Stochastic Programming Applications in FinanceStochastic Programming Applications in Finance

Hercules Vladimirou

Center for Banking and Financial ResearchUniversity 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 FinanceCase 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 decision-making 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. Description of the underlying (multi-variate) discrete stochastic process for the uncertain parameters;

Dynamic information structure (Scenario-tree Generation).

2. 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:• 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)

– 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:• Max expected terminal value/return (Risk neutral)

– Supplemented with (piecewise linear) penalties or (ad-hoc) bounding constraints to reflect risk aversion preferences

• 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• Common Risk Measures:

– 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

• 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

• 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:– 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

• 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:– 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.

– 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 risk-neutral (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:– 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 institutionsExpenditures 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 sector4% insurance sector24% 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– 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:

• Non-exhaustive. Communication with clairvoyant garbled.

• Scenario Generation– Consistency with fundamental financial principles & market data

• Assessing Comparative Effectiveness of Scenario Generation Methods– Empirical assessment of predictive power of density estimations (F. Diebold)

• Risk Measurement & Management in Dynamic Decision Processes• Coping with Estimation Errors

– Michaud’s heuristic, Shrinkage, Bayesian approaches, Robust Optimization, Regret models

• 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.)

• Coping with Estimation Errors:– 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

• 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)

• 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.)• Pension Fund Management:

– Pension Funds face severe challenges:• 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• 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.)• Scenario Generation:

– 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:• Capturing empirical features of dynamic stochastic processes• Assessing effectiveness of density (distribution) estimations (F. Diebold)

– 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.)• Risk Management:

– Particularly Credit Risk Measurement & Management• 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• 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• 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.)• Adopting Financial Techniques in Other

Applications (e.g., SCM, Production):– 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– 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.)

• Public Sector Finances:– Public Debt Management

• Timing & structuring issuance of public debt instruments among alternatives (currency, term/duration, zero-, fixed- or floating- coupon rates) – Tying with macroeconomic prospects– Meeting funding needs– Stabilization of taxes, avoiding concentration of debt payments, Risk

management

– Management of Strategic Reserves (J. Kreuser, S. Claessens, R.J-B Wets)

• Currencies, hedging, derivatives

Case Study: International Portfolio Management

Risk factors:Risk factors: Market Risk and Currency RiskMarket Risk and Currency Risk

The Objectives: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.

•Allocation of funds to international assets•Dynamic management of portfolio•Controlling risk exposures (hedging)

International DiversificationInternationally 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 anintegrated riskmanagement 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 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

Motivation

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 AimResearch Aim

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)

Strangle Strategy

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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.

Risk/Return Efficient Frontiers Of Portfolios with and without Options, Strangle Strategy

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Ex post Realized Returns of Portfolios with Quantos and Simple options, Strangle Strategy, μ=1%

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Μαϊ-99

Αυγ-99

Νοε-99

Φεβ-00

Μαϊ-00

Αυγ-00

Νοε-00

Φεβ-01

Μαϊ-01

Αυγ-01

Νοε-01

Time period

Rea

lized

Ret

urn

Quantos+forwards Simple+Forwards Forwards_WithoutOptions Totally Unhedged

Hedging the market risk using stock options (Dynamic tests)

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

0.95

1.00

1.05

1.10

1.15

1.20

Time Period

Cum

mul

ativ

e W

ealth

Tw o-Stage Single-stage(15000 scenarios) Single-stage(150 scenarios)

Single- vs Multi-stage models (Forward Contracts, Dynamic tests)

1-stage vs 2-stage Models (Minimum Risk Case)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

0.95

1.05

1.15

1.25

1.35

1.45

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

Real

ized

Ret

urn

Simple Options-Single

Without Options-Single

Simple Options-Two-Stage

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

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.

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.