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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

[email protected]@uncc.edu

Presentation at the 3rd Western Conference in Mathematical Finance

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

Outline

- Problem
- Motivation & Literature
- Solution in complete market
- Application to BS model
- Conclusion

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

Efficient Frontier and Capital Allocation Line (CAL):

- Standard deviation (variance) as risk measure
- Static (single step) optimization

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)

- 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)

- 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

- 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)

- 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 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)

- 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:
- 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)

- Stock dynamics:
- Definition:
- If we assume and , then “double-star-system” is optimal:

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

- 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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