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Theses

Méthodes numériques e fficaces pour la valorisation des GMWB

Abstract : This thesis deals with the problem of pricing GMWB contracts by efficient numerical methods using closed formulas or Monte Carlo method under the constraint of few simulations. GMWB products are highly complex products that have in recent years experienced a great success due to the guarantee given to the insured people on future withdrawals with an upside effect depending on the performance of the underlying fund contract. In addition, the subscriber has many attractive options that can be exercised at any time such as the option to lapse partially or totally his contract, the possibility of fund switching during the policy life and lastly the option to advance or postpone the withdrawal date. However, such options combined with the complexity of the product, the market risk and the mortality risk expose the insurer who should manage a lot of contracts under several operational constraints (computational time, few simulations, etc..) to a major challenge in terms of valuation and hedging. A large part of this thesis (Chapters 2, 4, 5 and 6) study the partial and total surrender option in the GMWB contracts from two points of view: the rational client and the hedger worst case. In this regard, in our general discrete time framework with a local volatility and Hull-White interest rates models, the optimal strategy determining the contract cost is in both cases the solution to an optimal stochastic control problem in discrete time. However, thanks to partial homogeneity property of prices and flows, we show that the strategy is Bang-Bang, hence explicit. Thus, the valuation problem is reduced to an optimal stopping problem, we propose a Monte Carlo method using Longstaff-Schwartz methodology whose empirical regression step was treated by the standard least squares method as well as a new method called VCP (preliminary control variates). The latter consists first to reduce the empirical variance of the flows to regress through an L2 projection on adapted and centered control variates, then to perform the standard least square regression methods on the new flows with reduced variance. A numerical study and a theoretical error analysis based on distribution-free nonparametric regression techniques confirme the VCP method efficiency in the context of few simulations. Chapter 3 gathers mathematical and numerical justifications of the hypothesis considering mortality risk as diversifiable. We achieve this in a simple American exercise style product sensitive to mortality risk. Finally, the last part (Chapter 7) is devoted to pricing GMWB contracts by closed analytical approximations in a Black Scholes model with Hull-White interest rates. By performing an asymptotic expansion on the amount of withdrawals, we obtain approximation price formulas as a Black-Scholes price plus a sum of explicit Greeks, faster to evaluate. Error estimates are provided when the payoff function is regular. The accuracy of the asymptotic formulas is experienced numerically and it shows a good behavior even for long maturity contracts (20 years).
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Submitted on : Thursday, March 14, 2013 - 12:36:23 PM
Last modification on : Wednesday, March 27, 2019 - 4:08:30 PM
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Tarik Ben Zineb. Méthodes numériques e fficaces pour la valorisation des GMWB. Probabilités [math.PR]. Ecole Polytechnique X, 2012. Français. ⟨pastel-00800761⟩

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