Is there an analogue to the "Delta" symbol for ratios?
The best symbol to use is $\exp\Delta\log$: $$\exp\Delta\log x = \exp(\log x_1-\log x_0) = \frac{x_1}{x_0}.$$ The point is that this operation isn't "qualitatively different" from $\Delta$, so it may be reduced to $\Delta$. So far, I haven't used any new symbols but if you want some multiplicative new creative symbols, see e-percentages and units of evidence:
http://motls.blogspot.com/2010/01/exponential-percentages-useful-proposed.html
http://motls.blogspot.com/2010/08/units-of-evidence.html
There is no compact symbol for $\exp\Delta\log$. If you want an ally who would endorse the idea to introduce such a symbol, you may count on me. What about $\Delta^\times$?
Not entirely standard, but in Peter Henrici's discussion of the (justly famous) quotient-difference (QD) algorithm in the books Elements of Numerical Analysis (see p. 163) and Essentials of Numerical Analysis (see p. 155), he defines the quotient operator as
$$Q\,x_n=\frac{x_{n+1}}{x_n}$$
in complete analogy with the (forward) difference operator $\Delta$.
Henrici's a pretty sharp mathematician, so I wouldn't mind borrowing notation from him if I were in your shoes...
Here's a screenshot of the relevant page of the first book (sorry, I don't have a digital copy of the other book):
