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.edu mxu2@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
• When
is
.
, the optimal is
, the optimal does not exist, but the infimum of CVaR
– Otherwise,
• If
and
, then “bar-system” is optimal:
• If
and
, then “star-system” is optimal:.
• If
and
• If
and
infimum of CVaR is
, then “double-star-system” is optimal:
, then optimal does not exist, but the
Application to BS Model (I)
• Stock dynamics:
• Definition:
• If we assume
is optimal:
and
, then “double-star-system”
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.
The End
Questions?
Thank you!