We consider the portfolio optimization problem for an investor whose consumption rate process and terminal wealth are subject to downside constraints. In the standard financial market model that consists of d risky assets and one riskless asset, we assume that the riskless asset earns a constant instantaneous rate of interest, r > 0, and that the risky assets are geometric Brownian motions. The optimal portfolio policy for a wide scale of utility functions is derived explicitly. The gradient operator and the Clark-Ocone formula in Malliavin calculus are used in the derivation of this policy. We show how Malliavin calculus approach can help us get around certain difficulties that arise in using the classical "delta hedging" approach.
Empirical studies show that correlation between national stock markets increased and the benefits of global portfolio diversification decreased significantly after the global stock market crash of 1987. The 1987 and 2008 crashes are the two most important global stock market crashes since the 1929 Great depression. Although the effects of the 1987 crash on the comovements of national stock markets have been investigated extensively, the effects of the 2008 crash have not been studied sufficiently. In this paper we study this issue with a research sample that includes the U.S stock market and twenty European stock markets. We find that correlation between the twenty-one stock markets increased and the benefits of portfolio diversification decreased significantly after the 2008 stock market crash.
Under the constraint that the initial capital is not enough for a perfect hedge, the problem of deriving an optimal partial hedging portfolio so as to minimize the shortfall risk is worked out by solving two connected subproblems sequentially. One subproblem is to find the optimal terminal wealth that minimizes the shortfall risk. The shortfall risk is quantified by a general convex risk measure to accommodate different levels of risk tolerance. A convex duality approach is used to obtain an explicit formula for the optimal terminal wealth. The second subproblem is to derive the explicit expression for the admissible replicating portfolio that generates the optimal terminal wealth. We show by examples that to solve the second subproblem, the Malliavin calculus approach outperforms the traditional delta-hedging approach even for the simplest claim. Explicit worked-out examples include a European call option and a standard lookback put option
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