Looking at it from their point of view, it's hard to see that the company is getting any hard benefit from your bicycling. So there's one less car in the parking lot, big deal. Any benefit you could argue, such as being a happier and healthier employee, is intangible and impossible to quantify. Senior management likes hard numbers. So I'm not sure the compensation angle is going to fly, it might just come off as an unrealistic sense of entitlement, which is sure to rub them the wrong way.

My company is not big on having us work from home either, although most managers will let us do it once in a while if there's a specific reason. Their argument, and I can see their point, is that being face to face is more convenient, even synergistic (buzzword, sorry), than everyone working in different physical places. Of course, you're not talking everyday, just three or four days per year. Make sure they understand that. And make sure they understand you have everything you need already, and also that you can make other arrangements to fulfill any obligations you have that day, such as being able to conference call into meetings and so forth. If their concern is that you won't be as productive, show them that you've thought through how you can be, don't just tell them you can be. (On the other hand, if they just don't trust people working from home, not much you can do to change that attitude, short of offering to install a camera on your computer so they can check on you throughout the day! )

Speaking of having everything you need, you might want to check with your desktop support people and make sure that's true. My company has a policy that only company-issue laptops are allowed to log into our VPN, and the software has to be installed on them to allow that. So if you are thinking of your home computer, make sure that it is in conformance with your company's policies and standards, or find out what that would take.

As noted, you don't say if you have any alternatives, such as a car or public transportation. If not, maybe you could argue a hardship angle: You don't own a car, and it makes no financial sense to buy one, plus get the insurance, for only 3-4 days a year. If there are any other alternatives, such as public transportation or car-commuting with someone else at your company who lives nearby, they might ask you why you can't just do that, so it would be good to consider that, and your answer, before you go in. Also find out how much it would cost to take a taxi, and if that would be a hardship. It's probably just a few dollars out of what you're saving by not owning a car!

Hope this helps. Good luck!