Two studies in risk management: risk-constrained portfolio insurance and quadratic hedging in jump models.

Authors
Publication date
2012
Publication type
Thesis
Summary In the first part, I am interested in a portfolio insurance problem for a manager of an investment fund, who wants to structure a financial product for investors with a capital guarantee. If the value of the product is below a fixed threshold, the investor will be reimbursed up to this threshold by the fund's insurer. In exchange, the insurer imposes a constraint on the risk that the manager can tolerate, measured with a risk measure. I give the solution to this problem and prove that the choice of the risk measure is a crucial point for the existence of an optimal portfolio. I apply my results for the entropy, spectral and G-divergence risk measures. Next, I focus on the quadratic hedging problem. The market is described by a three-dimensional Markov process with jumps. The first variable models the hedging instrument that is tradable on the market, the second one a financial asset that disturbs the dynamics of the hedging instrument and that is not tradable, such as a volatility factor. The third one represents a risk source that affects the option to be hedged and that is also not tradable. I prove that the value function of the problem is characterized by the unique solution of a system of three integro-differential equations, one of which is semilinear and does not depend on the option to be hedged, and the other two are linear. This allows me to characterize the optimal strategy. This result is demonstrated if the process is non-degenerate (strictly elliptic Brownian component) and if it is pure jump. I conclude with an application in the electricity market.
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