Risk Minimizing Portfolio Optimization and Hedging with
Conditional Value-at-Risk
Jing Li Mingxin XuDepartment of Mathematics and StatisticsUniversity of North Carolina at Charlotte
[email protected] [email protected]
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 ProblemMinimizing Conditional Value at Risk with Expected Return Constraint
wherePortfolio 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 & MotivationEfficient 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.