six examples from the following topics:
Dynamic asset allocation under risk measures or capital requirement.
Dynamic asset allocation under model risk.
Asset pricing under model risk.
Asset pricing under VaR and other risk measures.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 7 / 27
28. Introduction
Outlines
We discuss six examples from the following topics:
Dynamic asset allocation under risk measures or capital requirement.
Dynamic asset allocation under model risk.
Asset pricing under model risk.
Asset pricing under VaR and other risk measures.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 7 / 27
29. Introduction
Outlines
We discuss six examples from the following topics:
Dynamic asset allocation under risk measures or capital requirement.
Dynamic asset allocation under model risk.
Asset pricing under model risk.
Asset pricing under VaR and other risk measures.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 7 / 27
30. Introduction
Outlines
We discuss six examples from the following topics:
Dynamic asset allocation under risk measures or capital requirement.
Dynamic asset allocation under model risk.
Asset pricing under model risk.
Asset pricing under VaR and other risk measures.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 7 / 27
31. Dynamic asset allocation under risk measures or capital requirement
The economy
A financial market with asset prices S1, · · · , SN
An investor’s trading (percentage of the wealth) strategy (process) is
π1, · · · , πN; and consumption rate c
The wealth process W satisfies
dW = π1W
dS1
S1
+ · · · + πNW
dSN
SN
− cdt
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 8 / 27
32. Dynamic asset allocation under risk measures or capital requirement
Objective function
Markowitz’s mean-variance setting:
max E[WT] −
A
2
Var[WT]
Merton’s dynamic portfolio choice setting:
max E
T
0
e−ρt
u(ct)dt + e−ρT
V(WT)
Roy’s safey-first setting:
max Prob {WT ≥ LT}
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 9 / 27
33. Dynamic asset allocation under risk measures or capital requirement
Constraints from the risk measure requirement
Minimal wealth requirement
WT ≥ KT
Minimal capital requirement or VaR requirement
Var(p) ≤ LT
or Expected shortfall constraint
ES(p) ≤ MT
Ratio constraints (leverage ratio, liquidity ratio etc): The position on the
risk-free asset or liquid asset is higher enough.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 10 / 27
34. Dynamic asset allocation under risk measures or capital requirement
Example 1. Mean-variance under VaR measure
In a complete financial market with unique state price density process
(ζt).
Pre-commitment optimal strategy
max
E[ζT WT ]≤W0,P(WT ≥K)≥α
E[WT] −
A
2
Var(WT)
Consider a sequence of the optimal variance problem for each x
min
E[ζT WT ]≤W0,P(WT ≥K)≥α,E[WT ]=x
E[W2
T]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 11 / 27
35. Dynamic asset allocation under risk measures or capital requirement
Example 1. Mean-variance under VaR measure
In a complete financial market with unique state price density process
(ζt).
Pre-commitment optimal strategy
max
E[ζT WT ]≤W0,P(WT ≥K)≥α
E[WT] −
A
2
Var(WT)
Consider a sequence of the optimal variance problem for each x
min
E[ζT WT ]≤W0,P(WT ≥K)≥α,E[WT ]=x
E[W2
T]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 11 / 27
36. Dynamic asset allocation under risk measures or capital requirement
Example 1. Mean-variance under VaR measure
In a complete financial market with unique state price density process
(ζt).
Pre-commitment optimal strategy
max
E[ζT WT ]≤W0,P(WT ≥K)≥α
E[WT] −
A
2
Var(WT)
Consider a sequence of the optimal variance problem for each x
min
E[ζT WT ]≤W0,P(WT ≥K)≥α,E[WT ]=x
E[W2
T]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 11 / 27
37. Dynamic asset allocation under risk measures or capital requirement
Example 1 (continued)
The corresponding unconstrained problem is to maximize
E[−W2
T] − λ1(E[ζTWT] − W0) − λ2 (α − E [1WT ≥K]) − λ3 (E[WT] − x)
The static optimization problem at each scenario is WT(λ1, λ2, λ3)
satisfying three budget constraint equations.
WT(λ1, λ2, λ3) is the optimal wealth under the VaR constraint.
Reference. Basak and Shapiro (RFS, 2001); Basak, Shapiro and Tepla
(MS, 2006); Boyle and Tian (MF, 2007)
In general, time-consistent strategy for the dynamic mean-variance
preference. See Basak and Chabakauri (RFS, 2010). A Nash equilibrium
for all shelves.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 12 / 27
38. Dynamic asset allocation under risk measures or capital requirement
Example 1 (continued)
The corresponding unconstrained problem is to maximize
E[−W2
T] − λ1(E[ζTWT] − W0) − λ2 (α − E [1WT ≥K]) − λ3 (E[WT] − x)
The static optimization problem at each scenario is WT(λ1, λ2, λ3)
satisfying three budget constraint equations.
WT(λ1, λ2, λ3) is the optimal wealth under the VaR constraint.
Reference. Basak and Shapiro (RFS, 2001); Basak, Shapiro and Tepla
(MS, 2006); Boyle and Tian (MF, 2007)
In general, time-consistent strategy for the dynamic mean-variance
preference. See Basak and Chabakauri (RFS, 2010). A Nash equilibrium
for all shelves.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 12 / 27
39. Dynamic asset allocation under risk measures or capital requirement
Example 1 (continued)
The corresponding unconstrained problem is to maximize
E[−W2
T] − λ1(E[ζTWT] − W0) − λ2 (α − E [1WT ≥K]) − λ3 (E[WT] − x)
The static optimization problem at each scenario is WT(λ1, λ2, λ3)
satisfying three budget constraint equations.
WT(λ1, λ2, λ3) is the optimal wealth under the VaR constraint.
Reference. Basak and Shapiro (RFS, 2001); Basak, Shapiro and Tepla
(MS, 2006); Boyle and Tian (MF, 2007)
In general, time-consistent strategy for the dynamic mean-variance
preference. See Basak and Chabakauri (RFS, 2010). A Nash equilibrium
for all shelves.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 12 / 27
40. Dynamic asset allocation under risk measures or capital requirement
Example 1 (continued)
The corresponding unconstrained problem is to maximize
E[−W2
T] − λ1(E[ζTWT] − W0) − λ2 (α − E [1WT ≥K]) − λ3 (E[WT] − x)
The static optimization problem at each scenario is WT(λ1, λ2, λ3)
satisfying three budget constraint equations.
WT(λ1, λ2, λ3) is the optimal wealth under the VaR constraint.
Reference. Basak and Shapiro (RFS, 2001); Basak, Shapiro and Tepla
(MS, 2006); Boyle and Tian (MF, 2007)
In general, time-consistent strategy for the dynamic mean-variance
preference. See Basak and Chabakauri (RFS, 2010). A Nash equilibrium
for all shelves.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 12 / 27
41. Dynamic asset allocation under risk measures or capital requirement
Example 1 (continued)
The corresponding unconstrained problem is to maximize
E[−W2
T] − λ1(E[ζTWT] − W0) − λ2 (α − E [1WT ≥K]) − λ3 (E[WT] − x)
The static optimization problem at each scenario is WT(λ1, λ2, λ3)
satisfying three budget constraint equations.
WT(λ1, λ2, λ3) is the optimal wealth under the VaR constraint.
Reference. Basak and Shapiro (RFS, 2001); Basak, Shapiro and Tepla
(MS, 2006); Boyle and Tian (MF, 2007)
In general, time-consistent strategy for the dynamic mean-variance
preference. See Basak and Chabakauri (RFS, 2010). A Nash equilibrium
for all shelves.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 12 / 27
42. Dynamic asset allocation under risk measures or capital requirement
Example 2. Safety-first under VaR measure
Consider the problem
max
E[ζT WT ]≤W0,P(WT ≥L)≥α
P(WT ≥ K)
The static unconstrained problem for scenario ω is
max 1WT (ω)≥K + λ1 (W0 − ζTWT) + λ2 1WT (ω)≥L − α
The optimal one is written as WT(λ1, λ2) under budget constraints.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 13 / 27
43. Dynamic asset allocation under risk measures or capital requirement
Example 2. Safety-first under VaR measure
Consider the problem
max
E[ζT WT ]≤W0,P(WT ≥L)≥α
P(WT ≥ K)
The static unconstrained problem for scenario ω is
max 1WT (ω)≥K + λ1 (W0 − ζTWT) + λ2 1WT (ω)≥L − α
The optimal one is written as WT(λ1, λ2) under budget constraints.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 13 / 27
44. Dynamic asset allocation under risk measures or capital requirement
Example 2. Safety-first under VaR measure
Consider the problem
max
E[ζT WT ]≤W0,P(WT ≥L)≥α
P(WT ≥ K)
The static unconstrained problem for scenario ω is
max 1WT (ω)≥K + λ1 (W0 − ζTWT) + λ2 1WT (ω)≥L − α
The optimal one is written as WT(λ1, λ2) under budget constraints.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 13 / 27
45. Dynamic asset allocation under risk measures or capital requirement
Example 2 (continued)
Given another feasible wealth WT,
1WT (λ1,λ2;ω)≥K + λ1(W0 − ζTWT(λ1, λ2; ω)) + λ2(1WT (λ1,λ2;ω)≥L − α
≥ 1WT (ω)≥K + λ1 (W0 − ζTWT) + λ2 1WT (ω)≥L − α
Taking expectation on both sides, we have
P(WT(λ1, λ2) ≥ K) ≥ P(WT ≥ K) + λ1 (W0 − E[ζTWT])
+λ2 (P(WT ≥ L) − α)
≥ P(WT ≥ K).
References: Browne (MS, 2000); Cvitanic and Karatzas (FS, 1992);
Follmer and Leukert (FS, 1999); Spivak and Cvitanic (AAP, 1999).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 14 / 27
46. Dynamic asset allocation under risk measures or capital requirement
Example 2 (continued)
Given another feasible wealth WT,
1WT (λ1,λ2;ω)≥K + λ1(W0 − ζTWT(λ1, λ2; ω)) + λ2(1WT (λ1,λ2;ω)≥L − α
≥ 1WT (ω)≥K + λ1 (W0 − ζTWT) + λ2 1WT (ω)≥L − α
Taking expectation on both sides, we have
P(WT(λ1, λ2) ≥ K) ≥ P(WT ≥ K) + λ1 (W0 − E[ζTWT])
+λ2 (P(WT ≥ L) − α)
≥ P(WT ≥ K).
References: Browne (MS, 2000); Cvitanic and Karatzas (FS, 1992);
Follmer and Leukert (FS, 1999); Spivak and Cvitanic (AAP, 1999).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 14 / 27
47. Dynamic asset allocation under risk measures or capital requirement
Example 2 (continued)
Given another feasible wealth WT,
1WT (λ1,λ2;ω)≥K + λ1(W0 − ζTWT(λ1, λ2; ω)) + λ2(1WT (λ1,λ2;ω)≥L − α
≥ 1WT (ω)≥K + λ1 (W0 − ζTWT) + λ2 1WT (ω)≥L − α
Taking expectation on both sides, we have
P(WT(λ1, λ2) ≥ K) ≥ P(WT ≥ K) + λ1 (W0 − E[ζTWT])
+λ2 (P(WT ≥ L) − α)
≥ P(WT ≥ K).
References: Browne (MS, 2000); Cvitanic and Karatzas (FS, 1992);
Follmer and Leukert (FS, 1999); Spivak and Cvitanic (AAP, 1999).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 14 / 27
48. Dynamic asset allocation under risk measures or capital requirement
Equilibrium under (dynamic) measures
Equilibrium under minimal capital wealth constraint. Grossman and
Zhou (JF, 1996); El Karoui, Jeanbalc-Piques and Lacoste (JEDC 2005);
Equilibrium under Value at risk measure. Jiang and Tian (JFE, 2016).
Equilibrium under liquidity constraint, leverage constraint. Detemple
and Murthy (RFS, 1996), Detemple and Serrat (RFS, 2003).
Equilibrium under margin constraint, Chabakauri (JME 2016), Rytchkov
(JF, 2014); capital requirement, Chabakauri and Han (2016); operational
risk constraint, Basak and Buffa (2016).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 15 / 27
49. Dynamic asset allocation under risk measures or capital requirement
Equilibrium under (dynamic) measures
Equilibrium under minimal capital wealth constraint. Grossman and
Zhou (JF, 1996); El Karoui, Jeanbalc-Piques and Lacoste (JEDC 2005);
Equilibrium under Value at risk measure. Jiang and Tian (JFE, 2016).
Equilibrium under liquidity constraint, leverage constraint. Detemple
and Murthy (RFS, 1996), Detemple and Serrat (RFS, 2003).
Equilibrium under margin constraint, Chabakauri (JME 2016), Rytchkov
(JF, 2014); capital requirement, Chabakauri and Han (2016); operational
risk constraint, Basak and Buffa (2016).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 15 / 27
50. Dynamic asset allocation under risk measures or capital requirement
Equilibrium under (dynamic) measures
Equilibrium under minimal capital wealth constraint. Grossman and
Zhou (JF, 1996); El Karoui, Jeanbalc-Piques and Lacoste (JEDC 2005);
Equilibrium under Value at risk measure. Jiang and Tian (JFE, 2016).
Equilibrium under liquidity constraint, leverage constraint. Detemple
and Murthy (RFS, 1996), Detemple and Serrat (RFS, 2003).
Equilibrium under margin constraint, Chabakauri (JME 2016), Rytchkov
(JF, 2014); capital requirement, Chabakauri and Han (2016); operational
risk constraint, Basak and Buffa (2016).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 15 / 27
51. Dynamic asset allocation under risk measures or capital requirement
Equilibrium under (dynamic) measures
Equilibrium under minimal capital wealth constraint. Grossman and
Zhou (JF, 1996); El Karoui, Jeanbalc-Piques and Lacoste (JEDC 2005);
Equilibrium under Value at risk measure. Jiang and Tian (JFE, 2016).
Equilibrium under liquidity constraint, leverage constraint. Detemple
and Murthy (RFS, 1996), Detemple and Serrat (RFS, 2003).
Equilibrium under margin constraint, Chabakauri (JME 2016), Rytchkov
(JF, 2014); capital requirement, Chabakauri and Han (2016); operational
risk constraint, Basak and Buffa (2016).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 15 / 27
52. Dynamic asset allocation under model risk
Example 3. Robust approach for mean-variance investor
Mean-variance objective in terms of Sharpe ratio
f(x, µ, Σ) =
x µ
√
x Σx
µ is the expected return vector, Σ is the covariance matrix.
The model risk is that we do not know (µ, Σ) precisely. We have a
confidence level that (µ, Σ) belongs to convex, compact region A.
To address the model risk, the robust approach is to maximize
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 16 / 27
53. Dynamic asset allocation under model risk
Example 3. Robust approach for mean-variance investor
Mean-variance objective in terms of Sharpe ratio
f(x, µ, Σ) =
x µ
√
x Σx
µ is the expected return vector, Σ is the covariance matrix.
The model risk is that we do not know (µ, Σ) precisely. We have a
confidence level that (µ, Σ) belongs to convex, compact region A.
To address the model risk, the robust approach is to maximize
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 16 / 27
54. Dynamic asset allocation under model risk
Example 3. Robust approach for mean-variance investor
Mean-variance objective in terms of Sharpe ratio
f(x, µ, Σ) =
x µ
√
x Σx
µ is the expected return vector, Σ is the covariance matrix.
The model risk is that we do not know (µ, Σ) precisely. We have a
confidence level that (µ, Σ) belongs to convex, compact region A.
To address the model risk, the robust approach is to maximize
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 16 / 27
55. Dynamic asset allocation under model risk
Example 3 (continued)
We apply the Sion’s theorem,
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ) = inf
(µ,Σ)∈A
max f(x, µ, Σ)
which is equivalents to
inf
(µ,Σ)∈A
µ Σµ
1/2
A zero-sum game interpretation; Parameter uncertainty
The maximin expected (Gilboa and Schmeidler) utility for ambiguity
averse agent
Reference: Kim and Boyd (SIAM J Optim, 2008); Garlappi, Uppal and
Wang (RFS, 2007); Grinblatt and Linnainman (RFS, 2011).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 17 / 27
56. Dynamic asset allocation under model risk
Example 3 (continued)
We apply the Sion’s theorem,
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ) = inf
(µ,Σ)∈A
max f(x, µ, Σ)
which is equivalents to
inf
(µ,Σ)∈A
µ Σµ
1/2
A zero-sum game interpretation; Parameter uncertainty
The maximin expected (Gilboa and Schmeidler) utility for ambiguity
averse agent
Reference: Kim and Boyd (SIAM J Optim, 2008); Garlappi, Uppal and
Wang (RFS, 2007); Grinblatt and Linnainman (RFS, 2011).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 17 / 27
57. Dynamic asset allocation under model risk
Example 3 (continued)
We apply the Sion’s theorem,
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ) = inf
(µ,Σ)∈A
max f(x, µ, Σ)
which is equivalents to
inf
(µ,Σ)∈A
µ Σµ
1/2
A zero-sum game interpretation; Parameter uncertainty
The maximin expected (Gilboa and Schmeidler) utility for ambiguity
averse agent
Reference: Kim and Boyd (SIAM J Optim, 2008); Garlappi, Uppal and
Wang (RFS, 2007); Grinblatt and Linnainman (RFS, 2011).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 17 / 27
58. Dynamic asset allocation under model risk
Example 3 (continued)
We apply the Sion’s theorem,
max
x
inf
(µ,Σ)∈A
f(x, µ, Σ) = inf
(µ,Σ)∈A
max f(x, µ, Σ)
which is equivalents to
inf
(µ,Σ)∈A
µ Σµ
1/2
A zero-sum game interpretation; Parameter uncertainty
The maximin expected (Gilboa and Schmeidler) utility for ambiguity
averse agent
Reference: Kim and Boyd (SIAM J Optim, 2008); Garlappi, Uppal and
Wang (RFS, 2007); Grinblatt and Linnainman (RFS, 2011).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 17 / 27
59. Dynamic asset allocation under model risk
Example 4. Hansen-Sargent robust approach under model
uncertainty
A standard Black-Scholes economy: constant risk-free interest rate r and
a risky asset S with lognormal return.
In Merton’s model
E
T
0
e−δt c1−A
t
1 − A
dt
The wealth process
dWt = [Wt(r + πt(µ − r)) − ct]dt + σtπtWtdZt
= µ(Wt)dt + σ(Wt)dZt
Two control variables πt, ct. The value function V(W, t) satisfies
0 = supπ,c
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 18 / 27
60. Dynamic asset allocation under model risk
Example 4. Hansen-Sargent robust approach under model
uncertainty
A standard Black-Scholes economy: constant risk-free interest rate r and
a risky asset S with lognormal return.
In Merton’s model
E
T
0
e−δt c1−A
t
1 − A
dt
The wealth process
dWt = [Wt(r + πt(µ − r)) − ct]dt + σtπtWtdZt
= µ(Wt)dt + σ(Wt)dZt
Two control variables πt, ct. The value function V(W, t) satisfies
0 = supπ,c
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 18 / 27
61. Dynamic asset allocation under model risk
Example 4. Hansen-Sargent robust approach under model
uncertainty
A standard Black-Scholes economy: constant risk-free interest rate r and
a risky asset S with lognormal return.
In Merton’s model
E
T
0
e−δt c1−A
t
1 − A
dt
The wealth process
dWt = [Wt(r + πt(µ − r)) − ct]dt + σtπtWtdZt
= µ(Wt)dt + σ(Wt)dZt
Two control variables πt, ct. The value function V(W, t) satisfies
0 = supπ,c
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 18 / 27
62. Dynamic asset allocation under model risk
Example 4. Hansen-Sargent robust approach under model
uncertainty
A standard Black-Scholes economy: constant risk-free interest rate r and
a risky asset S with lognormal return.
In Merton’s model
E
T
0
e−δt c1−A
t
1 − A
dt
The wealth process
dWt = [Wt(r + πt(µ − r)) − ct]dt + σtπtWtdZt
= µ(Wt)dt + σ(Wt)dZt
Two control variables πt, ct. The value function V(W, t) satisfies
0 = supπ,c
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 18 / 27
63. Dynamic asset allocation under model risk
Example 4 (continued)
D(π,c)
V(W, t) =
dE[V]
dt
= Vw[W(r + π(µ − r)) − c] + Vt +
1
2
Vwwπ2
σ2
W2
What if the investor has concerns about the model of the wealth dWt?
The agent accepts it as a “reference model" but it might be “model
mispecification". Then the agent considers alternative models and the
agent guard against an adverse alternative model because of model
uncertainty concern.
Alternative models
dWt = µ(Wt)dt + σ(Wt)[σ(Wt)u(Wt)dt + dZt]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 19 / 27
64. Dynamic asset allocation under model risk
Example 4 (continued)
D(π,c)
V(W, t) =
dE[V]
dt
= Vw[W(r + π(µ − r)) − c] + Vt +
1
2
Vwwπ2
σ2
W2
What if the investor has concerns about the model of the wealth dWt?
The agent accepts it as a “reference model" but it might be “model
mispecification". Then the agent considers alternative models and the
agent guard against an adverse alternative model because of model
uncertainty concern.
Alternative models
dWt = µ(Wt)dt + σ(Wt)[σ(Wt)u(Wt)dt + dZt]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 19 / 27
65. Dynamic asset allocation under model risk
Example 4 (continued)
D(π,c)
V(W, t) =
dE[V]
dt
= Vw[W(r + π(µ − r)) − c] + Vt +
1
2
Vwwπ2
σ2
W2
What if the investor has concerns about the model of the wealth dWt?
The agent accepts it as a “reference model" but it might be “model
mispecification". Then the agent considers alternative models and the
agent guard against an adverse alternative model because of model
uncertainty concern.
Alternative models
dWt = µ(Wt)dt + σ(Wt)[σ(Wt)u(Wt)dt + dZt]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 19 / 27
66. Dynamic asset allocation under model risk
Example 4 (continued)
D(π,c)
V(W, t) =
dE[V]
dt
= Vw[W(r + π(µ − r)) − c] + Vt +
1
2
Vwwπ2
σ2
W2
What if the investor has concerns about the model of the wealth dWt?
The agent accepts it as a “reference model" but it might be “model
mispecification". Then the agent considers alternative models and the
agent guard against an adverse alternative model because of model
uncertainty concern.
Alternative models
dWt = µ(Wt)dt + σ(Wt)[σ(Wt)u(Wt)dt + dZt]
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 19 / 27
67. Dynamic asset allocation under model risk
Example 4 (continued)
The agent chooses the adjustment u(Wt) to minimize the expected payoff
but adjusted to reflect an entropy penalty (penalty control term on model
mispecification)
inf
u
DV + u(Wt)σ(Wt)2
Vw +
1
2θ
u(Wt)2
σ(Wt)2
The parameter θ ≥ 0 measures the strength of the reference model for
robustness (θ = 0 corresponds to Merton’s model). θ = θ(Wt, t).
The equation is
0 = supπ,c inf
u
[
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
+Vwπ2
σ2
W2
u +
1
2θ(W, t)
π2
σ2
W2
u2
].
Reference: Hansen and Sargent (AER 2001); Maenhout (RFS, 2004);
Uppal and Wang (JF, 2003)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 20 / 27
68. Dynamic asset allocation under model risk
Example 4 (continued)
The agent chooses the adjustment u(Wt) to minimize the expected payoff
but adjusted to reflect an entropy penalty (penalty control term on model
mispecification)
inf
u
DV + u(Wt)σ(Wt)2
Vw +
1
2θ
u(Wt)2
σ(Wt)2
The parameter θ ≥ 0 measures the strength of the reference model for
robustness (θ = 0 corresponds to Merton’s model). θ = θ(Wt, t).
The equation is
0 = supπ,c inf
u
[
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
+Vwπ2
σ2
W2
u +
1
2θ(W, t)
π2
σ2
W2
u2
].
Reference: Hansen and Sargent (AER 2001); Maenhout (RFS, 2004);
Uppal and Wang (JF, 2003)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 20 / 27
69. Dynamic asset allocation under model risk
Example 4 (continued)
The agent chooses the adjustment u(Wt) to minimize the expected payoff
but adjusted to reflect an entropy penalty (penalty control term on model
mispecification)
inf
u
DV + u(Wt)σ(Wt)2
Vw +
1
2θ
u(Wt)2
σ(Wt)2
The parameter θ ≥ 0 measures the strength of the reference model for
robustness (θ = 0 corresponds to Merton’s model). θ = θ(Wt, t).
The equation is
0 = supπ,c inf
u
[
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
+Vwπ2
σ2
W2
u +
1
2θ(W, t)
π2
σ2
W2
u2
].
Reference: Hansen and Sargent (AER 2001); Maenhout (RFS, 2004);
Uppal and Wang (JF, 2003)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 20 / 27
70. Dynamic asset allocation under model risk
Example 4 (continued)
The agent chooses the adjustment u(Wt) to minimize the expected payoff
but adjusted to reflect an entropy penalty (penalty control term on model
mispecification)
inf
u
DV + u(Wt)σ(Wt)2
Vw +
1
2θ
u(Wt)2
σ(Wt)2
The parameter θ ≥ 0 measures the strength of the reference model for
robustness (θ = 0 corresponds to Merton’s model). θ = θ(Wt, t).
The equation is
0 = supπ,c inf
u
[
c1−A
t
1 − A
− δV(W, t) + D(π,c)
V(W, t)
+Vwπ2
σ2
W2
u +
1
2θ(W, t)
π2
σ2
W2
u2
].
Reference: Hansen and Sargent (AER 2001); Maenhout (RFS, 2004);
Uppal and Wang (JF, 2003)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 20 / 27
71. Asset pricing under model risk
Asset Pricing under No-arbitrage approach
(Fundamental theorem of asset pricing) A financial market is absence of
arbitrage if and only if there exists one equivalent martingale measure.
Traded assets S1, · · · , SN, one numeaire asset B (which is always
positive). Q is an equivalent martingale measure if {Si
B } is a martingale
under Q for each i = 1, · · · , N.
Reference: Delbaen and Schachermayer, “The Mathematics of
Arbitrage, Springer Finance, 2006.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 21 / 27
72. Asset pricing under model risk
Asset Pricing under model uncertainty
Consider the economy with N traded assets i = 1, · · · , N and one
risk-free asset (as a numeaire) B,.
The agent has several models about the risky asset, say Sα
i (t)
representing the asset i’s price at time t in model α ∈ A.
How to compute the “right" price of a derivative X under this model
uncertainty?
What is arbitrage under model uncertainty? A trading strategy is
arbitrage if this strategy yields “arbitrage" in each model since the agent
is not certain which model is a right model.
Replication principle: It holds in all feasible models at the same time.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 22 / 27
73. Asset pricing under model risk
Example 5. Asset Pricing under model uncertainty
An equivalent martingale measure in this setting is one average of
measures under each feasible model (model average principle).
The market is free of arbitrage under model uncertainty if there exists
one such equivalent martingale measure
All available price of X are bounded by infQ α EQα [Xα/B] and
supQ α EQα [Xα/B].
The model risk measure can be measured by the difference of these
bounds.
Reference: Jiang and Tian (IRF, 2017); Cont (MF, 2006); Beissner (ET,
2017); Beissner and Riedel (Econometrica, 2019).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 23 / 27
74. Asset pricing under model risk
Example 5. Asset Pricing under model uncertainty
An equivalent martingale measure in this setting is one average of
measures under each feasible model (model average principle).
The market is free of arbitrage under model uncertainty if there exists
one such equivalent martingale measure
All available price of X are bounded by infQ α EQα [Xα/B] and
supQ α EQα [Xα/B].
The model risk measure can be measured by the difference of these
bounds.
Reference: Jiang and Tian (IRF, 2017); Cont (MF, 2006); Beissner (ET,
2017); Beissner and Riedel (Econometrica, 2019).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 23 / 27
75. Asset pricing under model risk
Example 5. Asset Pricing under model uncertainty
An equivalent martingale measure in this setting is one average of
measures under each feasible model (model average principle).
The market is free of arbitrage under model uncertainty if there exists
one such equivalent martingale measure
All available price of X are bounded by infQ α EQα [Xα/B] and
supQ α EQα [Xα/B].
The model risk measure can be measured by the difference of these
bounds.
Reference: Jiang and Tian (IRF, 2017); Cont (MF, 2006); Beissner (ET,
2017); Beissner and Riedel (Econometrica, 2019).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 23 / 27
76. Asset pricing under model risk
Example 5. Asset Pricing under model uncertainty
An equivalent martingale measure in this setting is one average of
measures under each feasible model (model average principle).
The market is free of arbitrage under model uncertainty if there exists
one such equivalent martingale measure
All available price of X are bounded by infQ α EQα [Xα/B] and
supQ α EQα [Xα/B].
The model risk measure can be measured by the difference of these
bounds.
Reference: Jiang and Tian (IRF, 2017); Cont (MF, 2006); Beissner (ET,
2017); Beissner and Riedel (Econometrica, 2019).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 23 / 27
77. Asset pricing under model risk
Example 5. Asset Pricing under model uncertainty
An equivalent martingale measure in this setting is one average of
measures under each feasible model (model average principle).
The market is free of arbitrage under model uncertainty if there exists
one such equivalent martingale measure
All available price of X are bounded by infQ α EQα [Xα/B] and
supQ α EQα [Xα/B].
The model risk measure can be measured by the difference of these
bounds.
Reference: Jiang and Tian (IRF, 2017); Cont (MF, 2006); Beissner (ET,
2017); Beissner and Riedel (Econometrica, 2019).
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 23 / 27
78. Asset pricing under constraint and VaR measures
Asset Pricing under “Convex-type" constraint
One-period economy with finite nature of states Ω = {ω1, · · · , ωK} and
a subjective probability P
N assets S1(ω), · · · , SN(ω), ω ∈ Ω, with time zero price
S1(0), · · · , SN(0). One asset is always positive (for instance, the first
asset).
One investor’s trading strategy H1, · · · , HN.
By a convex-type constraint we mean the range of the trading strategy
belongs to a “convex" subset of RN.
Fundamental theorem of asset pricing under convex-type constraint.
No-arbitrage price of a general contingent claim X by all “feasible
trading strategies" (short-sell, capital requirement, leverage, margin,
transaction-cost, etc).
References: Jouini and Kallal (MF, 1995, JET 1996); Garleanu and
Pedersen (RFS, 2011)
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 24 / 27
79. Asset pricing under constraint and VaR measures
Example 6. A no-arbitrage problem under VaR measure
The investor’s loss portfolio is
V0 − V1(ω) =
N
i=1
Hi (Si(0) − Si(ω))
The investor’s admissible strategy satisfies
Prob {ω : V1(ω) − V0(ω) ≥ K} ≥ 95%
H is an arbitrage if V1 − V0 ≥ 0, and E[V1 − V0] > 0.
What is the fundamental theorem in the presence of VaR constraint?
Characterization of no-arbitrage and feasible trading strategy in this
market.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 25 / 27
80. Asset pricing under constraint and VaR measures
Example 6. A no-arbitrage problem under VaR measure
The investor’s loss portfolio is
V0 − V1(ω) =
N
i=1
Hi (Si(0) − Si(ω))
The investor’s admissible strategy satisfies
Prob {ω : V1(ω) − V0(ω) ≥ K} ≥ 95%
H is an arbitrage if V1 − V0 ≥ 0, and E[V1 − V0] > 0.
What is the fundamental theorem in the presence of VaR constraint?
Characterization of no-arbitrage and feasible trading strategy in this
market.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 25 / 27
81. Asset pricing under constraint and VaR measures
Example 6. A no-arbitrage problem under VaR measure
The investor’s loss portfolio is
V0 − V1(ω) =
N
i=1
Hi (Si(0) − Si(ω))
The investor’s admissible strategy satisfies
Prob {ω : V1(ω) − V0(ω) ≥ K} ≥ 95%
H is an arbitrage if V1 − V0 ≥ 0, and E[V1 − V0] > 0.
What is the fundamental theorem in the presence of VaR constraint?
Characterization of no-arbitrage and feasible trading strategy in this
market.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 25 / 27
82. Asset pricing under constraint and VaR measures
Example 6. A no-arbitrage problem under VaR measure
The investor’s loss portfolio is
V0 − V1(ω) =
N
i=1
Hi (Si(0) − Si(ω))
The investor’s admissible strategy satisfies
Prob {ω : V1(ω) − V0(ω) ≥ K} ≥ 95%
H is an arbitrage if V1 − V0 ≥ 0, and E[V1 − V0] > 0.
What is the fundamental theorem in the presence of VaR constraint?
Characterization of no-arbitrage and feasible trading strategy in this
market.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 25 / 27
83. Asset pricing under constraint and VaR measures
Example 6 (continued)
In a dynamic model with assets Si(t), t = 1, · · · , T. For simplicity we
assume that one risk-free asset B (money market account)
In the absence of VaR constraint, there is no arbitrage if and only if there
exists an equivalent martingale measure Q.
The feasible trading strategy satisfies, at each time t, the conditional
probability that Vt − Vt+1 that across a VaR limit is smaller than 5%.
The characterization of no-arbitrage feasible trading strategy.
The feasible trading strategy in terms of other risk measures, ratio
requirement, or capital requirement.
Challenge: Non-convex issue in the optimization problem
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 26 / 27
84. Asset pricing under constraint and VaR measures
Example 6 (continued)
In a dynamic model with assets Si(t), t = 1, · · · , T. For simplicity we
assume that one risk-free asset B (money market account)
In the absence of VaR constraint, there is no arbitrage if and only if there
exists an equivalent martingale measure Q.
The feasible trading strategy satisfies, at each time t, the conditional
probability that Vt − Vt+1 that across a VaR limit is smaller than 5%.
The characterization of no-arbitrage feasible trading strategy.
The feasible trading strategy in terms of other risk measures, ratio
requirement, or capital requirement.
Challenge: Non-convex issue in the optimization problem
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 26 / 27
85. Asset pricing under constraint and VaR measures
Example 6 (continued)
In a dynamic model with assets Si(t), t = 1, · · · , T. For simplicity we
assume that one risk-free asset B (money market account)
In the absence of VaR constraint, there is no arbitrage if and only if there
exists an equivalent martingale measure Q.
The feasible trading strategy satisfies, at each time t, the conditional
probability that Vt − Vt+1 that across a VaR limit is smaller than 5%.
The characterization of no-arbitrage feasible trading strategy.
The feasible trading strategy in terms of other risk measures, ratio
requirement, or capital requirement.
Challenge: Non-convex issue in the optimization problem
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 26 / 27
86. Asset pricing under constraint and VaR measures
Example 6 (continued)
In a dynamic model with assets Si(t), t = 1, · · · , T. For simplicity we
assume that one risk-free asset B (money market account)
In the absence of VaR constraint, there is no arbitrage if and only if there
exists an equivalent martingale measure Q.
The feasible trading strategy satisfies, at each time t, the conditional
probability that Vt − Vt+1 that across a VaR limit is smaller than 5%.
The characterization of no-arbitrage feasible trading strategy.
The feasible trading strategy in terms of other risk measures, ratio
requirement, or capital requirement.
Challenge: Non-convex issue in the optimization problem
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 26 / 27
87. Asset pricing under constraint and VaR measures
Example 6 (continued)
In a dynamic model with assets Si(t), t = 1, · · · , T. For simplicity we
assume that one risk-free asset B (money market account)
In the absence of VaR constraint, there is no arbitrage if and only if there
exists an equivalent martingale measure Q.
The feasible trading strategy satisfies, at each time t, the conditional
probability that Vt − Vt+1 that across a VaR limit is smaller than 5%.
The characterization of no-arbitrage feasible trading strategy.
The feasible trading strategy in terms of other risk measures, ratio
requirement, or capital requirement.
Challenge: Non-convex issue in the optimization problem
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 26 / 27
88. Asset pricing under constraint and VaR measures
Example 6 (continued)
In a dynamic model with assets Si(t), t = 1, · · · , T. For simplicity we
assume that one risk-free asset B (money market account)
In the absence of VaR constraint, there is no arbitrage if and only if there
exists an equivalent martingale measure Q.
The feasible trading strategy satisfies, at each time t, the conditional
probability that Vt − Vt+1 that across a VaR limit is smaller than 5%.
The characterization of no-arbitrage feasible trading strategy.
The feasible trading strategy in terms of other risk measures, ratio
requirement, or capital requirement.
Challenge: Non-convex issue in the optimization problem
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 26 / 27
89. Conclude
Conclusion
Financial risks are modelled in a stochastic setting.
Financial risk management measures for risks
Dynamic asset allocations under risk control
No-arbitrage asset pricing under risk control
There are more challenge than what we know in both theory and practice.
Weidong Tian University of North Carolina at Charlotte Dynamic Decisions GDRR-SAMSI 27 / 27