Can I frame a new window without adding jack studs?
Yes, but you'd need to repurpose existing common studs as jack studs. This may mean using a wider (longer) header than would ordinarily be needed. Obviously this means opening up the wall more than you'd maybe like, but it's all kept up above the tub where it's more easily repairable.

_________________________________________________
_________________________________________________  &lt;-- double plate
_________________________________________________  &lt;-- top plate


                   header

________________________________________________
| |    | | |                        | | |    | |
| |    | | |                        | | |    | |
| |    | | |                        | | |    | |
| |    | | |                        | | |    | | &lt;-- existing common
| |    | | |        rough           | | |    | |     stud, cut to 
| |    | | |       opening          | | |    | |     header height
| |    | | |                        | | |    | |
| |    | | |                        | | |    | |
| |    | | |                        | | |    | |
| |    | | |                        | | |    | |
| |____|_|_|________________________|_|_|____| |
| |__________________________________________| |
| |             | |            | |           | |
| |             | |            | | &lt;-------- | | --- existing common studs
| |             | |            | |           | |     cut off at sill height


If this is a load-bearing wall, you probably want to install a temporary wall just inside the tub to support the joists. Chances are the wall as a system will carry the weight, but you want to prevent drywall cracks, etc. 


Pre-cut a few studs to the ceiling height minus 3", and cut two plates suitable for the space.
Pad a 2x4 plate with a blanket or other padding to protect your ceiling. Have an assistant hold the plate against the ceiling, perpendicular to the joists. 
Set a plate on the floor below it.
Tip your pre-cut studs in place, setting one end and driving the other with a hammer until plumb (or snug, whichever comes first). Spare your ceiling and drive the bottom end where possible.
Try to position one stud roughly under each joist.
Friction will more or less hold everything in place, but run some 3" gold screws in on a toenail angle to be sure.


Keep an eye on things as you start to work on the window opening. Watch upper floors for movement at doorways, etc. There shouldn't be any.