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Two studies in risk management: portfolio insurance under risk measure constraint and quadratic hedge for jump processes.

Abstract : In this thesis I'm interested in two aspects of portfolio management: the portfolio insurance under a risk measure constraint and quadratic hedge in incomplete markets. Part I. I study the problem of portfolio insurance from the point of view of a fund manager, who guarantees to the investor that the portfolio value at maturity will be above a fixed threshold. If, at maturity, the portfolio value is below the guaranteed level, a third party will refund the investor up to the guarantee. In exchange for this protection, for which the investor pays a given fee, the third party imposes a limit on the risk exposure of the fund manager, in the form of a convex monetary risk measure. The fund manager therefore tries to maximize the investor's utility function subject to the risk measure constraint. I give a full solution to this nonconvex optimization problem in the complete market setting and show in particular that the choice of the risk measure is crucial for the optimal portfolio to exist. An interesting outcome is that the fund manager's maximization problem may not admit an optimal solution for all convex risk measures, which means that not all convex risk measures may be used to limit fund's exposure in this way. I provide conditions for the existence of the solution. Explicit results are provided for the entropic risk measure (for which the optimal portfolio always exists), for the class of spectral risk measures (for which the optimal portfolio may fail to exist in some cases) and for the G-divergence. Key words: Portfolio Insurance; utility maximization; convex risk measure; VaR, CVaR and spectral risk measure; entropy and G-divergence. Part II. In the second part I study the problem of quadratic hedge in incomplete markets. I work with a three-dimensional Markov jump process: the first component is the state variable representing the hedging instrument traded in the market, the second component models a risk factor which "perturbs" the dynamics of the hedging instrument and is not traded in the market (as a volatility factor for example in stochastic volatility models); the third one is another source of risk which affects the option's payoff at maturity and is also not traded in the market. The problem can be seen then as a constrained quadratic hedge problem. I privilege here the dynamic programming approach which allows me to obtain the HJB equation related to the value function. This equation is semi linear and non local due the presence of jumps. The main result of this thesis is that this value function, as a function of the initial wealth, is a second order polynomial whose coefficients are characterized as the unique smooth solutions of a triplet of PIDEs, the first of which is semi linear and does not depend on the particular choice of option one wants to hedge, the other two being simply linear. This result is stated when the Markov process is assumed to be a non-generate jump-diffusion and when it is a pure jump process. I finally apply my theoretical results to an example of quadratic hedge in the context of electricity markets.
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Submitted on : Friday, June 15, 2012 - 8:10:25 AM
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Carmine de Franco. Two studies in risk management: portfolio insurance under risk measure constraint and quadratic hedge for jump processes.. Probability [math.PR]. Université Paris-Diderot - Paris VII, 2012. English. ⟨tel-00708397⟩

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