Explain why it has proved impossible to derive an analytical formula for valuing
American Puts, and outline the main techniques that are used to produce
approximate valuations for such securities

Investing in stock options is a way used by investors to hedge against risk. It is
simply because all the investors could lose if the option is not exercised before the
expiration rate is just the option price (that is the premium) that he or she has paid
earlier. Call options give the investor the right to buy the underlying stock at the
exercise price, X; while the put options give the investor the right to sell the
underlying security at X. However only America options can be exercised at any time
during the life of the option if the holder sees fit while European options can only be
exercised at the expiration rate, and this is the reason why American put options are
normally valued higher than European options. Nonetheless it has been proved by
academics that it is impossible to derive an analytical formula for valuing American
put options and the reason why will be discussed in this paper as well as 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 can
happen to it is that it expires worthless so this can be expressed as
where X is the strike price
Therefore for an American put with price P, , must always hold since the
investor can execute immediate exercise any time prior to the expiry date. As shown
in Figure 1,

Here provided that r > 0, exercising an American put immediately always seems to be
optimal when the stock price is sufficiently low which means that the value of the
option is X - S. The graph representing the value of the put therefore merges into the
put’s intrinsic value, X - S, for a sufficiently small value of S which is shown as point
A in the graph. When volatility and time to expiration increase, the value of the put
moves in the direction indicated by the arrows.

In other words, according to Cox and Rubinstein, there must always be some critical
value, S`(z), for every time instant z between time t and time T, at which the investor
will exercise the put option if that critical value, S(z), falls to or below this value (this
is when the investor thinks it is the optimal decision to follow). More importantly,
this critical value, S`(z) will depend on the time left to expiry which therefore also
implies that S`(z) is actually a function of the time to expiry. This function is referred
to, according to Walker, 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 turns as follows.

Cox et al claim that a more realistic model for option valuation is one that assumes
stock price movements are composed of a large number of small binomial
movements, which is the so-called Binomial Trees (Hull, p343, 3rd Ed).