Manuel Guerra

Optimal Reinsurance for Variance Related Premium Calculation Principles

This paper deals with numerical computation of the optimal form of reinsurance from the ceding company point of view, when the cedent seeks to maximize the adjustment coefficient of the retained risk and the reinsurance loading is an increasing function of the variance. We compare the optimal treaty with the best stop loss policy. The optimal arrangement can provide a significant improvement in the adjustment coefficient when compared to the best stop loss treaty. Further, it is substantially more robust with respect to choice of the retention level than stop-loss treaties.

Existence and Lipschitzian Regularity for Relaxed Minimizers

In this contribution we follow two main goals: to reconstruct a result announced in [4] about existence of relaxed minimizers for (nonconvex) Lagrange problems of optimal control (Theorem 1); to derive conditions for Lipschitzian regularity of trajectories corresponding to relaxed minimizers (Theorem 3). In passing, elaborating on the approach used in [10], we provide a condition for Lipschitzian regularity of non relaxed minimizers (Theorem 2).

Approximation of Generalized Minimizers and Regularization of Optimal Control Problems

An open problem, set by Yu. Orlov in his contribution to the volume “Open Problems in Mathematical Systems and Control Theory”, V. Blondel, A. Megretski Eds., 2004, regards regularization of optimal control-affine problems with control-independent state-quadratic cost. It is asked whether the infima of the regularized (by adding squared L 2-norm of controls) functionals converge to the infimum of the original functional?

Hamiltonian flows for impulsive control systems

Abstract Under appropriate commutativity assumptions, smooth control systems that are affine with respect to controls can be extended into classes of generalized controls that contain impulses. We show how to construct generalized Hamiltonian trajectories for the extended system that lift both the continuous and the discontinuous components of candidate optimal trajectories into the cotangent bundle. This construction gives useful insights into the structure of generalized extremal trajectories. lt is also useful from the computational point of view. An example is discussed.

Barrier Option Pricing under the 2-Hypergeometric Stochastic Volatility Model

We investigate the pricing of financial options under the 2-hypergeometric stochastic volatility model. This is an analytically tractable model that reproduces the volatility smile and skew effects observed in empirical market data. Using a regular perturbation method from asymptotic analysis of partial differential equations, we derive an explicit and easily computable approximate formula for the pricing of barrier options under the 2-hypergeometric stochastic volatility model. The asymptotic convergence of the method is proved under appropriate regularity conditions, and a multi-stage method for improving the quality of the approximation is discussed. Numerical examples are also provided.

Indifference Pricing in a Market with Transaction Costs and Jumps

We present an approach for pricing a European call option in presence of proportional transaction costs, considering the dynamics of the stock price following a general exponential Levy process. The model is a generalization of the celebrated work of Davis, Panas and Zariphopoulou, where the value of the option is defined as the utility indifference price. This approach requires the solution of two stochastic singular control problems in finite time, satisfying the same Hamilton-Jacobi-Bellman equation with different terminal conditions. Numerical results are obtained by Markov chain approximation methods. Option prices are computed for both writer and buyer, when the returns follow a Brownian motion and a Variance Gamma process.

Stochastic Dynamic Programming and Control of Markov Processes

This chapter contains a brief discussion of the basic mathematical ideas behind dynamic programming methods for optimal control of Markov processes. It is based on lectures given by the author at the Summer School on Computational Finance held at Smolenice Castle, Slovakia, in September 2014.

Exit option for a class of profit functions

ABSTRACT In this paper we propose a formula to derive the value of a firm which is currently producing a certain product and faces the option to exit the market, whose demand follows a geometric Brownian motion. The problem of optimal exiting is an optimal stopping problem that can be solved using the dynamic programming principle. This is a free-boundary problem. We propose an approximation for the original model and, using the Implicit Function Theorem, we obtain the solution of the original problem. Finally we show, analytically, that the exit threshold is decreasing with the volatility as well as the drift of the geometric Brownian motion.

On Nonautonomous Singular L-Q Problems

Abstract We show how to apply to singular nonautonomous linear-quadratic optimal control problems some tools previously used to solve the autonomous case. We deal with both the constant, and the time-variant order of singularity cases