Why is there still market value when a company goes bankrupt?

Title says most of it, but why is there value - I know book value of equity can go negative but market value can't because you can't have negative shares/negative share price. But believe the better answer comes down to "option value"? What does that mean? Is it just that people are buying call options?

Bonus: How do you define liquidity?

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It is because put options don’t expire or don’t go away if a company goes bankrupt. Put option is when you can sell at a particular price. However, because of high transaction costs, lack of buyers (limited liquidity), you won’t be able to exercise the put options hence they remain in place. And the company continues to trade at the value of those put options.

 
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The equity can be viewed as a call option on the firm with the strike price being the face value of the debt and the expiration being when the debt matures. Just like a call option, the equity's downside is limited to what you put in and the upside is potentially unlimited.

So if a firm is in distress, the reason that the equity still retains some value and does not go to zero immediately is because of the possibility that the firm can make it through the woods whereby its value can end up greater than the face value of the debt when the debt matures -- in other words, the equity is not worthless because, as a call option, it still has "time value".

 

So I'm trying to understand this. Does this mean that every dollar increase in the debt's face value in the future goes to the equity holder? Therefore there's value in betting on the potential rise in that face value?

Or is it more indirect? As in, if the face value of the debt increases, that implies that the firm is doing better, and in turn that means the firm might make money, which flows to shareholders?

 

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