３－５、３－６章で述べた、定率賭けで毎回資産の
を賭けるとき、
の極限で損益の分布関数が”極限の正規分布”と一致することを、より正確に示す。
公式
を利用する。
株価が1期間で確率1/2で
倍か
倍になるとき、当初円の全資産があり、そのうちの
円を最初に投資し、その後は全資産の定率
を毎回賭けるときの、
期間後の全資産の分布は、本文で述べた通り対数正規分布
![=LN\left(\log\genfrac{}{}{}{0}{S_I}{\alpha}+\genfrac{}{}{}{0}{n}{2}\log\left\{\left[1+(r-1)\alpha\right]\left[1+\left(\genfrac{}{}{}{0}{1}{r}-1\right)\alpha\right]\right\} , \genfrac{}{}{}{0}{n}{4}\left(\log\genfrac{}{}{}{0}{1+(r-1)\alpha}{1+\left(\genfrac{}{}{}{0}{1}{r}-1\right)\alpha}\right)^2\right)](https://chart.apis.google.com/chart?cht=tx&chl=%3DLN%5Cleft%28%5Clog%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7BS_I%7D%7B%5Calpha%7D%2B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%7D%7B2%7D%5Clog%5Cleft%5C%7B%5Cleft%5B1%2B%28r-1%29%5Calpha%5Cright%5D%5Cleft%5B1%2B%5Cleft%28%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B1%7D%7Br%7D-1%5Cright%29%5Calpha%5Cright%5D%5Cright%5C%7D%20%2C%C2%A0%20%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%7D%7B4%7D%5Cleft%28%5Clog%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B1%2B%28r-1%29%5Calpha%7D%7B1%2B%5Cleft%28%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B1%7D%7Br%7D-1%5Cright%29%5Calpha%7D%5Cright%29%5E2%5Cright%29)
となる。その確率密度関数は
![\genfrac{}{}{}{0}{1}{\sqrt{2\pi}sx}\exp\left[-\genfrac{}{}{}{0}{(\log x-m)^2}{2s^2}\right]](https://chart.apis.google.com/chart?cht=tx&chl=%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B1%7D%7B%5Csqrt%7B2%5Cpi%7Dsx%7D%5Cexp%5Cleft%5B-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%28%5Clog%20x-m%29%5E2%7D%7B2s%5E2%7D%5Cright%5D)
であるが、初期投資からの損益を見るために分布全体をだけ
軸左方向にずらした分布
を考えると、
![f(x)=\genfrac{}{}{}{0}{1}{\sqrt{2\pi}s\left(x+\genfrac{}{}{}{0}{S_I}{\alpha}\right)}\exp\left[-\genfrac{}{}{}{0}{\left(\log\left(x+\genfrac{}{}{}{0}{S_I}{\alpha}\right)-m\right)^2}{2s^2}\right]](https://chart.apis.google.com/chart?cht=tx&chl=f%28x%29%3D%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B1%7D%7B%5Csqrt%7B2%5Cpi%7Ds%5Cleft%28x%2B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7BS_I%7D%7B%5Calpha%7D%5Cright%29%7D%5Cexp%5Cleft%5B-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Cleft%28%5Clog%5Cleft%28x%2B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7BS_I%7D%7B%5Calpha%7D%5Cright%29-m%5Cright%29%5E2%7D%7B2s%5E2%7D%5Cright%5D)
となる。
、
とおいて、
、
の値を代入すると、
![f(x)=\genfrac{}{}{}{0}{\exp\left[-\genfrac{}{}{}{0}{\left(\log\left(x+\genfrac{}{}{}{0}{S_I}{\alpha}\right)-\log\genfrac{}{}{}{0}{S_I}{\alpha}-\genfrac{}{}{}{0}{n}{2}\log\left\{(1+A\alpha)(1+B\alpha)\right\} \right)^2}{\genfrac{}{}{}{0}{n}{2}\left(\log\genfrac{}{}{}{0}{1+A\alpha}{1+B\alpha}\right)^2}\right]}{\sqrt{\genfrac{}{}{}{0}{n\pi}{2}} \left(\log\genfrac{}{}{}{0}{1+A\alpha}{1+B\alpha} \right)\left(x+\genfrac{}{}{}{0}{S_I}{\alpha}\right)}](https://chart.apis.google.com/chart?cht=tx&chl=f%28x%29%3D%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Cexp%5Cleft%5B-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Cleft%28%5Clog%5Cleft%28x%2B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7BS_I%7D%7B%5Calpha%7D%5Cright%29-%5Clog%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7BS_I%7D%7B%5Calpha%7D-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%7D%7B2%7D%5Clog%5Cleft%5C%7B%281%2BA%5Calpha%29%281%2BB%5Calpha%29%5Cright%5C%7D%20%5Cright%29%5E2%7D%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%7D%7B2%7D%5Cleft%28%5Clog%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B1%2BA%5Calpha%7D%7B1%2BB%5Calpha%7D%5Cright%29%5E2%7D%5Cright%5D%7D%7B%5Csqrt%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%5Cpi%7D%7B2%7D%7D%20%5Cleft%28%5Clog%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B1%2BA%5Calpha%7D%7B1%2BB%5Calpha%7D%20%5Cright%29%5Cleft%28x%2B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7BS_I%7D%7B%5Calpha%7D%5Cright%29%7D)
公式を利用できるように変形すると、
![f(x)=\genfrac{}{}{}{0}{\exp\left[-\genfrac{}{}{}{0}{\left(\log\left(\genfrac{}{}{}{0}{\alpha x}{S_I}+1\right)-\genfrac{}{}{}{0}{n\alpha}{2}\left\{\genfrac{}{}{}{0}{\log(1+A\alpha)}{A\alpha}A+\genfrac{}{}{}{0}{\log(1+B\alpha)}{B\alpha}B \right\}\right)^2}{\genfrac{}{}{}{0}{n}{2}\left(\genfrac{}{}{}{0}{\log(1+A\alpha)}{A\alpha}A-\genfrac{}{}{}{0}{\log(1+B\alpha)}{B\alpha}B\right)^2\alpha^2}\right]}{\sqrt{\genfrac{}{}{}{0}{n\pi}{2}} \left(\genfrac{}{}{}{0}{\log(1+A\alpha)}{A\alpha}A-\genfrac{}{}{}{0}{\log(1+B\alpha)}{B\alpha}B\right)\left(x\alpha+S_I\right)}](https://chart.apis.google.com/chart?cht=tx&chl=f%28x%29%3D%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Cexp%5Cleft%5B-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Cleft%28%5Clog%5Cleft%28%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Calpha%20x%7D%7BS_I%7D%2B1%5Cright%29-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%5Calpha%7D%7B2%7D%5Cleft%5C%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BA%5Calpha%29%7D%7BA%5Calpha%7DA%2B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BB%5Calpha%29%7D%7BB%5Calpha%7DB%20%5Cright%5C%7D%5Cright%29%5E2%7D%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%7D%7B2%7D%5Cleft%28%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BA%5Calpha%29%7D%7BA%5Calpha%7DA-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BB%5Calpha%29%7D%7BB%5Calpha%7DB%5Cright%29%5E2%5Calpha%5E2%7D%5Cright%5D%7D%7B%5Csqrt%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%5Cpi%7D%7B2%7D%7D%20%5Cleft%28%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BA%5Calpha%29%7D%7BA%5Calpha%7DA-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BB%5Calpha%29%7D%7BB%5Calpha%7DB%5Cright%29%5Cleft%28x%5Calpha%2BS_I%5Cright%29%7D)
![=\genfrac{}{}{}{0}{\exp\left[-\genfrac{}{}{}{0}{\left(\left\{\genfrac{}{}{}{0}{\log\left(\genfrac{}{}{}{0}{\alpha x}{S_I}+1\right)}{\genfrac{}{}{}{0}{\alpha x}{S_I}}\genfrac{}{}{}{0}{\alpha x}{S_I}\right\}-\genfrac{}{}{}{0}{n\alpha}{2}\left\{\genfrac{}{}{}{0}{\log(1+A\alpha)}{A\alpha}A+\genfrac{}{}{}{0}{\log(1+B\alpha)}{B\alpha}B \right\}\right)^2}{\genfrac{}{}{}{0}{n}{2}\left(\genfrac{}{}{}{0}{\log(1+A\alpha)}{A\alpha}A-\genfrac{}{}{}{0}{\log(1+B\alpha)}{B\alpha}B\right)^2\alpha^2}\right]}{\sqrt{\genfrac{}{}{}{0}{n\pi}{2}} \left(\genfrac{}{}{}{0}{\log(1+A\alpha)}{A\alpha}A-\genfrac{}{}{}{0}{\log(1+B\alpha)}{B\alpha}B\right)\left(x\alpha+S_I\right)}](https://chart.apis.google.com/chart?cht=tx&chl=%3D%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Cexp%5Cleft%5B-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Cleft%28%5Cleft%5C%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%5Cleft%28%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Calpha%20x%7D%7BS_I%7D%2B1%5Cright%29%7D%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Calpha%20x%7D%7BS_I%7D%7D%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Calpha%20x%7D%7BS_I%7D%5Cright%5C%7D-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%5Calpha%7D%7B2%7D%5Cleft%5C%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BA%5Calpha%29%7D%7BA%5Calpha%7DA%2B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BB%5Calpha%29%7D%7BB%5Calpha%7DB%20%5Cright%5C%7D%5Cright%29%5E2%7D%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%7D%7B2%7D%5Cleft%28%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BA%5Calpha%29%7D%7BA%5Calpha%7DA-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BB%5Calpha%29%7D%7BB%5Calpha%7DB%5Cright%29%5E2%5Calpha%5E2%7D%5Cright%5D%7D%7B%5Csqrt%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%5Cpi%7D%7B2%7D%7D%20%5Cleft%28%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BA%5Calpha%29%7D%7BA%5Calpha%7DA-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BB%5Calpha%29%7D%7BB%5Calpha%7DB%5Cright%29%5Cleft%28x%5Calpha%2BS_I%5Cright%29%7D)
expのなかのを約分して、
![=\genfrac{}{}{}{0}{\exp\left[-\genfrac{}{}{}{0}{\left(\left\{\genfrac{}{}{}{0}{\log\left(\genfrac{}{}{}{0}{\alpha x}{S_I}+1\right)}{\genfrac{}{}{}{0}{\alpha x}{S_I}}\genfrac{}{}{}{0}{x}{S_I}\right\}-\genfrac{}{}{}{0}{n}{2}\left\{\genfrac{}{}{}{0}{\log(1+A\alpha)}{A\alpha}A+\genfrac{}{}{}{0}{\log(1+B\alpha)}{B\alpha}B \right\}\right)^2}{\genfrac{}{}{}{0}{n}{2}\left(\genfrac{}{}{}{0}{\log(1+A\alpha)}{A\alpha}A-\genfrac{}{}{}{0}{\log(1+B\alpha)}{B\alpha}B\right)^2}\right]}{\sqrt{\genfrac{}{}{}{0}{n\pi}{2}} \left(\genfrac{}{}{}{0}{\log(1+A\alpha)}{A\alpha}A-\genfrac{}{}{}{0}{\log(1+B\alpha)}{B\alpha}B\right)\left(x\alpha+S_I\right)}](https://chart.apis.google.com/chart?cht=tx&chl=%3D%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Cexp%5Cleft%5B-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Cleft%28%5Cleft%5C%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%5Cleft%28%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Calpha%20x%7D%7BS_I%7D%2B1%5Cright%29%7D%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Calpha%20x%7D%7BS_I%7D%7D%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bx%7D%7BS_I%7D%5Cright%5C%7D-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%7D%7B2%7D%5Cleft%5C%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BA%5Calpha%29%7D%7BA%5Calpha%7DA%2B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BB%5Calpha%29%7D%7BB%5Calpha%7DB%20%5Cright%5C%7D%5Cright%29%5E2%7D%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%7D%7B2%7D%5Cleft%28%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BA%5Calpha%29%7D%7BA%5Calpha%7DA-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BB%5Calpha%29%7D%7BB%5Calpha%7DB%5Cright%29%5E2%7D%5Cright%5D%7D%7B%5Csqrt%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%5Cpi%7D%7B2%7D%7D%20%5Cleft%28%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BA%5Calpha%29%7D%7BA%5Calpha%7DA-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Clog%281%2BB%5Calpha%29%7D%7BB%5Calpha%7DB%5Cright%29%5Cleft%28x%5Calpha%2BS_I%5Cright%29%7D)
の極限をとると、
![\displaystyle\lim_{\alpha\to 0}f(x)=\genfrac{}{}{}{0}{\exp\left[-\genfrac{}{}{}{0}{\left(x-\genfrac{}{}{}{0}{n}{2}(A+B)S_I\right)^2}{\genfrac{}{}{}{0}{n}{2}(A-B)^2S_I^2} \right]}{\sqrt{\genfrac{}{}{}{0}{n\pi}{2}}(A-B)S_I}](https://chart.apis.google.com/chart?cht=tx&chl=%5Cdisplaystyle%5Clim_%7B%5Calpha%5Cto%200%7Df%28x%29%3D%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Cexp%5Cleft%5B-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Cleft%28x-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%7D%7B2%7D%28A%2BB%29S_I%5Cright%29%5E2%7D%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%7D%7B2%7D%28A-B%29%5E2S_I%5E2%7D%20%5Cright%5D%7D%7B%5Csqrt%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%5Cpi%7D%7B2%7D%7D%28A-B%29S_I%7D)
、
を元に戻して、
![\displaystyle\lim_{\alpha\to 0}f(x)=\genfrac{}{}{}{0}{1}{\sqrt{\genfrac{}{}{}{0}{n\pi}{2}}\left(r-\genfrac{}{}{}{0}{1}{r}\right)S_I}\exp\left[-\genfrac{}{}{}{0}{\left\{x-\genfrac{}{}{}{0}{n}{2}\left(r+\genfrac{}{}{}{0}{1}{r}-2\right)S_I\right\}^2}{\genfrac{}{}{}{0}{n}{2}\left(r-\genfrac{}{}{}{0}{1}{r}\right)^2S_I^2} \right]](https://chart.apis.google.com/chart?cht=tx&chl=%5Cdisplaystyle%5Clim_%7B%5Calpha%5Cto%200%7Df%28x%29%3D%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B1%7D%7B%5Csqrt%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%5Cpi%7D%7B2%7D%7D%5Cleft%28r-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B1%7D%7Br%7D%5Cright%29S_I%7D%5Cexp%5Cleft%5B-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B%5Cleft%5C%7Bx-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%7D%7B2%7D%5Cleft%28r%2B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B1%7D%7Br%7D-2%5Cright%29S_I%5Cright%5C%7D%5E2%7D%7B%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7Bn%7D%7B2%7D%5Cleft%28r-%5Cgenfrac%7B%7D%7B%7D%7B%7D%7B0%7D%7B1%7D%7Br%7D%5Cright%29%5E2S_I%5E2%7D%20%5Cright%5D)
となり、これは”極限の正規分布"をだけ左にずらした正規分布（=損益の分布）
と一致する。
－目次へ－
