Correctly positioning the start of projectile position

I have the following code;

     if (Bullet == null)
         Bullet = Instantiate(Resources.Load("PrefabBullet")) as GameObject;

     GameObject o = GameObject.FindGameObjectWithTag("Player");
     var player = o.transform;

     var shootPos = player.position + player.forward * 0.9f;
     //Position the bullet to come from the middle of the player
     //shootPos.z = shootPos.z - 1.0f;
     //shootPos.x = shootPos.x - 0.5f;

     var instance = (GameObject)Instantiate(Bullet, shootPos, player.rotation);

     Vector3 bulletDirection = new Vector3(target.position.x-transform.position.x, 
     0.0f, 
     target.position.z-transform.position.z ) * bulletSpeed;

     //Debug Statements
     //print(transform.position);
     //print(target.position);
     //print(bulletDirection);

     instance.rigidbody.velocity = bulletDirection;

Now when I trigger the shot button my projectile is drawn on the screen and moves towards my target. What I am struggling to understand is how to make the projectile start from the center of my 'player' model, and then to travel in a horizontal plane towards the target. (I have gravity behaviour off). It seems that no matter how I tweak the shootPos I cannot get it to be consistently at the center of the player, nor to move in a horizontal line. (It moves straight, but sometimes goes up at an angle to the horizontal plane.)

I guess this is either down to some bug in my above code, or more likely down to my understanding of the math involved. So can somoen explain to me what I am doing wrong, or what incorrect assumption I have made?