DEE - Otros documentos
http://hdl.handle.net/10016/6548
2015-03-02T23:10:06ZMinimax strategies and duality with applications in financial mathematics
http://hdl.handle.net/10016/18147
Minimax strategies and duality with applications in financial mathematics
Balbás, Alejandro; Balbás, Raquel
Many topics in Actuarial and Financial Mathematics lead to Minimax or Maximin problems (risk measures optimization, ambiguous setting, robust solutions, Bayesian credibility theory, interest rate risk, etc.). However, minimax problems are usually difficult to address, since they may involve complex vector (Banach) spaces or constraints. This paper presents an unified approach so as to deal with minimax convex problems. In particular, we will yield a dual problem providing necessary and sufficient optimality conditions that easily apply in practice. Both, duals and optimality conditions are significantly simplified by using a new Mean Value Theorem. Important applications in risk analysis are given.
2011-01-20T00:00:00ZCapital requirements, good deals and portfolio insurance with risk measures
http://hdl.handle.net/10016/18146
Capital requirements, good deals and portfolio insurance with risk measures
Balbás, Alejandro; Balbás, Beatriz; Balbás, Raquel
General risk functions are becoming very important for managers, regulators and
supervisors. Many risk functions are interpreted as initial capital requirements that
a manager must add and invest in a risk-free security in order to protect the wealth
of his clients.
This paper deals with a complete arbitrage free pricing model and a general expectation
bounded risk measure, and it studies whether the investment of the capital
requirements in the risk-free asset is optimal. It is shown that it is not optimal in
many important cases. For instance, if the risk measure is the CV aR and we consider
the assumptions of the Black and Scholes model. Furthermore, in this framework and under short selling restrictions, the explicit expression of the optimal strategy is provided,
and it is composed of several put options. If the confidence level of the CV aR is
close to 100% then the optimal strategy becomes a classical portfolio insurance. This
theoretical result seems to be supported by some independent and recent empirical
analyses.
If there are no limits to sale the risk-free asset, i.e., if the manager can borrow
as much money as desired, then the framework above leads to the existence of “good
deals” (i.e., sequences of strategies whose V aR and CV aR tends to minus infinite
and whose expected return tends to plus infinite). The explicit expression of the
portfolio insurance strategy above has been used so as to construct effective good
deals. Furthermore, it has been pointed out that the methodology allowing us to
build portfolio insurance strategies and good deals also applies for pricing models
beyond Black and Scholes, such as Heston and other stochastic volatility models
2010-01-01T00:00:00ZCAPM and APT like models with risk measures
http://hdl.handle.net/10016/18145
CAPM and APT like models with risk measures
Balbás, Alejandro; Balbás, Raquel; Balbás, Beatriz
The paper deals with optimal portfolio choice problems when risk levels are given
by coherent risk measures, expectation bounded risk measures or general deviations.
Both static and dynamic pricing models may be involved.
Unbounded problems are characterized by new notions such as compatibility and
strong compatibility between pricing rules and risk measures. Surprisingly, it is
pointed out that the lack of bounded optimal risk and/or return levels arises in practice
for very important pricing models (for instance, the Black and Scholes model)
and risk measures (V aR, CV aR, absolute deviation and downside semi-deviation,
etc.).
Bounded problems will present a Market Price of Risk and generate a pair of
benchmarks. From these benchmarks we will introduce APT and CAPM like analyses,
in the sense that the level of correlation between every available security and some economic factors will expalin the security expected return. On the contray,
the risk level non correlated with these factors will have no influence on any return,
despite we are dealing with very general risk functions that are beyond the standard
deviation.
2009-06-01T00:00:00ZExtending pricing rules with general risk
http://hdl.handle.net/10016/18138
Extending pricing rules with general risk
Balbás, Alejandro; Balbás, Raquel; Garrido, José
The paper addresses pricing issues in imperfect and/or incomplete markets
if the risk level of the hedging strategy is measured by a general risk function. Convex
Optimization Theory is used in order to extend pricing rules for a wide family of risk functions,
including Deviation Measures, Expectation Bounded Risk Measures and Coherent
Measures of Risk. For imperfect markets the extended pricing rules reduce the bid-ask
spread. The paper ends by particularizing the findings so as to study with more detail
some concrete examples, including the Conditional Value at Risk and some properties of
the Standard Deviation
2008-04-01T00:00:00Z