Explain Why It Is Impossible To Derive An Analytical Formula For Valu
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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
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,
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
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
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.