Coupled sum estimator |$Z$|; Black–Scholes model; |$r=0.05$|, |$\sigma =2$|, |$T=1$|, |$S(0)=1$|; sample size |$10^{6}$|
| |$Z$| . | |$m$| . | |$Var$| . | |$Time$| . | |$Var\times time$| . | |||
|---|---|---|---|---|---|---|---|
| |$F^{(1)}$| | – | |$6.13\times 10^{-4}$| | |$27.28$| | |$0.0167$| | |||
| |$F^{(2)}$| | |$7$| | |$1.01\times 10^{-4}$| | |$89.90$| | |$0.0091$| | |||
| |$F^{(3)}$| | |$3$| | |$1.45\times 10^{-4}$| | |$77.73$| | |$0.0112$| | |||
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| |$\beta _{n}$| | 11.93 | |$10.25$| | |$37.99$| | |$8.97$| | |$2.55$| | |$0.71$| | |$0.20$| |
| |$F^{(1)}$| | 1 | 0.3536 | 0.1250 | 0.0442 | 0.0156 | 0.0055 | 0.0020 |
| |$F^{(2)}$| | 1 | 0.8209 | 0.8209 | 0.3066 | 0.1156 | 0.0432 | 0.0163 |
| |$F^{(3)}$| | 1 | 0.8209 | 0.8209 | 0.3066 | 0.1084 | 0.0383 | 0.0135 |
| |$Z$| . | |$m$| . | |$Var$| . | |$Time$| . | |$Var\times time$| . | |||
|---|---|---|---|---|---|---|---|
| |$F^{(1)}$| | – | |$6.13\times 10^{-4}$| | |$27.28$| | |$0.0167$| | |||
| |$F^{(2)}$| | |$7$| | |$1.01\times 10^{-4}$| | |$89.90$| | |$0.0091$| | |||
| |$F^{(3)}$| | |$3$| | |$1.45\times 10^{-4}$| | |$77.73$| | |$0.0112$| | |||
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| |$\beta _{n}$| | 11.93 | |$10.25$| | |$37.99$| | |$8.97$| | |$2.55$| | |$0.71$| | |$0.20$| |
| |$F^{(1)}$| | 1 | 0.3536 | 0.1250 | 0.0442 | 0.0156 | 0.0055 | 0.0020 |
| |$F^{(2)}$| | 1 | 0.8209 | 0.8209 | 0.3066 | 0.1156 | 0.0432 | 0.0163 |
| |$F^{(3)}$| | 1 | 0.8209 | 0.8209 | 0.3066 | 0.1084 | 0.0383 | 0.0135 |
Coupled sum estimator |$Z$|; Black–Scholes model; |$r=0.05$|, |$\sigma =2$|, |$T=1$|, |$S(0)=1$|; sample size |$10^{6}$|
| |$Z$| . | |$m$| . | |$Var$| . | |$Time$| . | |$Var\times time$| . | |||
|---|---|---|---|---|---|---|---|
| |$F^{(1)}$| | – | |$6.13\times 10^{-4}$| | |$27.28$| | |$0.0167$| | |||
| |$F^{(2)}$| | |$7$| | |$1.01\times 10^{-4}$| | |$89.90$| | |$0.0091$| | |||
| |$F^{(3)}$| | |$3$| | |$1.45\times 10^{-4}$| | |$77.73$| | |$0.0112$| | |||
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| |$\beta _{n}$| | 11.93 | |$10.25$| | |$37.99$| | |$8.97$| | |$2.55$| | |$0.71$| | |$0.20$| |
| |$F^{(1)}$| | 1 | 0.3536 | 0.1250 | 0.0442 | 0.0156 | 0.0055 | 0.0020 |
| |$F^{(2)}$| | 1 | 0.8209 | 0.8209 | 0.3066 | 0.1156 | 0.0432 | 0.0163 |
| |$F^{(3)}$| | 1 | 0.8209 | 0.8209 | 0.3066 | 0.1084 | 0.0383 | 0.0135 |
| |$Z$| . | |$m$| . | |$Var$| . | |$Time$| . | |$Var\times time$| . | |||
|---|---|---|---|---|---|---|---|
| |$F^{(1)}$| | – | |$6.13\times 10^{-4}$| | |$27.28$| | |$0.0167$| | |||
| |$F^{(2)}$| | |$7$| | |$1.01\times 10^{-4}$| | |$89.90$| | |$0.0091$| | |||
| |$F^{(3)}$| | |$3$| | |$1.45\times 10^{-4}$| | |$77.73$| | |$0.0112$| | |||
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| |$\beta _{n}$| | 11.93 | |$10.25$| | |$37.99$| | |$8.97$| | |$2.55$| | |$0.71$| | |$0.20$| |
| |$F^{(1)}$| | 1 | 0.3536 | 0.1250 | 0.0442 | 0.0156 | 0.0055 | 0.0020 |
| |$F^{(2)}$| | 1 | 0.8209 | 0.8209 | 0.3066 | 0.1156 | 0.0432 | 0.0163 |
| |$F^{(3)}$| | 1 | 0.8209 | 0.8209 | 0.3066 | 0.1084 | 0.0383 | 0.0135 |
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