Pricing American Options using Monte Carlo Methods

The Journal of Computational Finance, 2001

A number of Monte Carlo simulation-based approaches have been proposed within the past decade to address the problem of pricing American-style derivatives. The purpose of this paper is to empirically test some of these algorithms on a common set of problems in order to be able to assess the strengths and weaknesses of each approach as a function of the problem characteristics. In addition, we introduce another simulation-based approach that parameterizes the early exercise curve and casts the valuation problem as an optimization problem of maximizing the expected payoff (under the martingale measure) with respect to the associated parameters, the optimization problem carried out using a simultaneous perturbation stochastic approximation (SPSA) algorithm.

Sadhana, 2005

Pricing financial options is amongst the most important and challenging problems in the modern financial industry. Except in the simplest cases, the prices of options do not have a simple closed form solution and efficient computational methods are needed to determine them. Monte Carlo methods have increasingly become a popular computational tool to price complex financial options, especially when the underlying space of assets has a large dimensionality, as the performance of other numerical methods typically suffer from the 'curse of dimensionality'. However, even Monte-Carlo techniques can be quite slow as the problem-size increases, motivating research in variance reduction techniques to increase the efficiency of the simulations. In this paper, we review some of the popular variance reduction techniques and their application to pricing options. We particularly focus on the recent Monte-Carlo techniques proposed to tackle the difficult problem of pricing American options. These include: regression-based methods, random tree methods and stochastic mesh methods. Further, we show how importance sampling, a popular variance reduction technique, may be combined with these methods to enhance their effectiveness. We also briefly review the evolving options market in India.

The Journal of Computational Finance, 2014

We investigate the performance of the Ordinary Least Squares (OLS) regression method in Monte Carlo simulation algorithms for pricing American options. We compare OLS regression against several alternatives and find that OLS regression underperforms methods that penalize the size of coefficient estimates. The degree of underperformance of OLS regression is greater when the number of simulation paths is small, when the number of functions in the approximation scheme is large, when European option prices are included in the approximation scheme, and when the number of exercise opportunities is large. Based on our findings, instead of using OLS regression we recommend an alternative method based on a modification of Matching Projection Pursuit.

Asia-Pacific journal of financial studies

This paper presents a new methodology to approximate the value of American options by least-squares Monte Carlo simulation. Whereas Longstaff and Schwartz's approach do not utilize the underlying asset price movement, we develop several methods that incorporates it into option pricing. One category improves the R-squares from the regressions by using, [1] the weighted regression with the same regressors and, [2] new regressors which are related to the discount factor from the current decision to exercise time. The other category improves the computational speed without sacrificing the convergence level by, [1] terminating early during the backwardation procedure and, [2] decreasing the number of observations for the regressors. Finally, combining both methods, we can get improved R-squares and computational speed in comparison to Longstaff and Schwartz's approach.

This paper proposes several improvements to the least squares Monte Carlo (LSMC) option valuation method. We test different regression algorithms and suggest a variation to the estimation of the option continuation value, which reduces the execution time of the algorithm without any significant loss in accuracy. We test the choice of varying polynomial families with different number of basis functions and various variance reduction techniques, using a large sample of vanilla American options, and find that the use of low discrepancy sequences with Brownian bridges can increase substantially the accuracy of the simulation method. We also extend our analysis to the valuation of portfolios of compound and mutually exclusive options. For the latter, we also propose an improved algorithm which is faster and more accurate.

2003

Abstract In this paper we discuss accuracy issues of the Monte-Carlo method for valuing American options. Two major error sources are discussed: the discretization error of numerical methods for simulating stochastic models and the statistical error of finite samples. As the explicit Euler method is dominant in the extant literature of computational finance, it is strongly recommended to use numerical methods with higher convergence order to reduce the discretization error.

This article presents a simple yet powerful new approach for approximating the value of America11 options by simulation. The kcy to this approach is the use of least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference techniques cannot be used. We illustrate this technique with several realistic exatnples including valuing an option when the underlying asset follows a jump-diffusion process and valuing an America11 swaption in a 20-factor string model of the term structure.

American option pricing is challenging in terms of numerical methods as they can be exercised anytime. There is a mixture of advantages and disadvantages of particular methods. Binomial trees are simpler, faster but may not approximate any diffusion process and may be difficult to implement for high-dimensional options. On the other hand, Monte Carlo methods are computationally complex due to the large number of iterations involved but can be successfully applied to any diffusion model as well as high-dimensional options. The purpose of this thesis is to assess the efficiency of pricing methods for American and Bermudan style option on several diffusion processes and in both one and two dimensions. The thesis shows that binomial trees give a good approximation for the Ornstein-Uhlenbeck and CEV processes. Monte Carlo can be applied for American options by approximating the optimal stopping strategy with cross-sectional regression over a set of basis functions yielding lower bound for the true price. Additionally, the upper bound is obtained through a minimization problem over a set of martingales, in a duality setting.

This paper develops a Monte Carlo simulation method for solving option valuation problems. The method simulates the process generating the returns on the underlying asset and invokes the risk neutrality assumption to derive the value of the option. Techniques for improving the efficiency of the method are introduced. Some numerical examples are given to illustrate the procedure and additional applications are suggested.

Numerical methods form an important part of options pricing and especially in cases where there is no closed form analytic formula. We discuss two of the primary numerical methods that are currently used by financial professionals for determining the price of an options namely Monte Carlo method and finite difference method. Then we compare the convergence of the two methods to the analytic Black-Scholes price of European option. Monte Carlo method is good for pricing exotic options while Crank Nicolson finite difference method is unconditionally stable, more accurate and converges faster than Monte Carlo method when pricing standard options.