Depends on the user and their intent. When you ask what their expectation is, if you do not keep their goals in mind you can’t get a definitive answer. I’ll try and propose some potential goals and then respond with some supposition around user expectations.
If it is purely about accessing content (the bank’s contact information, educational and product pages, etc.) there are many mobile users who are perfectly happy with accessing such information in a non-mobile version of the banks website. If they are frequent users then they will be somewhat familiar with the navigation and they will just use a mobile browser to view the site. If the bank offers a mobile-version of their site, then it should follow the same general structure of the normal site (same content areas and navigation labels) to support quick learnability and use.
When it comes to product research you should consider (if you have a mobile version of the site) the mobile content. Let users request product literature to view later when they are not “on the go” (though more and more people are using thier mobile devices when sedentary – at home, watching TV). Provide sharing tools.
When it comes to accessing customer’s personal financial accounts by logging in… Again, many users are NOT going to access it via a dedicated app, so you need to make sure that the experience doesn’t “break” when customers log in to see their information. If you can offer a mobile web version that the user can access, you will produce a faster page load and many customers will be appreciative of this.