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

1. 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

2. Outline • Problem • Motivation & Literature • Solution in complete market • Application to BS model • Conclusion

3. 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

4. Background & Motivation Efficient Frontier and Capital Allocation Line (CAL): • Standard deviation (variance) as risk measure • Static (single step) optimization

5. 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)

6. 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

7. 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.

8. 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

9. 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 ( , ).

10. 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.

11. 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:

12. 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

13. Application to BS Model (I) • Stock dynamics: • Definition: • If we assume and , then “double-star-system” is optimal:

14. 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.

15. 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.

16. The End Questions? Thank you!