Rotated page starts at the edge of the paper

I’m rotating a page because of a wide figure, but it then begins just at the edge of the page, with no margins. I’m puzzled as I expected the rotated page to inherit the margins of the document it is inserted in. Further, I’d like it to be centered in the page.

documentclass{amsbook}
usepackage{graphicx}
usepackage{subcaption}
usepackage[figuresright]{rotating}

begin{document}

begin{sidewaysfigure}
  subcaptionbox{Returns $alpha = 1$}{includegraphics[width=.33linewidth]{Rendimientos1}}%
  subcaptionbox{Returns $alpha = frac{1}{2}$}{includegraphics[width=.33linewidth]{Rendimientos1-2}}%
  subcaptionbox{Returns $alpha = 2$}{includegraphics[width=.33linewidth]{Rendimientos2}}%
  caption{Returns}
end{sidewaysfigure}


end{document}

Also, my subfigures are too close together. I’ve tried changing the width of the subcaptionbox as

  subcaptionbox{Returns $alpha = 1$}[.3linewidth]{includegraphics[width=.33linewidth]{Rendimientos1}}%

but this seems to affect the size of the figure, rather than that of the box.

Finally, I get subfigures labelled (A), (B), … and I’d rather have (a), (b). I’m guessing this has to do with the default option labelformat=parens, but I’m unsure how to modify it. I’ve tried

renewcommand{thesubfigure}{alph{subfigure}}

but it doesn’t seem to work.

Edit:
I’ve managed to get more space between the figures and include a margin. In the process, I’ve rewritten my code to something easier (for me) to understand. However, the solution is quite ad hoc and doesn’t seem too elegant. In addition, the output still uses uppercase letters to identify the subfigures.

documentclass{amsbook}
usepackage{amssymb}
usepackage{amsfonts}

usepackage{graphicx}
usepackage{subcaption}
usepackage[figuresright]{rotating}


begin{document}

begin{sidewaysfigure}
    vspace{13cm}
    begin{subfigure}[b]{.33linewidth}
         centering
         includegraphics[width=.9linewidth]{Rendimientos12}%
         caption{$alpha=frac{1}{2}$}
    end{subfigure}% 
    begin{subfigure}[b]{.33linewidth}
         centering
         includegraphics[width=.9linewidth]{Rendimientos1}%
         caption{$alpha=1$}
    end{subfigure}
    begin{subfigure}[b]{.33linewidth}
         centering
         includegraphics[width=.9linewidth]{Rendimientos2}%
         caption{$alpha=2$}
    end{subfigure}
    caption{General caption}
 end{sidewaysfigure}

 end{document} 

My questions:

1. Is there a more elegant alternative to the use of vspace?
2. How do I change the format of the subcaption to lowercase? I’ve tried alternatives proposed here or here to no avail. I’m guessing it may have to do with the document class.