Convex order for path-dependent derivatives: a dynamic programming approach

We investigate the (functional) convex order of for various continuous martingale processes, either with respect to their diffusions coefficients for L\'evy-driven SDEs or their integrands for stochastic integrals. Main results are bordered by counterexamples. Various upper and lower bounds can be derived for path wise European option prices in local volatility models. In view of numerical applications, we adopt a systematic (and symmetric) methodology: (a) propagate the convexity in a {\em simulatable} dominating/dominated discrete time model through a backward induction (or linear dynamical principle); (b) Apply functional weak convergence results to numerical schemes/time discretizations of the continuous time martingale satisfying (a) in order to transfer the convex order properties. Various bounds are derived for European options written on convex pathwise dependent payoffs. We retrieve and extend former results obtains by several authors since the seminal 1985 paper by Hajek . In a second part, we extend this approach to Optimal Stopping problems using a that the Snell envelope satisfies (a') a Backward Dynamical Programming Principle to propagate convexity in discrete time; (b') satisfies abstract convergence results under non-degeneracy assumption on filtrations. Applications to the comparison of American option prices on convex pathwise payoff processes are given obtained by a purely probabilistic arguments.Comment: 48

Bank incentives and optimal CDOs

The paper examines a delegated monitoring problem between investors and a bank holding a portfolio of correlated loans displaying “contagion.” Moral hazard prevents the bank from monitoring continuously unless it is compensated with the right incentive-compatible contract. The asset pool is liquidated when losses exceed a state-contingent cut-off rule. The bank bears a relatively high share of the risk initially, as it should have high-powered incentives to monitor, but its long term financial stake tapers off as losses unfold. Liquidity regulation based on securitization can replicate the optimal contract. The sponsor provides an internal credit enhancement out of the proceeds of the sale and extends protection in the form of weighted tranches of collateralized debt obligations. In compensation the trust pays servicing and rent-preserving fees if a long enough period elapses with no losses occurring. Rather than being detrimental, well-designed securitization seems an effective means of implementing the second best.Credit risk transfer, Default Risk, Contagion.

Multilevel Richardson-Romberg extrapolation

We propose and analyze a Multilevel Richardson-Romberg (MLRR) estimator which combines the higher order bias cancellation of the Multistep Richardson-Romberg method introduced in [Pa07] and the variance control resulting from the stratification introduced in the Multilevel Monte Carlo (MLMC) method (see [Hei01, Gi08]). Thus, in standard frameworks like discretization schemes of diffusion processes, the root mean squared error (RMSE) $\varepsilon > 0$ can be achieved with our MLRR estimator with a global complexity of $\varepsilon^{-2} \log(1/\varepsilon)$ instead of $\varepsilon^{-2} (\log(1/\varepsilon))^2$ with the standard MLMC method, at least when the weak error $\mathbf{E}[Y_h]-\mathbf{E}[Y_0]$ of the biased implemented estimator $Y_h$ can be expanded at any order in $h$ and $\|Y_h - Y_0\|_2 = O(h^{\frac{1}{2}})$. The MLRR estimator is then halfway between a regular MLMC and a virtual unbiased Monte Carlo. When the strong error $\|Y_h - Y_0\|_2 = O(h^{\frac{\beta}{2}})$, $\beta < 1$, the gain of MLRR over MLMC becomes even more striking. We carry out numerical simulations to compare these estimators in two settings: vanilla and path-dependent option pricing by Monte Carlo simulation and the less classical Nested Monte Carlo simulation.Comment: 38 page

Stochastic Approximation with Averaging Innovation Applied to Finance

The aim of the paper is to establish a convergence theorem for multi-dimensional stochastic approximation when the "innovations" satisfy some "light" averaging properties in the presence of a pathwise Lyapunov function. These averaging assumptions allow us to unify apparently remote frameworks where the innovations are simulated (possibly deterministic like in Quasi-Monte Carlo simulation) or exogenous (like market data) with ergodic properties. We propose several fields of applications and illustrate our results on five examples mainly motivated by Finance

A mathematical treatment of bank monitoring incentives

In this paper, we take up the analysis of a principal/agent model with moral hazard introduced in [17], with optimal contracting between competitive investors and an impatient bank monitoring a pool of long-term loans subject to Markovian contagion. We provide here a comprehensive mathematical formulation of the model and show using martingale arguments in the spirit of Sannikov [18] how the maximization problem with implicit constraints faced by investors can be reduced to a classical stochastic control problem. The approach has the advantage of avoiding the more general techniques based on forward-backward stochastic differential equations described in [6] and leads to a simple recursive system of Hamilton-Jacobi-Bellman equations. We provide a solution to our problem by a verification argument and give an explicit description of both the value function and the optimal contract. Finally, we study the limit case where the bank is no longer impatient