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We study the free boundary problem of the American type of options. We consider a continuous dividend paying put option and provide a much simpler way of approximating the option payoff and value. The essence of this study is to apply geometric techniques to approximate option values in the exercise boundary. This, being done with the nature of the exercise boundary in mind, more accurate results are guaranteed. We define a transformation (map) from a unit square to the free boundary. We then examine the transformation and its properties. We take a linear case for a transformation as well as a nonlinear case which would be more fitting for option values. We consider stochasticity (an Ito process) as we define this transformation and this yields better approximations for option values and payoffs. We also numerically compute optimal option prices by using the same transformation. We finally demonstrate that our transformation performs better than most semi-analytic results.

Option pricing is one of the major areas of Mathematical Finance and has over the years generated a series of interesting problems. Among the many questions that still exist in this area, the analytic valuation of American options remains key as one of the most outstanding. About half a century ago, one of the most fundamental results in option pricing was put forth by Black & Scholes in their seminal paper. The results of this paper have gone on to be embraced as the dogma of option pricing [

The first section introduces the problem, followed by a section that details the study of an American put option (the only case we consider in this work). In section 3, we handle the transformation. We derive both the linear and non-linear cases of the transformation having also defined some parts of the free boundary. The next section follows with numerical simulations which attempt to envisage the reliability of our results. The last section discusses the results before we sum it up with suggestions for further work in this line.

In this section we dig in depth information about the features of the American put option as well as its price.

Let us now study the desirable properties of the put price. These are the ones on which we base to justify the reliability of our transformation in the preceding sub section as well as move on to carry out simulation of the same in the forth coming section 4. First, we note that the price

where

Now some of the desirable properties about this price are such that;

・ For

・ Also for

・

Now we need to summarize all we have done in the preceding subsection into a theorem. This is a standard theorem and it puts forward the restriction on the fact that the value of an American put option is at least the value of its European counterpart.

Theorem 1. For any market, if the risk-free rate r is positive, then for every

where

This theorem is clearly reflected by the decomposition of the American option value into the European option value as well as the early exercise premium. Since the premium can at worst be zero but not negative, this establishes the theorem 1 as can be seen from the

We now embark on establishing a transformation for the optimal boundary. We define

this transformation from a unit square a set

whenever

and also not that

Now the standard American put option valuation equation is given by;

And

Set

Again, set

and 7 is satisfied.

Lets again set

and 7 is satisfied

where

and 7 is satisfied. The optimal exercise boundary is

Recall that

The aim is to locate

The next question now is how to compute or express

In this sub section, we apply quadrature techniques to obtain an approximate for the area of

and computing the value of the integral, we employ the Gaussian quadrature approximation technique.

and

and now we notice that for

and

and hence

Consequently with increase in the n value, there will be an increase in the terms on the expansion thus increasing accuracy. So then recall the Equation (12) which is;

which is the required exercise boundary i.e.

Now having obtained the area of the co-domain of our desired transformation, we now move on to establish this transformation from a unit square to the set

such that

and on the boundaries of the unit square, it is clear that

So a transformation that satisfies both of these Equations (17) and (18) would be our appropriate result to use in the analytic approximation of the option optimal prices alongside their corresponding optimal times.

Now note that;

and also that

Consequently, a similar expression can be obtained for various other values of y provided they are assumed constant and only x varying. In general, the transformation for values along the x-axis of the unit square is given by;

And also for the second piece of the transformation i.e. in the y-direction we apply interpolation. However here we note that the

Also, a similar expression can be obtained for various other values of x provided they are assumed constant and only y varying. hence in general, the transformation for values along the y-axis of the unit square is given by;

In a summary, the transformation would then be defined as in the next proposition which is one of the major results of this work.

Proposition 2. Linear transformation

Define

where

Proof. The first part of the proof is to prove that

Notice that

hence

Hence

Now, we have a linear transformation that could be used to approximate the payoff values of the option over time. Nevertheless, we remark that the approximations from it would be too inaccurate as option payoffs are known not to be linear over time otherwise. It is rather evident that option prices and their corresponding pay-offs follow Ito processes and not log normal processes (even though the two are somewhat related). So we need to consider this in the approximation of payoff values from the bound values of the same. We thus employ techniques borrowed from the area of stochastic interpolation1; the type of interpolation in which we approximate functional values for random (stochastic) data; with some modifications so as to suit our problem here. Consider an Ito process

where

where

the log-normal probability density function. Using the technique given in Equation (22), we can reliably approximate the payoff values and thus define a transformation from the unit square using this approximation. So then we now organize the preceding results into a proposition which follows right away.

Proposition 3. Nonlinear transformation

Suppose the price

where

and

Proof. Notice that the proof can be done in the very exact way as in proposition (2) with a change of

from the start of the interval. Also from the interval end it would then be given as;

Hence from either end; we have that

So the task remains to demonstrate that our results concur with this in all ways. But before embarking on that we desire to note some properties of a good approximation for the put price that we can perhaps look out for from our results.

In this section we show the numerical results of this work. We numerically compute optimal option prices from the transformation. We also compare the results of our transformation with some of the powerful known analytic approximations. Now we proceed to derive and demonstrate the numerical approximations of the method in sub section 3 and the exercise boundary is obtained graphically depicted in

Now for the exercise boundary plot, as the stock price grows over time there is at first a gradual expansion in the size of the exercise region with a concurrent reduction in the

holding region. Also at about

grow at an exponential rate. Now we next study a 3D plot of option prices against stock prices over time. This is depicted in

relationships among these three. Notice that the structure of the exercise boundary (region) is depicted along the x-axis (time) of this plot which is analogous to the structure obtained for the exercise boundary plot which is graphed as stock prices over time.

Also notice that the shape of the variation of the payoff for an American option is also reflected here in (Option prices axis) over time. This, in a nut shell is a plot that summarizes all the plots into one. So all conclusions made regarding the other previous two plots still hold under

Here, we have provided a far much simpler way of approximating option values as well as payoffs basing on a unit square. Most approximation techniques provided in literature tend to be sophisticated and somewhat cumbersome at specific times of the option. However, our method stays put in regard to application throughout the entire life of the option. We have demonstrated that an option value can be approximated through basing on the unit square to acquire far better accurate results. This beats most approximation techniques already in existence. This method also exceeds others in terms of simplicity of application coupled with accuracy of results. The major objective of this work has been achieved as it was majorly providing an easier way of approximating the payoff by using a transformation from a unit square to the exercise boundary. This has been superfluously achieved. The transcendence of our method is evidenced by the fact that when approximating payoffs, one works within a known set, the unit square. Moreover, our method can easily be run on a computer and the average running time is

so minimal. Further work may be needed to be done in this area to improve the results such as considering better and more efficient non-linear approximation (interpolation) schemes such as ordinary kriging, universal kriging. Notice that considering these approximation schemes would better the results (in terms of accuracy) as variance is minimized. However, such methods were beyond the scope of this work.

We thank African Union for the support towards this research.

Katende, R., Seck, D. and Ngare, P. (2016) On the Location of a Free Boundary for American Options. Journal of Mathematical Finance, 6, 930- 943. http://dx.doi.org/10.4236/jmf.2016.65062