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...