Structural model - analogy with call option or put option?

CFA Level II Fixed Income
#1

In structural models of corporate credit risk, I understand that shareholders could be both consider as :

  • buyers of a call option on the company’s assets (« stock of a company with risky debt outstanding can be viewed as a call option »)
  • buyers of a put option on the company’s assets (« debt investors can be thought of as being short a put option » meaning from what I understand shareholders can be thought og as being long a put option)

I am confused, which one is correct or why both would be correct ?

#2

Do the shareholders gain when the value of the assets increases (long call option), or when the value of the assets decreases (long put option)?

#3

I would say shareholders gain when the value of the assets increase (because they are owners of the assets). So it means shareholders can be viewed as buyers of a call option or long call option, correct?

Then why debtholders are viewed as sellers of a put option (as described in the material) / is it because they are not owners of the assets (as they are debtholders and not shareholders) but as well as shareholders they gain if assets increase?

Thank you S2000magician!