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Financial calculus Introduction to Financial Option Valuation_10

Tham khảo tài liệu 'financial calculus introduction to financial option valuation_10', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 21.5 Analysis of the uniform case 219 Table 21.1. Ninety-five per cent confidence intervals for 21.4 and 21.6 on problem 21.3 plus ratios of their widths M Standard Antithetic Ratio of widths 102 1.8841 2.0752 1.9875 2.0012 14.0 1o3 1.9538 2.0087 1.9976 2.0017 13.4 104 1.9890 2.0062 1.9997 2.0010 13.5 105 1.9969 2.0023 1.9998 2.0002 13.5 of the two confidence intervals. This is precisely the ratio of the square roots of the sample variances. As predicted by 21.11 it converges to Ự181.2485 13.5. As a practical note it is worth emphasizing that the confidence intervals for the antithetic variates estimate were computed via the sample variance of Ti i 1 which are independent and not U M 1 u e 1-U4i 1 which are highly correlated. 21.5 Analysis of the uniform case To understand how the antithetic variate technique works consider the more general case of approximating I E f U where U U 0 1 for some function f . The standard Monte Carlo estimate is 1 xM Im M f Ui withi.i.d. Ui - U 0 1 21.12 i 1 and the antithetic alternative is _ 1 f Ui f 1 Ui JJ I zn n Z01 p Im ----------------1---------- withi.i.d. Ui U 0 1 . 21.13 i 1 Copying the way that we derived 21.8 we find that f Ui f 1 Ui var VTJV--------------- 1 var f Ui cov f Ui f 1 Ui . 2 2 21.14 The success of the new scheme hinges on whether var 1 f Ui f 1 Ui is smaller than var f Ui . The identity 21.14 tells us that efficiency boils down to making cov f Ui f 1 Ui as negative as possible. We want f Ui to be big relative to its mean when f 1 Ui is small relative to its mean . Intuitively this approach will work when f is monotonic. Loosely the 220 Monte Carlo Part II variance reduction by antithetic variates antithetic variate technique attempts to compensate for samples that are above the mean by adding samples that are below the mean and vice versa. We may convert this intuition into a mathematical result. First we recall that to say a function f is monotonic increasing means x1 X2 f xt f xì . .