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Matematik simulation and monte carlo with applications in finance and mcmc phần 5

Sau đó trình tự X 0 X 1 p yx = xyq yx cho tất cả các x y ∈ S, với x = y. Lưu ý rằng xác suất có điều kiện còn lại trong trạng thái x tại một bước trong chuỗi này là một khối lượng xác suất bằng 1 - xyqyx dy | 126 Simulation and finance Table 6.3 Results for basket option using naive Monte Carlo basket and importance sampling with post stratification basketimppostratv2 basketa basketimppostratv2b Ơ K c ỷVar c c yVar 0 v.r.r.c 600 84.03 0.881 85.18 0.0645 96 306 660 47.21 0.707 48.03 0.0492 107 338 OÌ 720 23.49 0.514 24.04 0.0282 171 544 2 600 71.68 0.390 72.25 0.0390 51 164 2 660 22.74 0.283 23.09 0.0096 447 1420 ơ2 720 2.74 0.100 2.87 0.00297 585 1869 a 10000 paths. b 25 replications each consisting of 400 paths over 20 equiprobable strata. c Approximate 95 confidence interval for the variance reduction ratio. Since this is the expectation of a function of P Z only the ideal stratification variable for the option with price cg is X Z-i -N 0 1 . vp p 6.42 From Equation 6.38 for the original option with price c the estimator e-r T- x0 n we 1 - i 1 K eirr-rz x0 6.43 is used where Z N P I p is determined from Equations 6.40 and 6.41 and Equation 6.42 defines the stratification variable. The procedure basketimppoststratv2 in Appendix 6.7.2 implements this using post stratified sampling. Table 6.3 compares results using this and the naive method for a call option on an underlying basket of four assets. The data are r 0.04 x 5 2.5 4 3 q 20 80 60 40 T 0.5 t 0 and p as given in Equation 6.37 . Two sets of cases were considered one with ơ Ơ1 0.3 0.2 0.3 0.4 the other with ơ ơ2 0.05 0.1 0.15 0.05 . The spot price is q x 660. 6.6 Stochastic volatility Although the Black-Scholes model is remarkably good one of its shortcomings is that it assumes a constant volatility. What happens if the parameter Ơ is replaced by a known function of time ơ t Then p dt ơ t dB1 t Stochastic volatility 127 so using Ito s lemma z dX d ln X -X. - r t X2 2X2 di dí Ơ t dB t - Ơ2 t dt M - 2 2 0 dt ơ t dB1 t . 6.44 Now define an average squared volatility V t 1 t 0 Ơ2 u 2 du. Given that X 0 x0 Equation 6.44 can be integrated to give X t x0 exp x0 exp M - -V t t ơ ù dB1 u 20 Using the principle that the