Etienne Chevalier

An Optimal Dividend and Investment Control Problem under Debt Constraints

This paper concerns the problem of determining an optimal control on the dividend and investment policy of a firm under debt constraints. We allow the company to make investment by increasing its outstanding indebtedness, which would impact its capital structure and risk profile, thus resulting in higher interest rate debts. Moreover, a high level of debt is also a challenging constraint to any firm, as it is the threshold below which the firm value should never go to avoid bankruptcy. It is equally possible for the firm to divest parts of its business in order to decrease its financial debt owed to creditors. In addition, the firm may favor investment by postponing or reducing any dividend distribution to shareholders. We formulate this problem as a combined singular and multiswitching control problem and use a viscosity solution approach to get qualitative descriptions of the solution. We further enrich our studies with a complete resolution of the problem in the two-regime case and provide some numeric...

Max-Min optimization problem for Variable Annuities pricing

In this paper, we study the valuation of variable annuities for an insurer. We concentrate on two types of these contracts, namely guaranteed minimum death benefits and guaranteed minimum living benefits that allow the insured to withdraw money from the associated account. Here, the price of variable annuities corresponds to a fee, fixed at the beginning of the contract, that is continuously taken from the associated account. We use a utility indifference approach to determine the indifference fee rate. We focus on the worst case for the insurer, assuming that the insured makes the withdrawals that minimize the expected utility of the insurer. To compute this indifference fee rate, we link the utility maximization in the worst case for the insurer to a sequence of maximization and minimization problems that can be computed recursively. This allows to provide an optimal investment strategy for the insurer when the insured follows the worst withdrawal strategy and to compute the indifference fee. We finally explain how to approximate these quantities via the previous results and give numerical illustrations of parameter sensitivity.

Optimal Early Retirement Near the Expiration of a Pension Plan

In a recent paper, Friedman and Shen (Finance Stoch 6: 273–302, 2002) have considered a pension plan with the option of early retirement. They showed that the financial value V of the retirement benefits is the solution of a variational inequality and have studied the associated free boundary. A description of the free boundary near maturity is given, thanks to integral equation methods. However, V is also the solution of an optimal stopping problem very close to the American option valuation problem. Comparing V to specific options, we derive an expansion of the free boundary near the expiration of the pension plan.

Optimal Market Dealing Under Constraints

We consider a market dealer acting as a liquidity provider by continuously setting bid and ask prices for an illiquid asset in a quote-driven market. The market dealer may benefit from the bid–ask spread, but has the obligation to permanently quote both prices while satisfying some liquidity and inventory constraints. The objective is to maximize the expected utility from terminal liquidation value over a finite horizon and subject to the above constraints. We characterize the value function as the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation, and further enrich our study with numerical results. The contributions of our study concern both the modelling aspects and the dynamic structure of the control strategies. Important features and constraints characterizing market making problems are no longer ignored.

Exercise Boundary Near Maturity for an American Option on Several Assets

We consider an American put option on a linear function of d dividend-paying assets. The value function of this option is given as the solution of a free boundary problem. When d = 1, the behavior of the free boundary near the maturity of the option is well known. In this article, we extend to the case d > 1 the study of the free boundary near maturity. A parameterization of the stopping region at time t is given. That enables us to define and give a convergence rate for this region when t goes to the maturity.

Bermudean Approximation of the Free Boundary Associated with an American Option

American options valuation leads to solve an optimal stopping problem or a variational inequality. These two approaches involve the knowledge of a free boundary, boundary of the so-called exercise region. As we are not able to get a closed formula for the American option value function, we will approximate the free boundary by this of a Bermudean option. Indeed a Bermudean option value function is the solution of an optimal stopping problem which can be viewed as a free boundary problem. Thanks to a maximum principle, we evaluate the difference between Bermudean and American boundaries.