Given that X 0= x0, Equation 6.44 can be integrated to give Using the principle that the price at time zero of a European call with exercise time T and strike K is the discounted present
Trang 1i =1wi= 1 Therefore, the aim is to find a good and stratification variable for
an option with price cg, where
Following the same approach as for the Asian option, a good choice of is ∗where
∗= ln
exp √
Trang 2126 Simulation and finance
Table 6.3 Results for basket option, using naive Monte Carlo (basket) and importancesampling with post stratification (basketimppostratv2)
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 Z only, the ideal stratification variablefor the option with price cgis
is used, where Z∼ N∗ I ∗ 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 poststratified 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 as given in Equation (6.37) Two sets of cases were considered, one with = 1= 0 3 0 2 0 3 0 4, the other with
= 2= 0 05 0 1 0 ... models include those by Banks
et al (20 05) , Fishman (1978), Law and Kelton (2000), and Pidd (1998).
Simulation and Monte Carlo: With applications in finance and MCMC< /small>... obtained by sampling such a volatility path using Equations (6 .51 ) and (6 .52 ) with
= This is an example of conditional Monte Carlo If T = nh, there are usually 2nvariables in the integration... However, with independence, = This design integratesout n of the variables analytically The remaining n variables are integrated using MonteCarlo
f is as given inEquation