Capital requirements, good deals and portfolio insurance with risk measures

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Universidad Carlos III de Madrid
Universidad Rey Juan Carlos
Universidad Politécnica de Madrid
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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
Risk Measure, Capital Requirement, Good Deal, Portfolio Insurance
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