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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,

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,

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