If we assume that ln has a normal distribution with mean m and variance v 2 , then, for any integer n : and where E log stands for expected value assuming a risk-neutral lognormal distribution for the average. A suitable technique is the method of antithetic variates, described in Kleijnen et al. The exotic nature of this option type has important financial and mathematical implications for us. To generate the paths necessary for an Asian option we will use a procedural approach. Moreover, Akhiezer [ 2 ] , page 66, shows that the possible error can be bounded, and establishes the relevant relationships of the difference between two distributions that share the same 2 m moments. But before proceeding to the empirical analysis concerning the precision and robustness of these algorithms, let us review the simulation approach, which will serve as an ultimate check and basis of comparison for the Laplace transform method. This entails the usage of an ever larger summation, or in other words the value of precision parameter M and thus the calculation time needs to be increased as.

Since there exists no analytical valuation formula to date, one has to resort to other methods to price this commonly used derivative product. Finally we take the arithmetic mean of those values and use pointer dereferencing to call the operator for the PayOff object. Then we describe different approximations based on the moments. If the asset price is deep in the money, then it is more likely to be exercised. As the name suggests, this is the simplest form of Monte Carlo method, taking the arithmetic-mean as an unbiased estimator of the expected value:. For example, Lindsay and Roeder [ 23 ] show that if two distributions functions have the same first n moments, then they must cross each other at least n times. The table above contrasts the results from closed approximation models against price simulations implemented via CRR trees and Monte Carlo.

Volume 1 Issue 4 Dec , pp. This class will be called AsianOption. For this reason Turnbull and Wakeman [ 33 ] and Ritchken and al. Based on the figures, we claim that the control variate algorithm generally beats numerical algorithms in terms of calculation time for usual values of and T. As it turns out, it is possible to further improve on our simulation technique by using control variates instead of antithetic variates for further details refer to Kleijnen et al. We must point out that in the case of path-dependent derivatives simulation methods have two sources of estimation error:. Since there exists no analytical valuation formula to date, one has to resort to other methods to price this commonly used derivative product.

Other numerical methods that are of interest are [ 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 ]. Journal of Computational Finance, 2, Note that using the Edgeworth approximation, for high volatility levels the density becomes bimodal and besides it can assume negative values. The major benefit of this approach is that we can encapsulate multiple various types of pay-off functionality without the need to modify the remaining classes, such as our AsianOption class to be discussed below. Finally in section 6, we compare the different approximations and then we draw some conclusion. Journal of Political Economy, 81, Path Generation Header To generate the paths necessary for an Asian option we will use a procedural approach.

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