Risk minimizing portfolio optimization and hedging with conditional value at risk
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Risk Minimizing Portfolio Optimization and Hedging with Conditional Value-at-Risk. Jing Li Mingxin Xu Department of Mathematics and Statistics University of North Carolina at Charlotte jli16@uncc.edu mxu2@uncc.edu. Presentation at the 3 rd Western Conference in Mathematical Finance

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Risk minimizing portfolio optimization and hedging with conditional value at risk l.jpg

Risk Minimizing Portfolio Optimization and Hedging with Conditional Value-at-Risk

Jing Li Mingxin Xu

Department of Mathematics and Statistics

University of North Carolina at Charlotte

jli16@uncc.edumxu2@uncc.edu

Presentation at the 3rd Western Conference in Mathematical Finance

Santa Barbara, Nov. 13th~15th, 2009


Outline l.jpg
Outline Conditional Value-at-Risk

  • Problem

  • Motivation & Literature

  • Solution in complete market

  • Application to BS model

  • Conclusion


Dynamic problem l.jpg
Dynamic Problem Conditional Value-at-Risk

Minimizing Conditional Value at Risk with Expected Return Constraint

where

Portfolio dynamics:

Xt – Portfolio value

– Stock price

– Risk-free rate

– Hedging strategy

– Lower bound on portfolio value; no bankruptcy if

– Upper bound on portfolio value; no upper bound if

– Initial portfolio value


Background motivation l.jpg
Background & Motivation Conditional Value-at-Risk

Efficient Frontier and Capital Allocation Line (CAL):

  • Standard deviation (variance) as risk measure

  • Static (single step) optimization


Risk measures l.jpg
Risk Measures Conditional Value-at-Risk

  • Variance - First used by Markovitz in the classic portfolio optimization framework (1952)

  • VaR(Value-at-Risk) - The industrial standard for risk management, used by BASEL II for capital reserve calculation

  • CVaR(Conditional Value-at-Risk) - A special case of Coherent Risk Measures, first proposed by Artzner, Delbaen, Eber, Heath (1997)


Literature i l.jpg
Literature Conditional Value-at-Risk(I)

  • Numerical Implementation of CVaR Optimization

    • Rockafellar and Uryasev (2000) found a convex function to represent CVaR

    • Linear programming is used

    • Only handles static (i.e., one-step) optimization

  • Conditional Risk Mapping for CVaR

    • Revised measure defined by Ruszczynski and Shapiro (2006)

    • Leverage Rockafellar’s static result to optimize “conditional risk mapping” at each step

    • Roll back from final step to achieve dynamic (i.e., multi-step) optimization


Literature ii l.jpg
Literature (II) Conditional Value-at-Risk

  • Portfolio Selection with Bankruptcy Prohibition

    • Continuous-time portfolio selection solved by Zhou & Li (2000)

    • Continuous-time portfolio selection with bankruptcy prohibition solved by Bielecki et al. (2005)

  • Utility maximization with CVaR constraint. (Gandy, 2005; Gabih et al., 2009)

    • Reverse problem of CVaR minimization with utility constraint;

    • Impose strict convexity on utility functions, so condition on E[X] is not a special case of E[u(X)] by taking u(X)=X.

  • Risk-Neutral (Martingale) Approach to Dynamic Portfolio Optimization by Pliska (1982)

    • Avoids dynamic programming by using risk-neutral measure

    • Decompose optimization problem into 2 subproblems: use convex optimization theory to find the optimal terminal wealth; use martingale representation theory to find trading strategy.


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The Idea Conditional Value-at-Risk

  • Martingale approach with complete market assumption to convert the dynamic problem into a static one:

  • Convex representation of CVaR to decompose the above problem into a two step procedure:

    Step 1: Minimizing Expected Shortfall

    Step 2: Minimizing CVaR

     Convex Function


Solution i l.jpg
Solution (I) Conditional Value-at-Risk

  • Problem without return constraint:

  • Solution to Step 1: Shortfall problem

    • Define:

    • Two-Set Configuration .

    • is computed by capital constraint for every given level of .

  • Solution to Step 2: CVaR problem

    • Inherits 2-set configuration from Step 1;

    • Need to decide optimal level for ( , ).


Solution ii l.jpg
Solution (II) Conditional Value-at-Risk

  • Solution to Step 2: CVaR problem (cont.)

    • “star-system” : optimal level found by

      • Capital constraint:

      • 1st order Euler condition .

    • : expected return achieved by optimal 2-set configuration.

    • “bar-system” :

      • is at its upper bound,

      • satisfies capital constraint .

    • : expected return achieved by “bar-system”

      • Highest expected return achievable by any X that satisfies capital constraint.


Solution iii l.jpg
Solution (III) Conditional Value-at-Risk

  • Problem with return constrain:

  • Solution to Step 1: Shortfall problem

    • Define:

    • Three-Set Configuration

    • , are computed by capital and return constraints for every given level of .

  • Solution to Step 2: CVaR problem

    • Inherits 3-set configuration from Step 1;

    • Need to find optimal level for ( , , );

    • “double-star-system” : optimal level found by

      • Capital constraint:

      • Return constraint:

      • 1st order Euler condition:


Solution iv l.jpg
Solution (IV) Conditional Value-at-Risk

  • Solution:

    • If , then

      • When , the optimal is

      • When , the optimal does not exist, but the infimum of CVaR is .

    • Otherwise,

      • If and , then “bar-system” is optimal:

      • If and , then “star-system” is optimal:.

      • If and , then “double-star-system” is optimal:

      • If and , then optimal does not exist, but the

        infimum of CVaR is


Application to bs model i l.jpg
Application to BS Model (I) Conditional Value-at-Risk

  • Stock dynamics:

  • Definition:

  • If we assume and , then “double-star-system” is optimal:


Application to bs model ii l.jpg
Application to BS Model (II) Conditional Value-at-Risk

  • Constant minimal risk can be achieved when return objective is not high.

  • Minimal risk increases as return objective gets higher.

  • Pure money market account portfolio is no longer efficient.


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Conclusion & Future Work Conditional Value-at-Risk

  • Found “closed” form solution to dynamic CVaR minimization problem and the related shortfall minimization problem in complete market.

  • Applications to BS model include formula of hedging strategy and mean CVaR efficient frontier.

  • Like to see extension to incomplete market.


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The End Conditional Value-at-Risk

Questions?

Thank you!