Transcript Slide 1

Stochastic Programming Applications in Finance
Hercules Vladimirou
Center for Banking and Financial Research
University of Cyprus
SP-XI Tutorial
Vienna, 08/2007
Outline:
• SP models:
– Basic features, Fundamental components, Modelling flexibility Advantages
– Important modelling choices:
• Dynamic structure (length of planning horizon, stages, scenario tree
structure & size)
• Alternative objective functions, Risk measures
• Representation of uncertainty (Scenario generation)
• Features giving rise to MIPs
– Sample application domains in Finance
Case Study: International Portfolio Management
– Criticisms (Capabilities, Limitations, Practicality, Challenges)
– Issues: Model development, Solution alternatives
– Current trends (driving forces & positive signs)
– Future potential – Interesting prospects of practical importance
Stochastic Programs: Fundamental Characteristics
• Powerful & flexible framework to support sequential decisionmaking under uncertainty (discrete-time models).
• Random variables assumed to evolve according to discrete
stochastic processes (represented in terms of scenario trees).
• Capture the interrelationship/interaction between dynamic
information processes and dynamic/active decision processes.
• (Recourse) variables allow adaptations of decisions to
information flows.
• Account for and reveal/quantify the value of decision flexibility.
• Fundamental Assumptions:
– Underlying stochastic processes (distributions of r.v.s) are not
influenced/dependent by the values of the decision variables.
– Non-anticipativity: Decisions adapt to available information at the
time they are made, but do not depend (invariant wrt) on specific
projected future outcomes (no hindsight).
Fundamental Components of SP models:
1.
2.
Description of the underlying (multi-variate) discrete
stochastic process for the uncertain parameters;
Dynamic information structure (Scenario-tree Generation).
Discrete-time dynamic (multi-period) optimization program
capturing the structure of the decision process.
(1) & (2) often cannot be considered independently.
Linking:
3. Mapping (1) & (2) in a logically conformable way.
4. Defining appropriate performance & risk measures for the
decision problem under uncertainty.
Distinguishing Features of SPs:
• Deterministic dynamic optimization models:
– Reflect the decision process
– Consider its dynamics but not that of the information flow
(i.e., the times at which the decisions take effect, but not the times at
which decisions need to be made and the information available then
on which they can be based)
– Suffer from “tunnel vision”: determine optimal decisions under a
specific (deterministic) circumstance – assume perfect foresight
– SA of limited usefulness to assess the impact of uncertainty in inputs
(only when problem structure remains unchanged; which is not the
case with SP models)
• Dynamic (multi-stage) SPs:
– Reconcile the sequential decision process with the information
process – Capture their connection
– State-contingent decisions
– Reflect/explore/exploit the value of decision flexibility
– Incorporate risk measures and aversion/tolerance to risk
Why Dynamic (multi-stage) Models?
– Reflect/explore/exploit/capture the value of decision
flexibility in “long” horizon models
• Capitalize gains
• Dynamically revise decisions and risk exposures as appropriate
(e.g., to meet targets, respond to wealth/risk outcomes)
– Market timing
• State-contingent decisions responding to short-term market
movements
• Instruments with multi-period maturity (term): bonds, derivatives
• Exogenous cashflows (e.g., time dependent contributions,
liabilities, consumption)
– Consider time/state dependent outcomes
(e.g., wealth-dependent objective, state-dependent risk measures)
– Properly account (& reduce) for the effect of transaction
costs, taxes, etc. (“amortization” of effects)
– Stochastic dynamic programs:
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Big (exponential growth in # stages and states)
Size poses computational challenges – also interpretation/insights
General & Flexible (can suit many practical situations)
Structured (exhibit special structural forms that can be
computationally exploited in numerical solution)
– Exploitation of block structures & sparsity patterns
» Some efficiencies
– Exploitation of similarity of constituent components (subproblems)
» Higher benefits
• Other challenges:
– Consistency/coherence of objectives
– Model consistency (dynamic pricing (no-arbitrage), sufficient
coverage of possible events incl. extremes)
Potential of SP Applications in Finance
(Good Omens)
• Wide availability of challenging problems
• Rich theoretical background governing the problems
• Rich tradition of sophisticated quantitative & probabilistic
models in the field
• Sophistication of stakeholders – receptiveness to
innovative approaches/models (influence of
competitiveness).
• Availability of computing and networking resources & IT
sophistication (infrastructure)
• High-stakes (measurable/estimable financial
consequences of actions)
• Availability of real-time data to enable practical
application/adoption of models
• Availability of extensive data warehouses to support
model development, calibration, empirical validation
• Availability of alternative approaches for benchmarking
purposes & comparative assessment.
Basic Common Features
• Stage- (node-) wise Fundamental Constraints
– Asset balance
– Cash-flow balance
– Valuation equations (wealth state)
– Additional application-specific constraints:
• E.g., regulatory & managerial requirements
Advantages – Modeling Flexibility
• Can handle multi-asset problems
• Determine optimal positions at individual asset level – not just
by broad class –actively leverage assets to enhance net worth
• Modelling uncertainty (scenario trees)
– Not restricted by specific distributional assumptions
– Can capture general, discrete distributions for multi-variate r.v.s
and inter-temporal dependencies
– Admit multiple alternative scenario generation procedures
• Can handle “Imperfections”: (practical issues)
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Transaction costs (usually linear wrt transaction magnitude)
Legal/regulatory requirements
) esp. as linear
Institutional/managerial requirements
) constraints
Turnover & portfolio structure constraints )
Taxes
• Can flexibly use alternative risk measures or performance
objectives (e.g., utility functions, goals, benchmarks)
Advantages – Modeling Flexibility
• Consider longer time horizon
• Encompass explicit decision dynamics – state-dependent portfolio
rebalancing at multiple points in time
• Consequences of portfolio compositions in subsequent periods are
directly considered
• Potential for subsequent portfolio rebalancing and evolution of
uncertainty in later periods accounted for in decisions of earlier
periods
• Avoid myopic reactions:
– Manifested in lower portfolio turnover – improved stability
– Generally improved diversification
• Determine dynamic contingency decisions under changing
economic factors
• Account for exogenous cash-flows at future periods (contributions or
liabilities)
• Accommodate
– multi-period investment instruments (e.g., bonds, CDs)
– decisions taken in one time period and effected in subsequent period(s)
(e.g., forwards, futures, options)
Alternative Objectives:
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Max expected terminal value/return (Risk neutral)
– Supplemented with (piecewise linear) penalties or (ad-hoc) bounding constraints
to reflect risk aversion preferences
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Min cost of initial investment (conservative)
Max expected utility of terminal wealth (NLP)
Min tracking error against goal or benchmark; one- or two- sided error
measures – Deterministic & probabilistic benchmarks (e.g., index)
Regret models (avg., min-max) against goals or benchmarks
Probability of beating benchmark (coherence issue)
Mean-risk models (tradeoffs of dual criteria, parametric models)
– Max exp. portfolio return, min a measure of risk
– Either composite objective function, or one criterion relegated to parametric
constraint
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Common Risk Measures:
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Variance/Std. Dev.
Mean Absolute Deviation (MAD)
One-sided (downside) variants
Value-at-Risk (VaR) – max loss with given confidence (percentile measure)
Min max loss / drawdown
Coherent risk measures (Artzner, Delbaen, Eber, Heath, Math. Fin., 1999)
• Conditional-Value-at-Risk (CVaR) – “Exp. Excess shortfall”
• Return-at-Risk (RaR)
• Weighted combinations of measures (CVaR at different %)
Alternative Objectives:
• Choice of objective function or risk measure
affects problem form
– LP or NLP
– R. Mansini, W. Ogryczak, M.G. Speranza
(ANOR, 2007 and elsewehere) review LP-solvable model forms
• Axiomatic characterization of
Coherent & rational risk measures:
(Artzner, Delbaen, Eber, Heath, Math. Fin. 1999)
– Monotonicity, Positive homogeneity, Translation
invariance, Subadditivity
Issues of Consistency & Choice:
• Traditional Approaches:
– Expected Utility
– Probabilistic (Chance) Constraints
• New Approaches:
– Mean-Risk Models: (Coherent) risk measures
– Robust Optimization
• Issues:
– All consistent? Where is each more suitable?
– Consensus on “best” choices?
– How to choose? Criteria?
Issues of Consistency & Choice (resolutions)
Need appropriate theoretical framework
formalizing connections/similarities/distinctions
of model variants (incl. various forms of risk
measures)
• Relation of Mean-Risk Models with Stochastic
Dominance
• A. Ruszczynski, W. Ogryczak, D. Dentcheva
• Connection of Mean-Risk Models with Uncertainty
Sets in RO (based on duality analysis)
• D. Bertsimas, M. Sim, D. Brown
Important Modeling Choices:
• Length of planning horizon
• # and division of decision stages
• Scenario generation (procedure, risk factors
captured, branching factor, size of scenario tree)
– Co-variation of r.v.s & dynamic effects
• Objective function & risk measures
• Application specific requirements (constraints)
• Preservation of convexity is important
(linearity preferable, at least for constraints)
Representation of Uncertainty
– Scenario Generation
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Review (Consigli, Dupacova, Wallace, Annals of OR, 2000).
Bootstrapping historical data
Subjective scenarios
Linking scenarios with macro-economic models
Interest rate models – typically multinomial lattices (depending on #
of factors)
• Factor models (e.g., Principal Component Analysis)
– Reduction of dimensionality to uncorrelated factors while capturing
correlations of r.v.s)
– Typically lead to multinomial trees/lattices
– Exponential growth of tree/lattice with # of stages
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Moment Matching
Sampling (Importance, Antithetic, Directed)
Discretization of Continuous Distributions/Stochastic Processes
Econometric Approaches
– Vector Autoregressive Models, VAR (building long-term dynamics from
short-term/lag impacts)
– Vector Equilibrium Correction Models, VeqC (capturing convergence to
long-term equilibrium conditions)
• Hybrid models
Pricing Assets and Optimizing Decisions on Lattice Structures
Source: S.A. Zenios, et al., JEDC, 1998.
SP Applications in Finance (some examples)
• Asset Allocation (typically single decision stage, passive):
– Broad asset classes
• Active Asset Portfolio (Fund) Management:
– Equities (and other broad classes, indices)
– Index tracking
– International portfolios
• Asset-Liability Management:
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Fixed-income portfolios
Insurance (funding contracts, e.g., products with guarantees)
Pension funds
Public debt management
Personal portfolios
• Risk Management:
– Market, Credit, Operational, Liquidity risks
• Design of financial products
– Callable bonds, Options
Features giving rise to MIPs
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Cardinality constraints
Fixed costs
Indivisible decisions (e.g., security issuing)
Transaction lots (lot sizing)
Restricting successive occurrences of events
Chance constraints
– Probabilistic “quality/reliability” considerations
– Percentile risk measures (e.g., VaR)
Realities:
• Despite significant advancements in recent years:
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In algebraic modelling systems: facilitating model development
In computational capabilities: powerful computing technology
In algorithmic developments
Proliferation of model prototypes for various applications with empirically
demonstrated superior potential (research studies, practical implementations)
Adoptions of dynamic SP models in practical financial
applications still not as widespread as should be expected!
– Views of financial analysts wrt SP models:
• Interesting, potentially useful, nice theory, … BUT …
• Difficult to develop, implement, solve
• Extensive data needs
• Non-standardized structures/features (constraints, objective functions, risk
measures, representation of uncertainty)
• Difficult to interpret the results in terms of familiar concepts (utility terms,
downside risks, etc.) and understand their behaviour
• “Black-box” reservations
Perceptions/View of Practitioners:
• B. Riley (columnist), article on dynamic ALM models,
Financial Times, Dec. 2, 2002.
– “Practitioners view dynamic models as ‘excessively complex’ ”
– Difficulties with “explaining the downside risk of even an optimal
solution”
– “Consensus is that much more work remains to be done [before
widespread adoption]”
– Banks have used advanced financial models for pricing
derivatives
– Also reports of static LP-based models used in the banking
sector
– Industry emphasis on managing downside risk. Simulation
models hold sway in industry emphasizing scenario generation
and reporting.
Criticisms:
• M.S. Sodhi, Operations Research, 53(2), 2005.
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Complexity in development
Practicality of implementations
Problem customization issues
Computational/solution complexity
Difficulty in understanding solution behaviour and
interpreting/defending results in terms of common
financial concepts
– Proliferation of models in the literature – Plethora of
alternatives:
• Differing in treatment of uncertainty (risk factors, sampling
procedure, dynamics)
• Differing in structure, risk measures, constraints
• Differing in use of market or model prices for assets
• Lack of consistency among model alternatives
• No clear guidelines for important features, modelling choices
and their comparative value
• Lack of standardization and consensus
Criticisms:
(cont.)
• Major criticism:
– Scenario representations not consistent with
financial fundamentals and market prices
– Non-compliance with no-arbitrage conditions –
(Scenario/model) asset price sets must not
admit “free lunches”
– Unbounded problems or spurious profits
– Sometimes ignored or undetected due to
constraint structures (e.g., diversification
constraints or bounds)
– Internal consistency: Arbitrage-free asset price
scenarios
– External consistency: Model prices must closely
approximate market prices of assets
Criticisms:
(resolutions)
• Verifying absence of arbitrage
– Klaassen, EJOR, 1997; Man. Sci., 1998.
• Conditions for generating arbitrage-free price scenarios
• Procedure for scenario-tree reduction while maintaining
arbitrage-free conditions (aggregating nodes or stages)
• Absence of “free-lunch” => Existence of riskneutral (martingale) measure
– (Harrison & Kreps, 1979)
• Determining risk-neutral probabilities on discrete
distributions (duality analysis)
– A.J. King, Math. Progr. 2001.
• Issue of uniqueness of risk-neutral measure
(market completeness) – How to accommodate
non-unique solutions?
Criticisms:
(cont.)
• Scenario Reductions:
– Typically performed for reasons of practicality – Reduction in
problem size/complexity and computational effort
– Can “destroy” important characteristics of information process,
e.g.,
• May introduce arbitrage opportunities
• May affect other statistical properties (moments, dynamic features)
• Other simplifications:
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Collapsing/aggregating scenarios
Aggregating decision stages – reducing planning horizon
Simplifying decision process
Hybrid models
Simplifications can have important impacts on problem structure
and properties of solutions
– De Lange, Gaivoronski, Annals OR, 1999: preferable to simplify
the decision process rather than the information process
(scenario sets)
Market for Packaged ALM s/w:
Survey of 400 largest financial institutions
Expenditures for packaged ALM s/w
1998
$613 million (O’Connell, 1999)
1999-2003 slowdown
Projections:
2004
$600 million ) (Keppler, 2004)
2009
$850 million )
Global breakdown of spending: (Keppler, 2004)
79%
banking sector
4%
insurance sector
24%
pension funds
Still,
• Mostly simulation-based tools
• Static models & optimal control (e.g., Campbell & Viceira)
• Share by Algorithmics, Inc.
Prospects of SP Applications: (Opportunities)
• Risk management currently of central importance for financial
institutions:
– Realization of potential financial impacts
) Risk management
– Examples of spectacular failures
) is big business
– Requirements of reshaped regulatory framework (Basel II, 2003/41/EC, IAS,
Sarbanes-Oaxley)
– Intensifying competition
– Impact of financial innovations & new complex financial instruments
• Increasing demand for risk professionals:
– very good career prospects
– increasing number of specialized graduate programs
• (Financial Engineering, Financial Mathematics, Quantitative Finance)
– scarcity of combinations of skills
• Solid mathematical background (& probability/statistics)
• Solid understanding of financial theory (incl. operation of markets, financial
instruments - derivatives)
• Understanding of modern risk management tools
• Data analysis & computing skills
– Increasing influence (& community) of professional risk associations:
• GARP, PRMIA, CFA (offering professional exams & certification)
– Risk Management does not mean eliminating risk – Optimal leveraging
of resources to enhance net worth – Prudence rules
Driving Forces/Opportunities:
(Banking and Risk Management)
• Basel II Accord
– Bank for International Settlement (BIS)
– “Consensus” regulatory framework - Governing risk
measurement, monitoring, management functions of commercial
banks
– Affects reporting requirements and roles of
supervisory/regulatory bodies (Central Banks)
– Endorsed by EU institutions
– Banks must comply in 2007Q1
– Banks need to adopt or internally develop risk
measurement/management models – Competitive & Business
necessity – Opportunity cost from capital adequacy burden
– Current emphasis mostly on credit risk (most important aspect of
bank operations)
– Needs of regulating/supervisory bodies (Central Banks) to
develop capabilities for model validation/assessment and
benchmarking
Driving Forces/Opportunities:
(Pension Funds)
• EU Directive 2003/41/EC (European Parliament & Council)
– Governing activities and supervision of institutions for
occupational retirement provision
– Specifying changes in governance of Pension Funds
– Effects on fund management practices:
• Requires articulation/defence of investment strategies (plans,
objectives, risk levels)
• Monitoring, periodic assessment of investment strategies and
reporting
– Far-reaching socio-economic effects of challenges to pension
funds
– Balancing (conflicting) interests of multiple stakeholders
– Also influence of:
• OECD Recommendations on core principles of pension regulation
and Guidelines for fund governance
• International Social Security Association (ISSA) Guidelines for the
investment of social security funds
• Varying national requirements
Some Modeling Challenges:
• Pension Funds:
– Long-term horizons
• Reliable projections of assets & liabilities
– Modeling choices to accommodate/reflect (conflicting)
objectives of multiple classes of stakeholders
(sponsors, active members, retirees, regulators, etc.)
– Complex regulatory provisions
• Credit risk & insurance models:
– Capturing joint effects of multiple risk factors (interest
rates, credit migrations, etc.)
– Adequately capturing low-probability, high-impact
extreme events
– Choice of appropriate risk measures
Model Development Issues:
• Substantial advancements in modelling systems
(AMPL, GAMS, MPL, Fort-MP, etc.)
– Analyst can concentrate on model development, model
variations, comparative assessment of results instead of
data management and i/o interfacing with solvers
– Some interfacing with SP-specific s/w (e.g., IBM’s
SP_OSL)
– Still, much room for improvement:
Interfacing seamlessly modelling systems with
specialized solvers for transparent benchmarking of
solvers on common test sets (taking advantage of
problem structures) and adoption of most efficient
solution algorithms
– Some solvers interfacing with spreadsheets and other
packages (for data manipulation, statistical analysis,
visualization, etc., e.g., MS Excel, Matlab).
Computational Issues:
• Significant advancements in computing capabilities
– Faster computers, larger memory capacity
– Now feasible to solve practical SPs on conventional computers
with general-purpose solvers (e.g., CPLEX) in reasonable time
– Still, appetite for problem size and complexity outpaces
technological capabilities
– Operational needs (real-time trading support challenge the
envelope of “solvable” models)
• Significant algorithmic advancements
– Specialized solvers that take advantage of problem structures
(esp. for linearly-constrained problems)
– Most effective:
• Specialized interior point methods (for LP and NLP)
(J. Gondzio & A. Grothey, A. Ruszczynski & J.M. Mulvey & R. Vanderbei, etc.)
• Decomposition methods (Bender’s)
(M.A.H. Dempster et al., J.R. Birge et al., Pereira/Pinto, G. Mitra et al., & others)
Computational Issues:
• Specialized parallel computing efforts have wavered
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System/architecture specific implementations
Custom built – Some implementations rigid to specific model forms
Many systems are now “extinct”
Advancements of conventional computing systems have put to
question the necessity for the effort to port models to parallel systems
• Potential for “mass customization” of high-performance
computing capabilities
– Grid computing
– Exploitation of available (distributed) computing resources & network
infrastructures
– Middle-ware s/w is being developed
Interesting Problems
of Potential Practical Usefulness:
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Non-exhaustive. Communication with clairvoyant garbled. 
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Scenario Generation
– Consistency with fundamental financial principles & market data
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Assessing Comparative Effectiveness of Scenario Generation Methods
– Empirical assessment of predictive power of density estimations (F. Diebold)
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Risk Measurement & Management in Dynamic Decision Processes
Coping with Estimation Errors
– Michaud’s heuristic, Shrinkage, Bayesian approaches, Robust Optimization,
Regret models
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Design of Products (e.g., option baskets, structural features of pension
systems, insurance or mutual funds contracts – 2nd pillar of pensions
system)
Public sector: debt management (timing & structuring issuance of public
debt instruments among alternatives)
Incorporation of derivatives (contingent claims) in risk management
Interface with Real Options
– Accounting for the value of flexibility in multi-stage decision processes
– Path dependencies
– Combinatorial (IP) aspects
Interesting Problems
of Potential Practical Usefulness: (cont.)
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Coping with Estimation Errors:
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Solutions influenced by characteristics of distributions of r.v.s
Parameters of probability models typically calibrated on basis of
historical data – subject to estimation errors
Need robust solutions wrt different instances of parameter
values (distributions, scenario sets) – current studies restricted to
Mean-Variance setting
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Robust Optimization (Semi-definite, conic quadratic programming)
optimizing wrt “worst case” parameter instances
(D. Goldfarb, G. Iyengar, A. Ben-Tal, A. Nemirovski, L. El Ghaoui, R. Tutuncu)
“Averaging heuristic” (R. Michaud)
Adjustment of risk aversion factor (F.A. de Roon)
Robust parameter estimation (A.V. de Miguel, F.J. Nogales)
Bayesian estimation of parameters – need a prior distribution
(H. Markowitz, H.R. Campbell)
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Parameter “shrinkage” approaches (P. Jorion, J.B. Jobson, B. Korkie)
Contamination techniques (J. Dupacova)
Regret “coordination” models – Minimization of disutility measures
Interesting Problems
of Potential Practical Usefulness: (cont.)
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Pension Fund Management:
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Pension Funds face severe challenges:
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Aging population (longevity risk)
Low birth rates – Demographic changes
Dependency ratio (Working population/retirees) declines
Substantial under-funding (actuarial deficit) of many funds
Hit during equity market declines of early 2000s
Affected by low bond yields – insufficient returns. Also low interest rates affect
the discount factors of liabilities
Conservative practices – now materially relaxed
Influence of EU Directive 2003/41
In search of (low risk) higher yields of fund portfolios
Restructurings of pension systems
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Questions on DB sustainability
Conversions to DC (risk redistribution among stakeholders)
Challenges to public guaranteeing agencies
Restructuring wrt 3 pillars – Design of incentives
Potential of SP models for assessing restructuring alternatives not just fund
management strategies
Interesting Problems
of Potential Practical Usefulness: (cont.)
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Scenario Generation:
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Capturing effectively correlations of multi-variate r.v.s
Capturing effectively “non-standard” statistical characteristics (skewness,
heavy tails)
Capturing complex dynamics of multi-variate stochastic processes
Joint scenarios over multiple risk factors
Compliance with financial fundamentals – Ensuring arbitrage-free
price models
Assessing comparative effectiveness of alternative approaches in:
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Capturing empirical features of dynamic stochastic processes
Assessing effectiveness of density (distribution) estimations (F. Diebold)
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Much more complex in multivariate setting
Open issue for dynamic processes
Producing robust/stable solutions in SP models
(Kaut, Wallace, Pacific J. Optimization, 2007)
Interesting Problems
of Potential Practical Usefulness: (cont.)
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Risk Management:
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Particularly Credit Risk Measurement & Management
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Credit scoring models
Modelling transitions of credit ratings
Estimating default probabilities
Estimating recovery rates (loss given default)
Modelling extreme events (low probability, high impact)
Extreme Value Theory
Multivariate stable distributions
Incorporating Complex Derivatives in conventional SP models
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Pricing derivatives in incomplete markets
Consistent pricing with probability models of SPs
(A.J. King, Math. Progr.; M.A.H. Dempster et al., Math. Finance; A.J. King, T.
Pennanen; N. Topaloglou et al., J. Bank. & Finance)
Risk Measurement/Management in Dynamic Decision Processes
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Extensions of coherent risk measures in multi-period decision setting
(Artzner, Delbaen, Eber, Heath, Ku, Annals of OR, 152, 2007)
Value of information in risk management of multi-period decision frameworks
(Pflug, J. Bank. & Fin., 2006)
Interesting Problems
of Potential Practical Usefulness: (cont.)
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Adopting Financial Techniques in Other
Applications (e.g., SCM, Production):
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Appropriate pricing of optional decisions
Lack of markets (non-tradeable assets) to base development of
pricing models
Optimizing selections of optional decisions
(e.g., Birge, M&SOM, 2000; Birge, NRL, 2006,
van Delft, Vial, Automatica, 2004)
Adoption of SP Models in Real Options Analysis
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Optimal choices of “in-project” options
Valuing flexibility of optional decisions – decision timing
Path dependency of decisions & distributions
Lead to stochastic MIPs
(R. de Neufville, MIT)
Interesting Problems
of Potential Practical Usefulness: (cont.)
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Public Sector Finances:
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Public Debt Management
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Timing & structuring issuance of public debt instruments among
alternatives (currency, term/duration, zero-, fixed- or floating- coupon
rates)
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Tying with macroeconomic prospects
Meeting funding needs
Stabilization of taxes, avoiding concentration of debt payments, Risk
management
Management of Strategic Reserves
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(J. Kreuser, S. Claessens, R.J-B Wets)
Currencies, hedging, derivatives
Case Study: International Portfolio Management
•Allocation of funds to international assets
•Dynamic management of portfolio
•Controlling risk exposures (hedging)
Risk factors: Market Risk and Currency Risk
The Objectives:
Effective Management of Risk/Return Tradeoffs (parametric programs)
Modeling Steps:
• Representation of uncertainty capturing all risk factors and their correlations.
• Portfolio Optimization Models that address the risk elements in an integrated
manner.
International Diversification
Internationally diversified portfolios
• provide wider scope for diversification
• can improve the risk/return profile of portfolios
• positive empirical evidence for portfolios of equities and bonds
But, …
International portfolios are exposed to additional risks:
exchange rate fluctuations → Currency risk
Observations:
• Multi-dimensional nature of risk:
(Market risk and Currency risk)
• Higher correlation of intl. investments
in bear markets
• Diversity of financial instruments
(to hedge these risks)
Need to adopt an
integrated risk
management approach
International Diversification
• Diversification can help mitigate market and currency risks,
as long as correlations are fairly low
• Effects depend on correlation structure of international
markets and currency exchange rates
But,
• Diversification alone may not be sufficient to
serve the investor’s risk management objectives
• Risk reduction may be augmented with the use
Motivation
of suitable hedging instruments and strategies
• Risk management problem should be approached
in a unified manner
Aim:
 Development of effective and practical decision support
tools for total risk management of international portfolios
Research Aim
Development of integrated simulation and
optimization approaches that determine jointly:
 Capital allocation across markets
 Optimal portfolio composition (asset mix)
 Flexible hedging strategies using appropriate
derivative securities
Thus:
Market and currency risks of international portfolios
are considered in an integrated, total risk
management decision framework
Research Issues
 Development of an integrated risk management framework
where multiple risk factors are considered simultaneously.
 Development of multi-stage stochastic programming models
for dynamic management of portfolios through rebalancing
decisions.
 Adoption of appropriate methods for scenario generation –
without explicit assumption for a particular functional form for
the distribution of the random variables to accommodate
asymmetries and fat tails observed in market data.
 Adoption of appropriate measures to control for risks
(distributions of asset returns and exchange rates are not
normal).
Research Issues
 Pricing and incorporation of derivatives in scenario-based
stochastic programming models. Pricing methods account for
asymmetric and heavy-tailed distributions of the underlying,
consistently with postulated scenarios.
 Development of a framework for empirical evaluation of
alternative instruments and strategies in terms of their
effectiveness to control risks. Derivatives are used to hedge
the market and the currency risks.
Hedging
• Controlling Market risk:
 Simple Options – on domestic and foreign stock indices
 Quantos - fixed exchange rate foreign equity options
Asymmetric payoff profiles of options can effectively cover against
adverse price movements – Judicious choices of combinations of
options can help shape a desired payoff profile.
• Hedging Currency risk:
 Forward Contracts – protect against potential losses but
forgo potential gains from favorable rate movements
 Currency options – allow the possibility to benefit from
currency appreciation, but require payment of premium
International Portfolio Management Model (single-stage version):
Trading Strategies involving Stock options (Shaping payoff profiles)
Straddle Strategy
70
60
50
40
30
20
10
0
-10
-20
2800
200
150
Payoff
Payoff
Strangle Strategy
100
50
0
-50
2900
3000
3100
3200
3300
-100
2800
3400
2900
Underlying US Stock Index
300
300
Payoff
Payoff
400
200
100
3300
3400
3300
3400
200
100
0
0
-100
2800
-100
2800
3100
3200
Strap Strategy
400
3000
3100
Underlying US Stock Index
Strip Strategy
2900
3000
3200
Underlying US Stock Index
3300
3400
2900
3000
3100
3200
Underlying US Stock Index
Methodologies for Pricing Options
• We need to reconcile the pricing of options with the scenarios for the
underlying assets (internally consistent framework)
• We do not restrict to specific distributional assumptions
We adopt two methods:
• Method 1: Derive risk-neutral probability measure based on equilibrium
principles (Rubinstein; Bakshi, Kapadia, Madan)
• Method 2: Gram-Charlier series approximation of conditional density of
asset returns => Adds correction terms to B-S valuations to account for
higher moments (Corrado and Su, Jarrow and Radd).
Topaloglou, Vladimirou, Zenios,
“Pricing options on scenario trees, J. Bank. & Fin., (in print).
Observations
The two methods produce very similar option prices for ITM,
ATM, and OTM options
For ITM, ATM options, both methods result in prices that are
very close to B-S prices
The B-S formula overprices OTM call options and underprices
OTM put options
Both methods exhibit consistent behavior in terms of the
variation of options prices with respect to changes in the
higher moments
Both methods result in option prices that approximate more
closely than the B-S method market quotations of option
prices, especially for OTM options.
Hedging the market risk using stock options (Static tests)
Risk/Return Efficient Frontiers Of Portfolios with and without Options,
Strangle Strategy
2.5%
Expected Return
2.0%
1.5%
1.0%
0.5%
0.0%
-1%
1%
3%
5%
7%
9%
95%-CVaR
Quanto+Forw ards
Simple+Forw ards
Forw ards_WithoutOptions
Totally Unhedged
Quantos_Without_Forw ards
Simple_Without_Forw ards
11%
Hedging the market risk using stock options (Static tests)
Shaping portfolio risk using options
0,35
0,3
Without
Probability
0,25
Strangle
Straddle
0,2
Strip
0,15
0,1
0,05
0
-0,06
-0,04
-0,02
0
0,02
Return
0,04
0,06
0,08
0,1
0,12
Hedging the market risk using stock options (Dynamic tests)
Ex post Realized Returns of Portfolios with Quantos and Simple options, Strangle
Strategy, μ=1%
1.4
1.35
Realized Return
1.3
1.25
1.2
1.15
1.1
1.05
1
0.95
Μαϊ-
Αυγ-
Νοε-
Φεβ-
Μαϊ-
Αυγ-
Νοε-
Φεβ-
Μαϊ-
Αυγ-
Νοε-
Φεβ-
Μαϊ-
Αυγ-
Νοε-
98
98
98
99
99
99
99
00
00
00
00
01
01
01
01
Time period
Quantos+f orwards
Simple+Forwards
Forwards_WithoutOptions
Totally Unhedged
Hedging Currency Risk
• It pays to hedge the currency risk compared to unhedged
positions
• Forwards are more effective than single put currency options
(issue of recovering the hedging cost of options)
• Combination of options (e.g., BearSpread) result in
performance improvements
• Options allow benefits from favorable exchange rate
movements, in contrast to forwards which lock in a
prespecified forward rate
Hedging Market risk
• The impact from managing the market risk is substantial; much
more substantial than currency hedging.
• The inclusion of options provides an efficient and effective way
to control risk and to improve the performance of portfolios
• Options shape the portfolio return distribution (lower tails,
positively skewed distributions)
• Quantos provide effective instruments for risk hedging
purposes due to their integrative nature
Integrated Framework
• Incremental performance improvements as we gradually
move to a more integrative risk management framework
using options
• The holistic framework constitutes the most effective risk
management scheme
Single- vs Multi-stage models (Forward Contracts, Dynamic tests)
1.20
1.10
1.05
1.00
Time Period
Tw o-Stage
Single-stage(15000 scenarios)
Single-stage(150 scenarios)
ct01
O
Ju
l-0
1
Ap
r-0
1
Ja
n01
ct00
O
Ju
l-0
0
Ap
r-0
0
Ja
n00
ct99
O
Ju
l-9
9
Ap
r-9
9
Ja
n99
O
ct98
0.95
Ju
l-9
8
Cummulative Wealth
1.15
1-stage vs 2-stage Models (Minimum Risk Case)
Single- vs Multi-stage models (Stock Options, Dynamic tests)
Single vs Two-Stage models with stock options. Strangle strategy
1.45
Realized Return
1.35
1.25
1.15
Simple Options-Single
1.05
Without Options-Single
Simple Options-Tw o-Stage
0.95
Μαϊ-98 Αυγ-98 Νοε-98 Φεβ-99 Μαϊ-99 Αυγ-99 Νοε-99 Φεβ-00 Μαϊ-00 Αυγ-00 Νοε-00 Φεβ-01 Μαϊ-01 Αυγ-01 Νοε-01
Time period
Single- vs Multi-stage Models
• In all cases, regardless of the hedging strategy and the
hedging instruments that are used, multi-stage models
outperform their single-stage counterparts.
Scientific Contributions
 Development of an integrated simulation and optimization
framework for international portfolio management.
 Implementation of portfolio optimization models that jointly
select the appropriate investments across markets and
determine the levels of hedging.
 Development of multi-stage stochastic programming models
for optimal selection of international portfolios in a dynamic
setting.
 Adaptation of suitable methods for pricing derivatives in
scenario-based stochastic programming models.
Scientific Contributions
 Incorporation of derivatives in stochastic programming
models for portfolio risk management.
 Development of an integrated risk management
framework where all risk factors are considered
simultaneously.
 Empirical assessment of alternative instruments and
strategies for coping with market and currency risk of
international investments, either separately, or jointly.
 Empirical validation of the models through extensive
numerical tests using real market data.
References:
Vigorous research interest is evident in recent volumes:
E.g., (recent volumes)
•
G. Szego (ed.), J. Bank. & Finance, 26(7), 2002.
•
W.T. Ziemba, The Stochastic Programming Approach to Asset Liability and Wealth Management, AIMR, 2003.
•
W.T. Ziemba, S.W. Wallace (eds.), Applications of Stochastic Programming, MPS-SIAM, 2005.
•
J.R. Birge, V. Linetsky, Financial Engineering, Handbooks in OR & MS, Elsevier, 2006.
•
S.A. Zenios, W.T. Ziemba (eds.), Handbooks of Asset and Liability Modeling, Elsevier
– “Theory and Methodology”, 2006
– “Applications and Case Studies”, 2007
•
T. Rockafellar, S. Uryasev (eds.), J. Bank. & Finance, 30(2), 2006, “Risk management & optimization in
finance”
•
H. Vladimirou (ed.), Annals of OR, 151, 04/2007, “Financial Modeling”
•
H. Vladimirou (ed.), Annals of OR, 152, 07/2007, “Financial Optimization”
•
M. Dempster, G. Mitra, G. Pflug (eds.), Quant. Finance, 7(2), 04/2007, “Financial planning in a dynamic
setting”
•
M. Dempster, G. Mitra, G. Pflug (eds.), Quant. Finance, 7(4), 08/2007, “Portfolio construction & risk
management”
•
Y. kaniovski, Murgia, G. Pflug (eds.), J. Bank. & Finance, 31(8), 08/2007, “Optimization techniques in finance”
•
Other relevant volumes of Annals OR, J. Econ. Dynamics & Control, Math. Progr., J. Bank. & Finance, etc.
E.g., (forthcoming)
•
W. Roemisch, G.Ch. Pflug, Book on risk measurement/management.
•
S.A. Zenios, Practical Financial Optimization: Decision Making for Financial Engineers, Blackwell.
•
G. Infanger (ed.), Stochastic Programming: The State of the Art.
•
•
•
•
•
•
H. Foellmer, A. Scheid, Stochastic Finance: An Introduction in Discrete Time, Walter de Gruyter, 2002.
J. Dupacova, J. Hurt, and J. Stepan, Stochastic Modeling in Economics and Finance. Springer, 2002.
W.T. Ziemba, J.M. Mulvey (eds.), Worldwide Asset and Liability Modeling, Cambridge University Press, 1998.
G. Infanger, Planning under uncertainty -- Solving large-scale stochastic linear programs, Boyd & Fraser,
1994.
W.T. Ziemba, R.G. Vickson (eds.), Stochastic Optimization Models in Finance, World Scientific (1975,
reprinted 2006).
Series of Handbooks in Operations Research and Management Science, Elsevier, includes several relevant
References:
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
S.A. Zenios, Practical Financial Optimization: Decision Making for Financial Engineers, Blackwell
(forthcoming).
S.A. Zenios, W.T. Ziemba (eds.), Handbook of Asset and Liability Modeling: Applications and Case
Studies, Elsevier (forthcoming).
G. Infanger (ed.), Stochastic Programming: The State of the Art (in preparation).
H. Vladimirou (ed.), “Financial Optimization”, Annals of Operations Research, 152, 2007.
S.A. Zenios, W.T. Ziemba (eds.), Handbook of Asset and Liability Modeling: Theory and
Methodology, Elsevier, 2006.
J.R. Birge, V. Linetsky, Financial Engineering, Handbooks in Operations Research and
Management Science, Elsevier, 2006.
W.T. Ziemba, S.W. Wallace (eds.), Applications of Stochastic Programming, MPS-SIAM, 2005.
W.T. Ziemba, The Stochastic Programming Approach to Asset Liability and Wealth Management,
AIMR, 2003.
H. Foellmer, A. Scheid, Stochastic Finance: An Introduction in Discrete Time, Walter de Gruyter,
2002.
J. Dupacova, J. Hurt, and J. Stepan, Stochastic Modeling in Economics and Finance. Springer,
2002.
W.T. Ziemba, J.M. Mulvey (eds.), Worldwide Asset and Liability Modeling, Cambridge University
Press, 1998.
G. Infanger, Planning under uncertainty -- Solving large-scale stochastic linear programs, Boyd &
Fraser, 1994.
W.T. Ziemba, R.G. Vickson (eds.), Stochastic Optimization Models in Finance, World Scientific
(1975, reprinted 2006).
Series of Handbooks in Operations Research and Management Science, Elsevier, includes
several relevant volumes (e.g., Vol. 10, 2003, “Stochastic Programming”, A. Ruszczynski, A.
Shapiro (eds.), Vol. 9, 1995, “Finance”, Jarrow, Maksimovic, Ziemba (eds.); Also the Elsevier
Handbook Series in Finance, W.T. Ziemba (ed.)
Several dedicated issues of Annals of OR, J. Economic Dynamics and Control, Mathematical
Programming, J. of Banking and Finance, etc.