Explain Why It Is Impossible To Derive An Analytical Formula For Valuing American Puts.

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some main suggested
techniques that are used to value them.

According to Hull, exercising an American put option on a non-dividend-paying
stock early if it is sufficiently deeply in the money can be an optimal practice.
For example, suppose that the strike price of an American option is $20 and the
stock price is virtually zero. By exercising early at this point of time, an
investor makes an immediate gain of $20. On the contrary, if the investor waits,
he might not be able to get as much as $20 gain since negative stock prices are
impossible. Therefore it implies that if the share price was zero, the put
would have reached its highest possible value so the investor should exercise
the option early at this point of time.

Additionally, in general, the early exerices of a put option becomes more
attractive as S, the stock price, decreases; as r, the risk-free interest rate,
increases; and as , the volatility, decreases. Since the value of a put is
always positive as the worst ...

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as the Optimum Exercise Boundary (OEB).

However in order to be able to value an American put option, we need to solve
for the put valuation foundation and then optimum exercise boundary at the same
time. Yet up to now, no one has managed to produce an analytical solution to
this problem so we have to depend on numerical solutions and some techniques
which are considered to be good enough for all practical purposes. (Walker,
1996)

There are basically three main techniques in use for American put option
valuations, which are known as the Binomial Trees, Finite Difference Methods,
and the Analytical Approximations in Option Pricing. These three techniques
will be discussed in ...

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