Multilevel Monte Carlo methods for applications in finance

Since Giles introduced the multilevel Monte Carlo path simulation method [18], there has been rapid development of the technique for a variety of applications in computational finance. This paper surveys the progress so far, highlights the key features in achieving a high rate of multilevel variance convergence, and suggests directions for future research.Comment: arXiv admin note: text overlap with arXiv:1202.6283; and with arXiv:1106.4730 by other author

Conditional Quasi-Monte Carlo with Constrained Active Subspaces

Conditional Monte Carlo or pre-integration is a useful tool for reducing variance and improving regularity of integrands when applying Monte Carlo and quasi-Monte Carlo (QMC) methods. To choose the variable to pre-integrate with, one need to consider both the variable importance and the tractability of the conditional expectation. For integrals over a Gaussian distribution, one can pre-integrate over any linear combination of variables. Liu and Owen (2022) propose to choose the linear combination based on an active subspace decomposition of the integrand. However, pre-integrating over such selected direction might be intractable. In this work, we address this issue by finding the active subspaces subject to the constraints such that pre-integration can be easily carried out. The proposed method is applied to some examples in derivative pricing under stochastic volatility models and is shown to outperform previous methods

Smoothing the payoff for efficient computation of Basket option prices

We consider the problem of pricing basket options in a multivariate Black Scholes or Variance Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster compared to Monte Carlo or Quasi Monte Carlo in dimensions up to 35

Stochastic mesh approximations for dynamic hedging with costs

Cette thèse se concentre sur le calcul de la solution optimale d'un problème de couverture de produit dérivé en temps discret. Le problème consiste à minimiser une mesure de risque, définie comme l'espérance d'une fonction convexe du profit (ou perte) du portefeuille, en tenant compte des frais de transaction. Lorsqu'il y a des coûts, il peut être optimal de ne pas transiger. Ainsi, les solutions sont caractérisées par des frontières de transaction. En général, les politiques optimales et les fonctions de risque associées ne sont pas connues explicitement, mais une stratégie bien connue consiste à approximer les solutions de manière récursive en utilisant la programmation dynamique. Notre contribution principale est d'appliquer la méthode du maillage stochastique. Cela permet d'utiliser des processus stochastiques multi-dimensionels pour les dynamiques de prix. On obtient aussi des estimateurs biasés à la hausse et à la baisse, donnant une mesure de la proximité de l'optimum. Nous considérons différentes façons d'améliorer l'efficacité computationelle. Utiliser la technique des variables de contrôle réduit le bruit qui provient de l'utilisation de prix de dérivés estimés à même le maillage stochastique. Deux autres techniques apportent des réductions complémentaires du temps de calcul : utiliser une grille unique pour les états du maillage et utiliser une procédure de "roulette Russe". Dans la dernière partie de la thèse, nous présentons une application pour le cas de la fonction de risque exponentielle négative et un modèle à volatilité stochastique (le modèle de Ornstein-Uhlenbeck exponentiel). Nous étudions le comportement des solutions sous diverses configurations des paramètres du modèle et comparons la performance des politiques basées sur un maillage à celles d'heuristiques.This thesis focuses on computing the optimal solution to a derivative hedging problem in discrete time. The problem is to minimize a risk measure, defined as the expectation of a convex function of the terminal profit and loss of the portfolio, taking transaction costs into account. In the presence of costs, it is sometimes optimal not to trade, so the solutions are characterized in terms of trading boundaries. In general, the optimal policies and the associated risk functions are not known explicitly, but a well-known strategy is to approximate the solutions recursively using dynamic programming. Our central innovation is in applying the stochastic mesh method, which was originally applied to option pricing. It allows exibility for the price dynamics, which could be driven by a multi-dimensional stochastic process. It also yields both low and high biased estimators of the optimal risk, thus providing a measure of closeness to the actual optimum. We look at various ways to improve the computational efficiency. Using the control variate technique reduces the noise that comes from using derivative prices estimated on the stochastic mesh. Two additional techniques turn out to provide complementary computation time reductions : using a single grid for the mesh states and using a so-called Russian roulette procedure. In the last part of the thesis, we showcase an application to the particular case of the negative exponential risk function and a stochastic volatility model (the exponential Ornstein-Uhlenbeck model). We study the behavior of the solutions under various configurations of the model parameters and compare the performance of the mesh-based policies with that of well-known heuristics