Quasi-Monte Carlo Methods in Numerical Finance

Proceedings of the 2004 Winter Simulation Conference, 2004., 2004

We review the basic principles of Quasi-Monte Carlo (QMC) methods, the randomizations that turn them into variancereduction techniques, and the main classes of constructions underlying their implementations: lattice rules, digital nets, and permutations in different bases. QMC methods are designed to estimate integrals over the s-dimensional unit hypercube, for moderate or large (perhaps infinite) values of s. In principle, any stochastic simulation whose purpose is to estimate an integral fits this framework, but the methods work better for certain types of integrals than others (e.g., if the integrand can be well approximated by a sum of low-dimensional smooth functions). Such QMC-friendly integrals are encountered frequently in computational finance and risk analysis. We give examples and provide computational results that illustrate the efficiency improvement achieved.

2023

Pricing financial derivatives such as options with desired accuracy can be hard due to the nature of the functions and complicated integrals required by the pricing techniques. In this paper we investigate the pricing methodology of the European style options using two advanced numerical methods, namely, Quasi-Monte Carlo and Fourier Cosine (COS). For the RQMC method, we use the random-start Halton sequence. We use the Black-Scholes-Merton model to measure the pricing quality of both of the methods. For the numerical results we compute the option price of the call option and we found a few reasons to prefer the RQMC method over the COS method to approximate the European style options.

WIT transactions on modelling and simulation, 2004

For many applications of computational finance, the use of quasirandom sequences seems to provide a faster rate of convergence than pseudorandom sequences. However, at present there are only a few choices of quasirandom sequences. By scrambling a quasirandom sequence we can produce a family of related quasirandom sequences. Finding an optimal quasirandom sequence within this family is an interesting problem, as such optimal quasirandom sequences can be quite useful in applications. The process of finding such optimal quasirandom sequences is called the derandomization of a randomized (scrambled) family. In this paper, we summarize aspects of this technique and explore applications of optimal quasirandom sequences to evaluate a particular derivative security. We find that the optimal quasirandom sequences give promising results even for high dimensions.

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.

In this paper, an exposition is made on the use of Monto Carlo method in simulation of financial problems. Some selected problems in financial economics such as pricing of plain vanilla options driven by continuous and jump stochastic processes are simulated and results obtained.

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.

International Journal of Theoretical and Applied Finance, 2006

We propose a quasi-Monte Carlo (qMC) algorithm to simulate variates from the normal inverse Gaussian (NIG) distribution. The algorithm is based on a Monte Carlo technique found in Rydberg , and is based on sampling three independent uniform variables. We apply the algorithm to three problems appearing in finance. First, we consider the valuation of plain vanilla call options and Asian options. The next application considers the problem of deriving implied parameters for the underlying asset dynamics based on observed option prices. We employ our proposed algorithm together with the Newton Method, and show how we can find the scale parameter of the NIG-distribution of the logreturns in case of a call or an Asian option. We also provide an extensive error analysis for this method. Finally we study the calculation of Value-at-Risk for a portfolio of nonlinear products where the returns are modeled by NIG random variables.

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.

2013

In this paper, the pricing of a European call option on the underlying asset is performed by using a Monte Carlo method, one of the powerful simulation methods, where the price development of the asset is simulated and value of the claim is computed in terms of an expected value. The proposed approach, applied in Monte Carlo simulation, is based on the Black-Scholes equation which generally defined the pricing of European call options in a dynamic environment. Therefore, the main goal of this study is how can Monte Carlo be applied to finance? Although it is stated that because of being based on randomness, the Monte Carlo method has its obvious disadvantages and does not yield solutions for all possible stock prices, by applying Black-Scholes formula, it is efficient to use this method for calculating payoff. Hence, in the matter of this paper, we introduce the Black-Scholes model and Monte Carlo simulations as main tools to determine.