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).

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