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What is an American Option?

American options are different to European style options (Vanillas) in that the contract buyer has the right to exercise the option at any time before maturity  (t=T).  This additional privilege of an American option must come at a cost - know as the early exercise premium.  Therefore, the value of an American put option in relation to a vanilla must satisfy

How do the prices compare?

Below are the values of an American (red) and European (blue) Put option prior expiry as a function of a non dividend paying asset price S are shown.  Notice that cost for the American Put more than its European counterpart when in-the-money, and the two curves tend to fall on top of one another when out-of-the-money.  This is cause the early exercise premium tends to zero the more out-of-the-money it gets since the option is unlikely to be exercised early.  Therefore, a deep out-of-the-money American and Put option have almost the same value.

The freedom to exercise an American option whenever the holder wishes introduces a boundary problem to solving the Black-Scholes equation as done before for vanillas.

From a mathematical point of view:

The boundary condition is no longer fixed at t=T as for the European option case to solve the Black-Scholes equation (more precisely, the reduced heat equation) - we have now got a free boundary. In fact, only a couple of American option solutions can be found analytically, namely pepetual (independent of time) American Call and Put options.  Approximate analytic solution have been found for times subject a certain conditions, however, there are no general analytic solutions.  The Put-Call parity relationship for American options does not exist since the exercise date is no longer fixed.

From a financial point of view:

There are several factors to consider on a early exercise decision.  Firstly, the early exercise premium is lost if exercised early.  Secondly, early exercise leads to a loss or gain in time value of the asset depending on if a Put or Call.  Therefore, one immediately sees that early exercise on an America Call option tends to be less favourable, whereas for an American Put it is more likely (assuming r>0 here).  However, a holder of an American Put options on dividend paying assets generally prefers not be exercised early since the dividend payments are lost if the asset is sold early.  For a discrete dividend paying asset, a decision on early exercise is also influenced by the size of the dividend payments.  One may exercise an American Call option early if the dividend payments are higher than initially forecasted, if not exercised the ex-div value may fall below the expectation price due the loss in asset value through the payouts.  All these factors have an influence on early exercise of an American option, and the decision on when to exercise seems almost a subjective one.

The contract holder will ideally, of course, only exercise the option prior to the expiry date if the present payoff at time t exceeds the discounted expectation of the possible future values of the option from time t to T.  So only if what the holder of the options gets out of exercising early exceeds the markets view of the expected future return in keeping the option alive will early exercising result.  Otherwise, he or she will continue to hold on to the option. At every time t there will be a region of values of S whereby it is best to exercise the option (Exercise region) and a complimentary region whereby it is best to keep the option (Free region). There will also be a particular value S*(t) which defines the optimal exercise boundary separating the two regions.  We have already stated what factors can determine this boundary.

Simple No-Arbitrage consideration

Consider a non dividend paying American Call option worth C(S,t) at time t on an underlying stock S(t) and strike value K. We buy this together with a bond guaranteed to pay K at time the same maturity time T of the option.  Lets further consider the case that when you exercise the option early you are forced to keep the underlying up to maturity. Prior to maturity if S(t)>K we may be tempted to exercises. However, by exercising at time t the value of our portfolio is  which is less than S(t), and thus by keeping this to time T we would be left with S(T). Instead, if we wait to expiry our portfolio value may be worth Max[S-K,0]+K = Max[S(T),K]. Clearly, in this case it is best to keep the option alive up to maturity. Of course S(t) may well be greater than Max[S(T),K], and we are not forced to keep it up to time T, but can cash its value in. However, in this case it would be better to sell the option and cash in its value as an insurance at time t worth more than S(t).  

Why is it hard to solve the problem using a PDE method? Lets see this by considering BS equation  in the two regions:

1. Exercise Region:  The "fair game" (martigale) expected return on holding the option is lower than the options present payoff value (). The expected incremental change in value of holding the option is outperformed by the equivalent BS valuation on the present payoff value:

2. Free Region: Here the opposite is true and it is best to hold the option.

How do we go about solving the Black-Scholes with a free boundary?  What is the value of the option in the Free Region?

So how do we know the value of an American option at any time? We know that a fair no-arbitrage value of the option at time t will either be its present payoff value or the discounted expectation of the possible values from time t to T depending on which one is greater.

where   and is the expectation of X at time t.

So at time t, I would need to know to price the option, which means solving the above problem for future times and taking its average. This iterative nature of the problem makes it very hard to solve analytically and is a subject of research. It is also a numerically challenging problem that only recently has been shown to be solvable via Monte Carlo methods.  Lattice models such as the Binomial method form the standard way to computing the value of the option. This is because the different paths in the evolution of the underlying S are forced to pass through a pre-establish set of nodal points (uS,vS, uvS, etc..) which easily allow us to compute   by backward induction. However, exercise can only happen at these nodal point which are discretely spaced in time. Thus, it is more indicated for a Bermudan option which is an American option that can only be exercised at a pre-established list of times.

Example: Consider a six months American put on a non-dividend exercised paying stock. At t=0  ,  Strike price $100, risk free return 10% and volatility  40% . Lets use a binomial method with 4 time periods and force p=0.5 (this means that )

Knowing the final values of the payoff at time T we can compute the discounted expected return for the previous time step. Lets consider the nodes for when S(T) = $45.5 and S(T)=$60.6. The Put payoff for these final underlying values are $54.4 (= Max[100-45.5,0]) and $39.4(=Max[100-60.6,0]). Thus the expected return on holding the option at the previous time step is

Likewise lets go to the first time step. Here S(T/n)= ${92.5,69.5} with respective payoffs ${12.0, 30.5}. Thus the expected return on holding the option at the previous time step is

Written by Raffaello Vardavas and Alessio Farhadi