Early exercise of an option means exercising the option before its expiration date, and whether it is beneficial to do so depends on the type of option and the
current market conditions.

Time value is the portion of an option's premium that reflects the amount of time left until expiration. It represents the market's expectation of how much the option's price could potentially change due to various factors, such as changes in the underlying asset's price, changes in volatility, and changes in interest rates, over the remaining life of the option.

As an option approaches its expiration date, its time value decreases because there is less time for the option to move in a favorable direction.

On the other hand, early exercise of a deep in the money put option can be advantageous for different reasons:

First, because it allows the put option holder to lock in a profit and avoid potential losses. This is because a deep in the money put option gives the holder the right but not the
obligation to sell the underlying higher than its current market value.

Second, the price of the underlying is borne by zero.

A "deep in the money" put option implies that the price of the underlying is significantly lower than the strike price. In other words, the price of the underlying tends towards zero and the stock is more likely to rise than to fall before expiration.

Third and for the reason explained above a deep ITM put can have negative time value which decreases the premium value.

A negative time value reflects high pessimism the possibility of the underlying going down further.

By exercising early you will bear the cost equivalent to this negative time value.

Remember that option value= intrinsic value+ time value.

If you exercise the option you will get the whole difference between the strike price and the market value of the underlying which is the price at which you can buy back it.

In other words by exercising a deep in the money put option you will obtain the full intrinsic value and if you sell the option without exercising it you will get the intrinsic value LESS the time value.

Fourth, you can invest the proceeds at the risk free rate until expiration.

As a consequence, you would be better off exercising early a deep in the money put option.

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