This instructable shows how to make a tapered drill chuck for a lathe out of a bolt. You'll need the following materials:

Bench Grinder
Metal File
Bolt
Nut
Threaded Chuck
Prussian Blue
Epoxy Putty
Small Motor
Carbide Scribe

Step 1: Small Motor

To keep the run-out to a minimum when machining the bolt, it's necessary to keep it rotating as it's shaped. To do this, you'll need to find a small motor. This type of motor can be found inside microwave ovens and household ventilation fans. It's typically low power (watts) and uses low current, comparable to a typical household lightbulb. These type of motors are known as shaded-pole motors. They should be (fairly) easy to stop by hand. To prepare it for use, do the following:

Strip wire and slide it through the motor's tab holes. Wrap the wire around the tab.
Use hot glue to insulate the tabs so that you do not electrocute yourself.

Step 2: Ready the Bolt

Ready the bolt for grinding by doing the following:

Use a nut to jam the bolt to the drill chuck.
Insert the motor's shaft into the drill chuck.
Power up the motor.
While the motor is spinning, it causes the drill chuck and bolt to spin. Use the scribe (sharp point of tungsten carbide or high-speed steel) to mark the top of the bolt. The rotation will tend to move the scribe's tip towards the center of rotation which is the exact center of the bolt.
As a hole forms, apply more pressure to the scribe so that it forms a relatively deep pit in the center.

Step 3: Grind the Bolt

To grind the bolt into a rough taper, use a bench grinder:

Place a shallow dish of water below the grinder so that it captures and extinguishes sparks.
Use an ice pick and the center pit (from previous step) to keep the bolt center as the chuck-motor spins.
As the grinder is rotates, make sure that the motor rotates as well. If the motor stalls, change the angle slightly so that it works perpendicular to, or with the rotation of the grinder.
Use the coarse wheel of the bench grinder to remove the hex-head of the bolt.
Use the side of the smooth wheel to grind the taper. The area of the taper which is being removed will produce the most sparks while grinding.

Step 4: Taper Template

To shape the taper so that it doesn't wobble while inserted into a lathe's tail-stock, a template is needed. Moreover, because the angle of typical tapers is so small (less than 2°) there is no good way to measure it so a template is best. It's easy to make:

Use an existing taper or tail-stock to create the template.
Apply paste-wax to the existing taper so that it becomes difficult to glue.
Apply a quick-setting (5 minute) epoxy putty, such as Oatey's Fix-It Stick, to half the tail-stock.
When cured, it should not be difficult to remove the hardened epoxy template from the tail-stock.

Step 5: Blue & File

Finalize the shape of the taper:

Use gloves or a finger cots (they look like miniature condoms) to apply prussian blue marking paste to the taper.
Insert and rotate the taper into the epoxy template.
With an excess of prussian blue, the spots with too much metal will have less prussian blue and will show up as a lighter color than areas with too little metal. With the chuck motor rotating, use a a metal file to work away at the lighter colored areas. [With a small amount of prussian blue, the opposite is true.]
Retest using the template after every few minutes after filing. When the taper color is uniform enough, it should fit into the tail-stock without wobbling.
Wipe off prussian blue when done.

Step 6: Test & Use

Now that it's done test and use:

Insert the taper and chuck into the spindle (rotating part) of the lathe.
Turn the lathe on and check for run-out. A small drill bit will appear to rotate off-center if there is run-out.
Insert the taper and chuck into the tail-stock and use as needed.