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

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