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Operations Research

Handling Discontinuities in Financial Engineering: Good Path Simulation and Smoothing

Xiaoqun Wang

Published Feb 16, 2016• 11 citations
Publisher

Abstract

Discontinuities are common in the pricing and hedging of complex financial derivatives. Quasi-Monte Carlo (QMC) methods for high-dimensional finance problems with discontinuities can be inefficient because of the lack of good smoothness and high dimensionality. Interestingly, path simulation method (PSM) may affect both factors, implying its significance in QMC methods. What defines a “good” PSM for problems with discontinuities? The ability to align the discontinuities with the coordinate axes is a desirable property for a PSM. We show that for an arbitrary PSM, there exists a class of options with discontinuous payoff functions such that the transformed functions have only axis-parallel discontinuities, for which good QMC performance can be expected. In this sense, any PSM can be “good” in QMC methods for a specific class of problems. We analyze the structure of discontinuities for digital options using the new approach and show the superiority and the uniqueness (up to a permutation) of the standard construction. We develop a two-step procedure for pricing and hedging derivatives with discontinuous payoff functions. The first step is to design a good PSM that has the ability to align the discontinuities with the coordinate axes and the second step is to further exploit this nice property to remove the discontinuities completely. We prove that the new estimate is unbiased and has smaller variance. Numerical experiments demonstrate that the two-step procedure is very effective in QMC methods for pricing options and estimating Greeks, leading to a dramatic variance reduction. Both the path simulation step and the smoothing step are crucial and beneficial for QMC methods, with the contribution from each step varying depending on the severity of discontinuity.

DOI
10.1287/opre.2015.1470

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Discontinuities are common in the pricing and hedging of complex financial derivatives. Quasi-Monte Carlo (QMC) methods for high-dimensional finance problems with discontinuities can be inefficient because of the lack of good smoothness and high dimensionality. Interestingly, path simulation method (PSM) may affect both factors, implying its significance in QMC methods. What defines a “good” PSM for problems with discontinuities? The ability to align the discontinuities with the coordinate axes is a desirable property for a PSM. We show that for an arbitrary PSM, there exists a class of options with discontinuous payoff functions such that the transformed functions have only axis-parallel discontinuities, for which good QMC performance can be expected. In this sense, any PSM can be “good” in QMC methods for a specific class of problems. We analyze the structure of discontinuities for digital options using the new approach and show the superiority and the uniqueness (up to a permutation) of the standard construction. We develop a two-step procedure for pricing and hedging derivatives with discontinuous payoff functions. The first step is to design a good PSM that has the ability to align the discontinuities with the coordinate axes and the second step is to further exploit this nice property to remove the discontinuities completely. We prove that the new estimate is unbiased and has smaller variance. Numerical experiments demonstrate that the two-step procedure is very effective in QMC methods for pricing options and estimating Greeks, leading to a dramatic variance reduction. Both the path simulation step and the smoothing step are crucial and beneficial for QMC methods, with the contribution from each step varying depending on the severity of discontinuity.
Quasi–Monte Carlo (QMC) methods are important numerical tools in the pricing and hedging of complex financial instruments. The effectiveness of QMC methods crucially depends on the discontinuity and the dimension of the problem. This paper shows how the two fundamental limitations can be overcome in some cases. We first study how path-generation methods (PGMs) affect the structure of the discontinuities and what the effect of discontinuities is on the accuracy of QMC methods. The insight is that the discontinuities can be QMC friendly (i.e., aligned with the coordinate axes) or not, depending on the PGM. The PGMs that offer the best performance in QMC methods are those that make the discontinuities QMC friendly. The structure of discontinuities can affect the accuracy of QMC methods more significantly than the effective dimension. This insight motivates us to propose a novel way of handling the discontinuities. The basic idea is to align the discontinuities with the coordinate axes by a judicious design of a method for simulating the underlying processes. Numerical experiments demonstrate that the proposed method leads to dramatic variance reduction in QMC methods for pricing options and for estimating Greeks. It also reduces the effective dimension of the problem. This paper was accepted by Assaf Zeevi, stochastic models and simulation.
ABSTRACT For a trajectory simulated from s standardized independent normal variates ε = (ε1, . . . , εs )1, the payoff of a European option can be represented as max[g(ε), 0], where the function g(ε) is assumed to be differentiable and it relates to the nature of the derivative securities. In this paper, we develop a new simulation technique by introducing an orthogonal class of transformation to ε so that the function g is instead generated from g(Aε), where A is an s-dimensional orthogonal matrix. The matrix A is optimally determined so that the effective dimension of the underlying function is minimized, thereby enhancing the quasi-Monte Carlo (QMC) method. The proposed simulation approach has the advantage of greater generality and has a wide range of applications as long as the problem of interest can be represented by g(ε). The flexibility of our proposed technique is illustrated by applying it to two high-dimensional applications: Asian basket options and European call options with a stochastic volatility model. We benchmark our proposed method against well-known efficient simulation algorithms that have been advocated in these applications. The numerical results demonstrate that our proposed technique can be an extremely powerful simulation method when combined with QMC.
The pricing of American options is one of the most challenging problems in financial engineering due to the involved optimal stopping time problem, which can be solved by using dynamic programming (DP). But applying DP is not always practical, especially when the state space is high dimensional. However, the curse of dimensionality can be overcome by Monte Carlo (MC) simulation. We can get lower and upper bounds by MC to ensure that the true price falls into a valid confidence interval. During the recent decades, progress has been made in using MC simulation to obtain both the lower bound by least-squares Monte Carlo method (LSM) and the upper bound by duality approach. However, there are few works on pricing American options using quasi-Monte Carlo (QMC) methods, especially to compute the upper bound. For comparing the sample variances and standard errors in the numerical experiments, randomized QMC (RQMC) methods are usually used. In this paper, we propose to use RQMC to replace MC simulation to compute both the lower bound (by the LSM) and the upper bound (by the duality approach). Moreover, we propose to use dimension reduction techniques, such as the Brownian bridge, principal component analysis, linear transformation and the gradients based principle component analysis. We perform numerical experiments on American–Asian options and American max-call options under the Black–Scholes model and the variance gamma model, in which the options have the path-dependent feature or are written on multiple underlying assets. We find that RQMC in combination with dimension reduction techniques can significantly increase the efficiency in computing both the lower and upper bounds, resulting in better estimates and tighter confidence intervals of the true price than pure MC simulation.
Abstract We consider randomized QMC methods for approximating the expected recourse in two-stage stochastic optimization problems containing mixed-integer decisions in the second stage. It is known that the second-stage optimal value function is piecewise linear-quadratic with possible kinks and discontinuities at the boundaries of certain convex polyhedral sets. This structure is exploited to provide conditions implying that first and higher order terms of the integrand’s ANOVA decomposition (Math. Comp. 79 (2010), 953–966) have mixed weak first order partial derivatives. This leads to a good smooth approximation of the integrand and, hence, to good convergence rates of randomized QMC methods if the effective (superposition) dimension is low.
As one of the major methods for the simulation of option pricing, Monte Carlo method assumes random fluctuations in the distribution of asset prices. Under certain uncertainties process, different evolution paths could be simulated so as to finally yield the expectation value of the asset price, which requires a lot of simulations to ensure the accuracy based on huge and expensive calculations. In order to solve the above computational problem, quantum Monte Carlo (QMC) has been established and applied in the relevant systems such as European call options. In this work, both MC and QM methods are adopted to simulate European call options. Based on the preparation of quantum states in QMC algorithm and the construction of quantum circuits by simulating a quantum hardware environment on a traditional computer, the amplitude estimation (AE) algorithm is found to play a secondary role in accelerating the pricing of European options. More importantly, the payoff function and the time required for the simulation in QMC method show some improvements than those in MC method.
The quasi-Monte Carlo (QMC) methods use deterministic, usually low-discrepancy sequences, to improve the order of convergence when compared with Monte Carlo methods dealing with the same problem. Although this improved convergence depends on the smoothness properties of the functions under consideration and sometimes cannot be established theoretically, good practical results were obtained in many areas, notably in Financial Mathematics. Most of the QMC methods do not depend on particular properties of the sequences used beyond their good distribution in a suitable multidimensional space. Many families of low-discrepancy sequences have been studies, with the family of the Sobol sequences being not only one of the oldest but also the most popular in practical applications. Since these sequences have many free parameters, it is desirable to have methods to obtain a good set of parameters, suitable for a wide class of problems. In this work we propose a fully deterministic method for optimisation of the direction numbers of the Sobol sequences that is flexible in taking into account smoothness properties of the potential sub-integral functions. We demonstrate how by implementing this algorithm on modern GPUs it is possible to obtain good sets of direction numbers within a reasonable timeframe. The numerical tests demonstrate the significant advantage in terms of accuracy that is observed, which makes the corresponding QMC methods even more competitive.