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What is the relationship of the option value and the pay off to an option? What happens if the option value > pay off? or if it is equal to the pay off?

I read that in a put option , the option value is a solution to $$\sup_\tau E[\max\left\{K-S,0\right \}]$$, where $K$ is the exercise price. Does this imply that the possible option values are just 0 or K-S. How important is the option value?

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The put option allows you to decide to sell at the exercise price $K$ rather than the market price $S$.

So if you exercise the put option, in effect it is worth $K-S$ at that point as you could have sold at the market price.

You will only do that if it is positive. Hence the $\max\left\{K-S,0\right \}$ expression.

If you could only exercise at a particular future point in time, the option would now be worth the expected future benefit $\mathbb{E}[\max\left\{K-S,0\right \}]$

But with a so-called American put option you can exercise at any time up to a limit, making the value the highest of the future expectations, leading to an option value now of $$\sup_\tau \mathbb{E}[\max\left\{K-S,0\right \}]$$

Typically the option value now is higher than the payoff if exercised now of $K-S$. This difference is called the time value of the option and reflects the benefit from the future payoff being asymmetrical since it cannot be worse than $0$

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By option value, I suppose you mean the price of the options (i.e. The option premium)

The premium you pay for an option, say $10, consists of two parts - Intrinsic value and Time value.

Intrinsic value is simply as you described, the pay off of the option. For a call option, it is the market price - strike price and strike price - market price for a put option.

If you have a positive intrinsic value, it means that the option is "in-the-money" aka you are making money now (K < S). The other two results is that the option is "at-the-money", where the strike price is exactly the market price, and "out-of-the-money", where if you exercised the option now you would be making a loss (S < K). This is intended for call options. The opposite holds true for put options. Note that when you are ITM, intrinsic value is positive. The other two scenarios, your intrinsic value is simply 0 and cannot be negative.

The time value of the option is therefore, in this case, $10 - Intrinsic value. The reason there is a time value is because an investor would pay a premium based on the probability that the option can result in a ITM outcome before expiry.

To calculate the premium of an option, you can use the Black-Scholes model which takes into account Market price, Exercise price, interest rate, term to maturity, volatility as well as probability distribution curve of the underlying asset.

In reality, the importance of the value of options only comes into play when the market price is near the exercise price and that is where most of the volume of options trading is. When you are way in the money, it is highly unlikely that the market will turn that much resulting in a loss. In a way, your returns from the option are guaranteed and therefore requires no attention. Similarly, there is low importance when your options are way out of the money. It is highly unlikely that the market will turn by expiry such that you suddenly become in the money and take positive returns from the option.

Thereby lies an area when you are around and at the money where trading heavily takes place, and the importance of the option value comes into play. Financial institutions usually set risk limit values which their traders must not exceed. They usually calculate such risks based on the sensitivity of the value to changes in parameters, "Delta, Gamma, Vega, Theta and Rho". One example is Delta which measures the sensitivity of the option price relative to changes in the price of the underlying asset.

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