Entity Framework - bidirectional one-to-one relationship on a foreign key

Question

I have two entities like Hat and Owner:

+========+          +=========+
|Owner   |----------|Hat      |
+--------| 0/1    1 +---------|
|ID      |          |ID       |
|Name    |          |Size     |
+--------|          +---------|
|HatId   |          |OwnerId  |
+========+          +=========+

(Every Owner has his Hat. Some Hats does not have their owners.)

I've created models:

public class Owner
{
    [Key]
    public Int32 ID { get; set; }
    public String Name { get; set; }
    public virtual Hat Hat { get; set; }
}

public class Hat
{
    [Key]
    public Int32 ID { get; set; }
    public Int32 Size { get; set; }
    public virtual Owner Owner { get; set; }
}

At this point I figured out that:

• I can create one-to-one relationship on primary key - which I don't want because Hat can change his owner but ID of Hat and ID of Owner should never change after creation.
• I can create one-to-many relationship - but here I'm unable to create foreign key on "many" side, which will be auto updated.

So question in: Is it possible to create bidirectional one-to-one relationship on a foreign key in Entity Framework?