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How does one typically exit (close out) a large, in-the-money long put option position?
You are long the puts. By exercising them you force the underlying stock to be bought from you at your strike price. Let's say your strike it $100 and the stock is currently $25. Buy 100 shares and exercise 1 (bought/long) put. That gives you $7500 of new money, so do the previous sentence over again in as many 'units' as you can.