  We consider an analogue of Nakada's $\alpha$-continued fraction
transformation in the setting of continued fractions with odd partial
quotients. More precisely, given $\alpha \in
[\frac{1}{2}(\sqrt{5}-1),\frac{1}{2}(\sqrt{5}+1)]$, we show that every
irrational number $x\in I_\alpha=[\alpha-2,\alpha)$ can be uniquely represented
as $$ x= \cfrac{e_1 (x;\alpha)}{d_1 (x;\alpha)
+\cfrac{e_2(x;\alpha)}{d_2(x;\alpha)+\cdots}} , $$ with $e_i(x;\alpha) \in \{
\pm 1\}$ and $d_i(x;\alpha) \in 2{\mathbb N} -1$ determined by the iterates of
the transformation $$\varphi_\alpha (x) := \frac{1}{| x|} - 2 \bigg[
\frac{1}{2| x|} +\frac{1-\alpha}{2} \bigg]-1$$ of $I_\alpha$. We also describe
the natural extension of $\varphi_\alpha$ and prove that the endomorphism
$\varphi_\alpha$ is exact.
