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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 [email protected] [email protected] 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

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

[email protected]@uncc.edu

Presentation at the 3rd Western Conference in Mathematical Finance

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

outline
Outline
  • Problem
  • Motivation & Literature
  • Solution in complete market
  • Application to BS model
  • Conclusion
dynamic problem
Dynamic Problem

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
Background & Motivation

Efficient Frontier and Capital Allocation Line (CAL):

  • Standard deviation (variance) as risk measure
  • Static (single step) optimization
risk measures
Risk Measures
  • 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
Literature(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
Literature (II)
  • 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.
the idea
The Idea
  • 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
Solution (I)
  • 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
Solution (II)
  • 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
Solution (III)
  • 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
Solution (IV)
  • 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
Application to BS Model (I)
  • Stock dynamics:
  • Definition:
  • If we assume and , then “double-star-system” is optimal:
application to bs model ii
Application to BS Model (II)
  • 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.
conclusion future work
Conclusion & Future Work
  • 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.
the end
The End

Questions?

Thank you!

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