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Why are American-style options worth more than European-style options?
The value of an option has 2 components, the extrinsic or time value element and the intrinsic value from the difference in the strike price and the underlying asset price. With either an American or European option the intrinsic value of a call option can be 'locked in' any time by selling the same amount of the underlying asset (whether that be a stock, a future etc).  Further, the time value of any option can be monitised by delta hedging the option, i.e. buying or selling an amount of the underlying asset weighted by the measure of certainty (delta) of the option being in the money at expiry. Instead, the extra value of the American option comes from the financial benefit of being able to realise the value of the underlying asset early. For a dividend paying stock this will predominantly be the dividend. But for non-dividend paying stocks or futures, the buyer of an in-the-money option can realise their intrinsic gains on the option early and earn interest on the profits today. But what they sacrifice is the timevalue of the option.  However when an option becomes very in the money and the delta approaches 1 or -1, the discounting of the intrinsic value (i.e. the extra amount a future cash flow is worth each day as we draw closer to payment) becomes larger than the 'theta' or time value decay of the option. Then it becomes optimal to early exercise, abandon the optionality and realise the monetary gains upfront. For a non-dividend paying stock, the value of the American call option is actually the same as the European. The spot price of the stock will be lower than the forward price at expiry discounted by the risk free rate (or your cost of funding). This will exactly offset the monetary gain by exercising early and banking the proceeds. However for an option on a future, the value today of the underlying asset (the future) is the same as at expiry and its possible to fully realise the interest earned on the money received today. Hence the American call option is worth more. For both examples the American put option is worth more, slightly more so for the stock. As the stock's spot price is lower than the forward price, the owner of the put option realises a higher (undiscounted) intrinsic profit from selling the stock at the higher strike price today than waiting till expiry, as well as realising the interest earned. Liquidity may influence the perceived value of being able to exercise early but its not a tangible factor that is added to the commonly used maths of the option valuation, and isn't really a consideration for most of the assets that have tradeable option markets. It's also important to remember at any point in the life of the option, you don't know the future price path. You're only modelling the distribution of probable outcomes. What subsequently happens after you early exercise an American option no longer has any bearing on its value; this is now zero! Whether the stock subsequently crashes in price is irrelevent. What is relevant is that when you early exercise a call you 'give up' all potential upside protected by the limit to your downside from the strike price.