You may also remember that a recent trip to New Orleans provided Mr. Squareview and I with an idea to answer that question.
Here is that idea in practice (Sorry, I forgot to take any before pictures).
WHAT YOU'LL NEED ::
1. A 24 inch high PVC tube with a diameter slightly wider than the post of the umbrella
2. Metal or study plastic zip ties
HOW TO ::
1. Using the zip ties, attach the PVC tube to a secure section of the deck, preferably a corner like the two images below.
2. Place umbrella in the tube like the 3rd image.
3. Secure the post of the umbrella to the balcony with another zip tie.
Voila! Your done. Seriously, that's it. If you were uber creative, you would likely spray paint the PVC tube to match the balcony. I didn't do so because 1. I thought of it only after I had already secured it to the deck and 2. My furniture covered it enough that it is not too much of an eyesore.