Fast Estimation of True Bounds on Bermudan Option Prices under Jump-diffusion Processes

Fast pricing of American-style options has been a difficult problem since it was first introduced to financial markets in 1970s, especially when the underlying stocks' prices follow some jump-diffusion processes. In this paper, we propose a new algorithm to generate tight upper bounds on the Bermudan option price without nested simulation, under the jump-diffusion setting. By exploiting the martingale representation theorem for jump processes on the dual martingale, we are able to explore the unique structure of the optimal dual martingale and construct an approximation that preserves the martingale property. The resulting upper bound estimator avoids the nested Monte Carlo simulation suffered by the original primal-dual algorithm, therefore significantly improves the computational efficiency. Theoretical analysis is provided to guarantee the quality of the martingale approximation. Numerical experiments are conducted to verify the efficiency of our proposed algorithm

Multilevel dual approach for pricing American style derivatives

In this article we propose a novel approach to reduce the computational complexity of the dual method for pricing American options. We consider a sequence of martingales that converges to a given target martingale and decompose the original dual representation into a sum of representations that correspond to different levels of approximation to the target martingale. By next replacing in each representation true conditional expectations with their Monte Carlo estimates, we arrive at what one may call a multilevel dual Monte Carlo algorithm. The analysis of this algorithm reveals that the computational complexity of getting the corresponding target upper bound, due to the target martingale, can be significantly reduced. In particular, it turns out that using our new approach, we may construct a multilevel version of the well-known nested Monte Carlo algorithm of Andersen and Broadie (2004) that is, regarding complexity, virtually equivalent to a non-nested algorithm. The performance of this multilevel algorithm is illustrated by a numerical example

Unbiased Optimal Stopping via the MUSE

We propose a new unbiased estimator for estimating the utility of the optimal stopping problem. The MUSE, short for Multilevel Unbiased Stopping Estimator, constructs the unbiased Multilevel Monte Carlo (MLMC) estimator at every stage of the optimal stopping problem in a backward recursive way. In contrast to traditional sequential methods, the MUSE can be implemented in parallel. We prove the MUSE has finite variance, finite computational complexity, and achieves $\epsilon$-accuracy with $O(1/\epsilon^2)$ computational cost under mild conditions. We demonstrate MUSE empirically in an option pricing problem involving a high-dimensional input and the use of many parallel processors.Comment: 39 pages, add several numerical experiments and technical results, accepted by Stochastic Processes and their Application

On Nesting Monte Carlo Estimators

Many problems in machine learning and statistics involve nested expectations and thus do not permit conventional Monte Carlo (MC) estimation. For such problems, one must nest estimators, such that terms in an outer estimator themselves involve calculation of a separate, nested, estimation. We investigate the statistical implications of nesting MC estimators, including cases of multiple levels of nesting, and establish the conditions under which they converge. We derive corresponding rates of convergence and provide empirical evidence that these rates are observed in practice. We further establish a number of pitfalls that can arise from naive nesting of MC estimators, provide guidelines about how these can be avoided, and lay out novel methods for reformulating certain classes of nested expectation problems into single expectations, leading to improved convergence rates. We demonstrate the applicability of our work by using our results to develop a new estimator for discrete Bayesian experimental design problems and derive error bounds for a class of variational objectives.Comment: To appear at International Conference on Machine Learning 201

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

A multilevel Monte Carlo algorithm for SDEs driven by countably dimensional Wiener process and Poisson random measure

In this paper, we investigate the properties of standard and multilevel Monte Carlo methods for weak approximation of solutions of stochastic differential equations (SDEs) driven by the infinite-dimensional Wiener process and Poisson random measure with Lipschitz payoff function. The error of the truncated dimension randomized numerical scheme, which is determined by two parameters, i.e grid density $n \in \mathbb{N}_{+}$ and truncation dimension parameter $M \in \mathbb{N}_{+},$ is of the order $n^{-1/2}+\delta(M)$ such that $\delta(\cdot)$ is positive and decreasing to $0$. We derive complexity model and provide proof for the upper complexity bound of the multilevel Monte Carlo method which depends on two increasing sequences of parameters for both $n$ and $M.$ The complexity is measured in terms of upper bound for mean-squared error and compared with the complexity of the standard Monte Carlo algorithm. The results from numerical experiments as well as Python and CUDA C implementation are also reported.Comment: 23 pages, 4 figures, 2 code listing

Multilevel Ansatz zur Bewertung von Bermuda Optionen

The Multilevel approach has been introduced into stochastics by Heinrich 2001 and Giles 2008. It is an idea about how to reduce the complexity of Monte Carlo simulations, if the precision and computational time of these simulations depend on a parameter. In this work, the Multilevel approach will be applied to approximate the fair price of a Bermudan option. The latter is a financial option that gives the holder the right to get an amount of money depending on a stochastic process at one of finitely many exercise dates. The strategy when to exercise the option should therefore by optimized. The task is now to find stochastic methods that calculate a lower and an upper bound for the fair price of such an option. If the performance of such methods is measured via its mean-squared error epsilon, the calculation of lower bounds has a complexity of epsilon^{-3} in the usual case. Here, the usual case is the combination of a good-natured problem and the use of a stochastic mesh method. By using the Multilevel approach, we can reduce the complexity down to epsilon^{-2.5}. In other cases, the reduction of complexity can be even of order epsilon^{-1} instead of epsilon^{-0.5}. In order to find upper bounds, we use the dual method from Rogers 2002 and Haugh and Kogan 2004. It expresses the fair price of the option as a minimization problem over a set of martingales. Andersen and Broadie 2004 exploit this idea by approximating martingales with nested simulations. These nested simulations lead to a high computational complexity. Depending on the problem, this complexity can be of order epsilon^{-3} or even epsilon^{-4}. The Multilevel approach reduces this order in each case to epsilon^{-2} up to a logarithmic factor.Der Multilevel Ansatz wurde durch die Arbeiten von Heinrich 2001 und Giles 2008 in der Stochastik populär. Es handelt sich dabei um eine Technik, die die Komplexität einer Monte-Carlo Simulation reduzieren kann, wenn deren Rechenzeit und Präzision von einem Parameter abhängt. Dieser Ansatz wird im Folgenden angewendet, um die Berechnung des fairen Preises einer Bermuda-Finanzoption zu berechnen. Letztere ist ein Derivat, das dem Besitzer das Recht gibt, einen von einem stochastischen Prozess abhängigen Geldbetrag an einem von endlich vielen gegebenen Zeitpunkten zu erhalten. Deshalb sollte der Zeitpunkt, an dem der Besitzer die Option einlöst, optimal gewählt werden. Die Aufgabe besteht nun darin, mit stochastischen Methoden eine obere und eine untere Schranke für den fairen Preis einer solchen Option zu berechnen. Bewertet man die Qualität einer solchen Methode mithilfe des mittleren quadratischen Fehlers epsilon, so ergibt sich für die Berechnung unterer Schranken eine Komplexität von epsilon^{-3} im gewöhnlichen Fall. Unter dem gewöhnlichen Fall ist ein gut gestelltes Problem und die Verwendung eines stochastischen Netzes zu verstehen. Mit Hilfe des Multilevel Ansatzes lässt sich in diesem Fall eine Komplexität von epsilon^{-2.5} erreichen. Die Verbesserung um den Faktor epsilon^{-0.5} kann in anderen Fällen sogar bis zu epsilon^{-1} betragen. Um obere Schranken für den Wert einer Bermuda Option zu berechnen, kann die duale Formulierung nach Rogers 2002 und Haugh und Kogan 2004 benutzt werden. Diese drückt den Wert der Option als Minimierungsproblem über einer Menge adaptierter Martingale aus. Andersen und Broadie 2004 nutzen in ihrer Arbeit diese Formulierung, indem sie das optimale Martingal mit Hilfe von geschachtelten Simulationen approximieren. Diese geschachtelten Simulationen führen zu hohem Rechenaufwand, welcher erneut durch den Multilevel Ansatz reduziert werden kann. Je nach Problemstellung kann das Problem eine Komplexität von epsilon^{-3} oder sogar epsilon^{-4} aufweisen. Der Multilevel Ansatz senkt die Komplexität in jedem Fall (bis auf einen logarithmischen Faktor) auf epsilon^{-2}