The buyer of an American Option has the right to exercise the option at any time before and including the maturity date of the option while the buyer of a European Option has the right to exercise the option at the maturity date only.

We assume that there is no arbitrage and we place ourself in discrete time in the Cox-Ross-Rubinstein binomial model framework.

We know that in this framework the price of an European option is equal to the discounted expectation of its final payoff under the risk-neutral probability Q.

We don’t have such simple expression for American options.

But we can calculate the price of an American option with a backward recursion.

The price of the option at maturity is equal to its final payoff. At the previous dates, the option’s holder will decide to exercise the option if it is interesting for him, so if the value of exercising the option is higher than the value of keeping it.

So the price of the American option at a given date t is equal to the maximum between the value of exercising it at t, and the discounted expectation of the value of the option at the next date.

The underlying asset price can go up by a factor u with a probability q and down by a factor d with a probability 1 – q. We know the expression of the risk-neutral probability q, and the upward and downward factors u and d are determined from the volatility sigma of the underlying asset price.

So we can determine the price of the American option starting from the boundary condition with the final payoff of the option and applying a recursive backward algorithm.

An American option, which gives more rights than a European option, has a higher price. It can be easily shown by using the backward recursion formula for the American option price, and the fact that the European option price is equal to the discounted expectation of the future price.

For call options, the price of American and European options are the same if we assume no dividend payments and a positive risk free interest rate. There is indeed no benefit for the American call option holder to exercise the option before it expires. The price of the option is always higher than the exercise price.

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