350 research outputs found
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
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) can
be achieved with our MLRR estimator with a global complexity of
instead of with the standard MLMC method, at least when the weak
error of the biased implemented estimator
can be expanded at any order in and . The MLRR estimator is then halfway between a regular MLMC
and a virtual unbiased Monte Carlo. When the strong error , , 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
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