An introduction to multilevel Monte Carlo for option valuation

Monte Carlo is a simple and flexible tool that is widely used in computational finance. In this context, it is common for the quantity of interest to be the expected value of a random variable defined via a stochastic differential equation. In 2008, Giles proposed a remarkable improvement to the approach of discretizing with a numerical method and applying standard Monte Carlo. His multilevel Monte Carlo method offers an order of speed up given by the inverse of epsilon, where epsilon is the required accuracy. So computations can run 100 times more quickly when two digits of accuracy are required. The multilevel philosophy has since been adopted by a range of researchers and a wealth of practically significant results has arisen, most of which have yet to make their way into the expository literature. In this work, we give a brief, accessible, introduction to multilevel Monte Carlo and summarize recent results applicable to the task of option evaluation.Comment: Submitted to International Journal of Computer Mathematics, special issue on Computational Methods in Financ

Pricing American Options using Monte Carlo Method

This thesis reviewed a number of Monte Carlo based methods for pricing American options. The least-squares regression based Longstaff-Schwartz method (LSM) for approximating lower bounds of option values and the Duality approach through martingales for estimating the upper bounds of option values were implemented with simple examples of American put options. The effectiveness of these techniques and the dependencies on various simulation parameters were tested and discussed. A computing saving technique was suggested to reduce the computational complexity by constructing regression basis functions which are orthogonal to each other with respect to the natural distribution of the underlying asset price. The orthogonality was achieved by using Hermite polynomials. The technique was tested for both the LSM approach and the Duality approach. At the last, the Multilevel Mote Carlo (MLMC) technique was employed with pricing American options and the effects on variance reduction were discussed. A smoothing technique using artificial probability weighted payoff functions jointly with Brownian Bridge interpolations was proposed to improve the Multilevel Monte Carlo performances for pricing American options

Pricing exotic options using strong convergence properties?

In finance, the strong convergence properties of discretisations of stochastic differential equations (SDEs) are very important for the hedging and valuation of exotic options. In this paper we show how the use of the Milstein scheme can improve the convergence of the multi-level Monte Carlo method, so that the computational cost to achieve an accuracy of O(e) is reduced to O($\epsilon^{−2}$) for a Lipschitz payoff. The Milstein scheme gives first order strong convergence for all 1−dimensional systems (one Wiener process). However, for processes with two or more Wiener processes, such as correlated portfolios and stochastic volatility models, there is no exact solution for the iterated integrals of second order (Lévy area) and the Milstein scheme neglecting the Lévy area gives the same order of convergence as the Euler-Maruyama scheme. The purpose of this paper is to show that if certain conditions are satisfied, we can avoid the calculation of the Lévy area and obtain first convergence order by applying an orthogonal transformation. We demonstrate when the conditions of the 2−Dimensional problem permit this and give an exact solution for the orthogonal transformation. We present examples of pricing exotic options to demonstrate that the use of both the orthogonal Milstein scheme and the Multi-level Monte Carlo give a substantial reduction in the computation cost

Stochastic ordinary differential equations in applied and computational mathematics

Using concrete examples, we discuss the current and potential use of stochastic ordinary differential equations (SDEs) from the perspective of applied and computational mathematics. Assuming only a minimal background knowledge in probability and stochastic processes, we focus on aspects that distinguish SDEs from their deterministic counterparts. To illustrate a multiscale modelling framework, we explain how SDEs arise naturally as diffusion limits in the type of discrete-valued stochastic models used in chemical kinetics, population dynamics, and, most topically, systems biology. We outline some key issues in existence, uniqueness and stability that arise when SDEs are used as physical models, and point out possible pitfalls. We also discuss the use of numerical methods to simulate trajectories of an SDE and explain how both weak and strong convergence properties are relevant for highly-efficient multilevel Monte Carlo simulations. We flag up what we believe to be key topics for future research, focussing especially on nonlinear models, parameter estimation, model comparison and multiscale simulation

Pricing American Options by Exercise Rate Optimization

We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black-Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black-Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model