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2001, Journal of computational finance
We explain how a carefully chosen scheme can lead to competitive Monte Carlo algorithms for the computation of the price of Asian options. We give evidence of the efficiency of these algorithms with a mathematical study of the rate of convergence and a numerical comparison with some existing methods.
Price manipulation is reserved for commodity products with low trading volumes, Asian options play an important in pricing in such cases. Since there is no systematic solutions to arithmetic average options, iterative or numerical methods are used. Computer Simulation using Monte Carlo methods plays an important in this case. Various reduction techniques are also used to improve accuracy. This paper deals with Monte Carlo method's use in pricing options and also comparison with other options. Moreover paper also gives a thought to using Quasi Monte Carlo methods in pricing Asian options.
1998 Winter Simulation Conference. Proceedings (Cat. No.98CH36274), 1998
When pricing options via Monte Carlo simulations, precision can be improved either by performing longer simulations, or by reducing the variance of the estimators. In this paper, two methods for variance reduction are combined, the control variable and the change of measure (or likelihood) methods. We specifically consider Asian options, and show that a change of measure can very significantly improve the precision when the option is deeply out of the money, which is the harder estimation problem. We also show that the simulation method itself can be used to find the best change of measure. This is done by incorporating an updating rule, based on an estimate of the gradient of the variance. The paper includes simulation results.
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.
2011
We concentrate in this paper on Asian options. As defined, the value of those options depends on the average I = ∫ T 0 Sudu of the price of underlying during all the life of the option. This price is written in integral form, from where the benefits to use Monte Carlo method (MC ) to evaluate it. MC techniques have proved to be flexible, robust and easily computed. But on other side, MC require a big number of calculations. So, we turn it on parallel machine to reduce time execution. As known, results obtained by this method depends on the quality of the underlying pseudo-random number generator PRNG. In [17] we considered the standard C generator to compute algorithms (sequential and parallel). This generator gives good results for the three examples cited, but in higher number of iteration, it’s become uncontrolled. We consider here a modified LCG, and in parallel algorithm, we use a splitting approach to produce independent sub-sequences. To evaluate Asian option, in preference t...
The pricing of options is a very important problem encountered in financial markets today. Many problems in mathematical finance entail the computation of a particular integral. In many cases these integrals can be valued analytically, and in still more cases they can be valued using numerical integration, or computed using a partial differential equation (PDE). The famous Black-Scholes model, for instance, provides explicit closed form solutions for the values of certain (European style) call and put options. However when the number of dimensions in the problem is large, PDEs and numerical integrals become intractable, the formulas exhibiting them are complicated and difficult to evaluate accurately by conventional methods. In these cases, Monte Carlo methods often give better results, because they have proved to be valuable and flexible computational tools to calculate the value of options with multiple sources of uncertainty or with complicated features.
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.
Journal of Computational Finance, 2001
The author describes a modi®ed binomial method that provides a simple and uni®ed framework for the valuation of various kinds of Asian options (American or European, arithmetic or geometric, ®xed or¯oating strike, discrete or continuous sampling and dividends, and partial Asians). The greeks can also be calculated accurately and stably. The method is a re®nement of that of , where at each node of a standard binomial tree one also considers a table of possible values of the average. To avoid the exponential explosion of the size of this table in the arithmetic average case, one considers a smaller set of representative values for the average, interpolates when necessary, and otherwise uses standard backward recursion to value the option. In this paper, an ef®cient implementation of this idea is presented. In particular, option values are insured to be smooth as a function of the number N of binomial time periods, so that Richardson extrapolation can be applied to eliminate 1=N (and sometimes higherorder) corrections, dramatically increasing the speed of the method. Detailed checks and illustrations are provided, showing that this approach can achieve any desired level of accuracy for convection-or diffusion-dominated regimes and for long or short maturities. It is typically much faster than standard PDE and Monte Carlo approaches.
Operations Research, 2003
American-Asian options are average-price options that allow early exercise. In this paper, we derive structural properties for the optimal exercise policy, which are then used to develop an efficient numerical algorithm for pricing such options. In particular, we show that the optimal policy is a threshold policy: The option should be exercised as soon as the average asset price reaches a characterized threshold, which can be written as a function of the asset price at that time. By exploiting this and other structural properties, we are able to parameterize the exercise boundary, and derive gradient estimators for the option payoff with respect to the parameters of the model. These estimators are then incorporated into a simulation-based algorithm to price American-Asian options. Computational experiments carried out indicate that the algorithm is very competitive with other recently proposed numerical algorithms.
Journal of Economic Dynamics and Control, 1997
The Monte Carlo approach has proved to be a valuable and flexible computational tool in modern finance. This paper discusses some of the recent applications of the Monte Carlo method to security pricing problems, with emphasis on improvements in efficiency. We first review some variance reduction methods that have proved useful in finance. Then we describe the use of deterministic low-discrepancy sequences, also known as quasi-Monte Carlo methods, for the valuation of complex derivative securities. We summarize some recent applications of the Monte Carlo method to the estimation of partial derivatives or risk sensitivities and to the valuation of American options. We conclude by mentioning other applications.
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.
Monte Carlo Methods and Applications, 2000
Taking advantage of the recent litterature on exact simulation algorithms (Beskos, Papaspiliopoulos and Roberts [1]) and unbiased estimation of the expectation of certain fonctional integrals (Wagner [27], Beskos et al. [2] and Fearnhead et al. [6]), we apply an exact simulation based technique for pricing continuous arithmetic average Asian options in the Black & Scholes framework. Unlike existing Monte Carlo methods, we are no longer prone to the discretization bias resulting from the approximation of continuous time processes through discrete sampling. Numerical results of simulation studies are presented and variance reduction problems are considered.
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.
Prices of two available stocks follow, under the risk neutral measure, the dynamics given by the following model δS 1 t = S 1 t rδt + S 1 t σ 1 1 + sin 4t δW 1 t , δS 2 t = S 2 t rδt + S 2 t σ 2 1 + sin 4t · ρδW 1 t + 1 − ρ 2 δW 2 t , where W 1 and W 2 are two independent Wiener processes, r is the interest rate and ρ ∈ [−1, 1] is the correlation coefficient. Two algorithms for pricing a call option on a portfolio of stocks will be implemented and analyzed; the payoff is given by: αS 1 T + (1 − α)S 2 T − K + where α ∈ [0, 1] specifies the composition of the portfolio. The first algorithm is a plain implementation of the Monte Carlo method for option pricing, the second one will implement an antithetic variates technique to reduce the variance of the price estimator. In both the programmes the Euler-Maruyama scheme is implemented to approximate the solution of the SDE that describe the price processes.
Monte Carlo Methods and Applications, 2014
The purpose of this paper is to study the problem of pricing Asian options using the multilevel Monte Carlo method recently introduced by Giles [8] and to prove a central limit theorem of Lindeberg Feller type for the obtained algorithm. Indeed, the implementation of such a method requires first a discretization of the integral of the payoff process. To do so, we use two well-known second order discretization schemes, namely, the Riemann scheme and the trapezoidal scheme. More precisely, for each one of these schemes we prove a stable law convergence result for the error on two consecutive levels of the algorithm. This allows us to go further and prove two central limit theorems on the multilevel algorithm providing us a precise description on the choice of the associated parameters with an explicit representation of the limiting variance. For this setting of second order schemes, we give new optimal parameters leading to the convergence of the central limit theorem. A complexity of the multilevel Monte Carlo algorithm is carried out.
The numerical methods form an important part of pricing options, especially in cases where there is no closed form analytic formula. The Monte Carlo method is one of the primary numerical methods that is currently used by financial professionals for determining the price of options and security pricing problems with emphasis on improvement in efficiency. We discuss the pricing of exotic options with special emphasis on path dependent options, like Asian and lookback options. Monte Carlo simulation technique is very versatile in cases where there is no closed form analytical formula. This method is slow and time consuming but very flexible even for multi-dimensional problems and has proved to be a valuable and flexible computational tool in modern finance. We compare the result of the Monte Carlo method with the analytic Black-Scholes results and the exact values of the options.
Journal of Business & Economics Research (JBER), 2011
The advantage of Monte Carlo simulations is attributed to the flexibility of their implementation. In spite of their prevalence in finance, we address their efficiency and accuracy in option pricing from the perspective of variance reduction and price convergence. We demonstrate that increasing the number of paths in simulations will increase computational efficiency. Moreover, using a t-test, we examine the significance of price convergence, measured as the difference between sample means of option prices. Overall, our illustrative results show that the Monte Carlo simulation prices are not statistically different from the Black-Scholes type closed-form solution prices.
Hybrid Monte Carlo (HMC) method is defined in this thesis as Monte Carlo method that utilizes conditional expectation so that the regular Monte Carlo method and other computational methods can be combined to price financial derivatives. This thesis introduces several hybrid Monte Carlo methods and studies the algorithm and efficiency of these methods, which include three methods combining Monte Carlo with fast Fourier transform, cosine series, and Black-Scholes formula respectively.
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.
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