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.edujli16@uncc.edu mxu2@uncc.edumxu2@uncc.edu Presentation at the 3 rd Western Conference in Mathematical Finance Santa Barbara, Nov. 13 th ~15 th, 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: X t – 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: 1 st 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: 1 st 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!