Difference in Value between General American & European Options

Question

I am trying to get a grip on the difference in value between American and European options. I have read up on and understand that the value of an American call option is equal to that of a European call, and the value of an American put is greater than that of a European put.

My problem is with determining whether the value of an American option is equal to that of a European option in general.

Take, as a simple example, an American option with payoff E - S(t) and compare this to the corresponding European option with maturity date T. My intuition tells me that the American option is more valuable, but I can't seem to justify this rigorously, because when I exercise at t < T, I don't know what will happen at T, or if E - S(t) > E - S(T). I also understand that if I exercise early I can earn interest on my profit but again am struggling to formulate a rigorous argument to this effect.

Any explanation/guidance would be greatly appreciated.

Answer 1

An option's value is tied up in its Intrinsic Value and its Time Value. And, Time Value [upper case T & V] consists of two elements, the DCF-related time value and the volatility-related time value [lower case t & v].

DCF-related time value for a call option is the present value of the interest that you can earn on the exercise monies in the time remaining until expiry. DCF-related time value for a put option is the present value of the interest lost on not being able to put the exercise monies on deposit for the time remaining until expiry.

Volatility-related time value is the value tied up in the probability to make more Intrinsic Value in the time remaining until expiry.

If you exercise an option early, you kill the Time Value and so an American option will be worth as much as its European equivalent, no more.

However, there are exceptions to this general rule. 1. For a call option, if the underlying stock is paying out a distribution that is worth more than the Time Value that would be destroyed by exercising early and taking the distribution, then the American option is worth more than its European equivalent by the differential amount. 2. For a put option, the DCF-related time value is negative, but the volatility-related time value is (always) positive. For a put option that is not deep in-the-money, the positive volatility-related time value outweighs the negative DCF-related time value and so Time Value (overall) is positive. However, when a put option becomes deep in-the-money, the positive volatility-related time value may not outweigh the negative DCF-related time value and so Time Value (overall) can be negative. In this instance, the American option to kill the Time Value before expiration is worth exercising and so the American option will be worth more that its European equivalent.