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American option

American options usually or of put or call type.
American call Payoff=$\max(S(\tau)-K,0)$ where we can choose $\tau~$ to be any time between zero and T.
American put Payoff=$\max(K-S(\tau),0)$ where we can choose $\tau~$ to be any time between zero and T.
The exercise time $\tau~$ should be a stopping time, that is given the information available up to time t we must be able to decide if $\tau\le t$.

Call Option

If the underlying asset S does not pay dividends and the risk free interest rate is positive it is not optimal to exercise the call option before T. The argument for this is quite straightforward:
First since the American call has more opportunities it is at least worth as much a European call option with strike K and maturity T. The European call option has a non-negative value since it is an expectation of a non-negative function of the underlying asset. Moreover due to Jensen's inequality we have that the European call option is worth more than $\max(S(t)-\exp(-r(T-t))K,0)\gt \max(S(t)-K, 0), 0\leq t< T$. Therefore the European call option is worth more than $\max(S(t)-K,0)$, $0\leq t\lt T$. But this is the pay-off we get if we exercise the American option at time t before T so therefore we should wait until T to exercise.

Put Option

If risk free interest rate is zero or negative one can with the same type of argument as for the call option show that it is not optimal to exercise before T. For the put option however it is if the interest rate is positive sometimes optimal exercise before T. One can show that there exist a level L(T-t) which depends on the time left to maturity (and other parameters) so that it is optimal to exercise as soon the underlying S(t) goes below L(T-t) for the first time. The function L is not known in closed form even for the Black Scholes model. One can however show that for the standard Black-Scholes model that L(0)=K and that $$\frac{2Kr}{2r+\sigma^2}\leq L(T-t)\leq K,$$ for $0\leq t \leq T,~T\gt 0$. The function L is decreasing and convex for $r\gt 0$. Its derivative tends to $-\infty$ as t tends to T. See figure for an illustration of the exercise level's dependence on the time to maturity.
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In the Black Scholes model we get the following values for $K = 100; T = 1; r = 0.05$ and $\sigma = 0.2$: []
We see that for low values of the stock the value approaches the pay off and for high values of the stock the value approaches the value of the European put.


The value of an American put option using the Carr and Jarrow decomposition (no dividends no upward jumps) \begin{eqnarray*} \Pi_A^p(t,s,T,K)&=&\Pi_E^p(t,s,T,K)\\ &&+\int_t^TrKe^{-r(u-t)}\mathbb{Q}(S(u)\leq L(T-u)|S(t)=s)\text{d} u \end{eqnarray*} L can be found be solving the non-linear integral equation \begin{eqnarray*} L(T)&=&K-\Pi_E^p(0,L(T),T,K)\\ &&-\int_0^TrKe^{-ru}\mathbb{Q}(S(u)\leq L(T-u)|S(0)=L(T))\text{d} u \end{eqnarray*} For diffusion type models the value is usually calculated by PDE methods. The value satisfies the Black-Scholes PDE above L(T-t) and equals $K-S$ below. It also possible to use binomial tree approximations to obtain the value.

Perpetual Put Option

An option with infinite time to maturity is usually called a perpetual option. Above the optimal exercise level $L(\infty)$ (the reason for the notation $L(\infty)$ for the optimal exercise level is that it is the limit of the finite time to maturity problem above as T tends to infinity.) the value $f$ satisfies the ODE \[ rSf'(S)+ \frac{1}{2}\sigma^2S^2f''(S)=rf(S),\] below it is equal to the exercise price $K-S$. Using the ansatz $f(S)=S^\gamma$ we obtain the general solution to the above ODE as \[ f(S)=c_1S+c_2S^{-\frac{2r}{\sigma^2}}.\] The first term can be disregarded due to the fact that we know that $f$ should be decreasing in $S$. Assuming that the value satisfies the so called high contact condition at $L(\infty)$, i.e. both the value and its derivative w.r.t. $S$ are continuous at $L(\infty)$, we can solve for $c_2$ and $L(\infty)$. The optimal exercise level is thus obtained as $$L(\infty)=\frac{2Kr}{2r+\sigma^2}.$$ Now putting everything together we have that the value at time zero (or any other time for that matter) is given by $$ \begin{cases}K-S & S< L(\infty),\\ \frac{\sigma^2}{2r}L(\infty)^{\frac{2r}{\sigma^2}+1}S^{-\frac{2r}{\sigma^2}} & S\geq L(\infty), \end{cases}$$ where $S$ is the current stock price.

References
Carr, Peter, Robert Jarrow, and Ravi Myneni. "Alternative characterizations of American put options." Mathematical Finance 2.2 (1992): 87-106. Journal Article

 

Questions: Magnus Wiktorsson
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