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Explain why it has proved impossible to derive an analytical

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). Binomial

trees assume that in each short interval of time, , over the life of the option a stock

price either moves "up" from its initial value of S to , or moves "down" to . In

general, > 1 and < 1. The probability of an up movement will be denoted by

thus, the probability for a down movement is . The basic model of this simple

binomial tree is shown in Figure 2. Furthermore, the risk-neutral valuation principle

is also in use when using a binomial tree, which states that any security dependent on

a stock price can be valued on the assumption that the world is risk neutral.

Therefore the risk-free interest rate is the expected return from all traded securities

and future cash flows can be valued by discounting their expected values at the

risk-free interest rate. The parameters p, u, and d must give correct values for the

mean and variance of stock price changes during a time interval of length .

By using the binomial tree, options are evaluated by starting at the end of the tree

(that is time T) and working backward. The value of the option is known at time T.

As a risk-neutral world is being assumed, the value at each node at time T - can be

calculated as the expected value at time T discounted at rate r for a time period .

Similarly the value at each node at time T - can be calculated as the expected

value at time T - discounted for a time period at rate r, and so on. When we are

dealing with American options, it is necessary to check at each node to see if early

exercise is optimal rather than holding the option for a longer while. Therefore by

working the binomial backward through all the nodes, the value of the option at time

zero is obtained.

For example, consider a five-month American put option on a non-dividend-paying

stock when the stock price is $50, the strike price is $50, the risk-free interest rate is

10% per annum, and the volatility is 40% per annum. With our usual notation, this

means that S = 50, X = 50, r = 0.10, = 0.40, and T = 0.4167. Suppose that we

break the life of the option into five intervals of length one month (= 0.0833 year) for

the purposes of constructing a binomial tree. Then = 0.0833 and using the

formulas,

The top value in the tree diagram above shows the stock price at the node while the

lower one shows the value of the option at the node. The probability of an up

movement is always 0.5076; the probability of a down movement is always 0.4924.

Here the stock price at the jth node (j = 0, 1, ..., i) at time is calculated as

. Also the option prices at the penultimate nodes are calculated from the

option prices at the first final nodes. First we assume no exercise of the option at the

nodes. This means that the option price is calculated as the present value of expected

option price in time . For example at node E the option price is calculated as

while at node A it is calculated as

Then it is possible to check if early exercise of the option is worthwhile. At node E,

the option has a value of zero as both the stock price and strike price are $50. Thus it

is best to wait and the correct value at node E is $2.66. Yet the option should be

exercised at node A if it is reached because the option would be worth $50.00 -

$39.69 or $10.31, which is obviously higher than $9.90. Options in earlier nodes are

calculated in a similar way. As we keep on calculating backward, we find the value

of the option at the initial node to be $4.48, which is the numerical estimate for the

option's current value. However in practice, a smaller value of would be used by

which the true value of the option would be $4.29. (Hull, p347, 3rd Ed)

The second technique that is commonly used is the so-called Finite Difference

Methods. These methods value a derivative by solving the differential equation that

the derivative satisfies. The differential equation is converted into a set of difference

equations and the difference equations are solved repeatedly. For instance, in order to

value an American put option on a non-dividend-paying stock by using this method,

the differential equation that the option must satisfy is

The Finite Difference Methods are similar to tree approaches in that the computations

work back from the end of the life of the derivative to the beginning. There are two

different methods involved; one is called the Explicit Finite Difference Method and

the other is the Implicit Finite Difference Method. The former is functionally the

same as using a trinomial tree. The latter is more complicated but has the advantage

that the user does not have to take any special precautions to ensure convergence.

The main drawback of these methods is they cannot easily be used in situations where

the pay-off from a derivative depends on the past history of the underlying variable.

Finally there is also an alternative to the numerical procedures which is known as a

number of analytic approximations to the valuation of American options. The best

known of these is a quadratic approximation approach proposed by MacMillan and

then extended by Barone-Adesi and Whalley. This method involves estimating the

difference, v, between the European option price and the American option price since

v must satisfy the differential equation for both. They then show that when an

approximation is made, the differential equation can be solved using standard

methods.

The techniques mentioned in this paper are those commonly used in practise.

Although they are not perfect, they are still considered good enough for practical

purposes. So far no one has managed to create a direct analytical valuation method

for valuing American put options.

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