Some Properties for the American Option-Pricing Model

In this paper we study global properties of the optimal excising boundary for the American option-pricing model. It is shown that a global comparison principle with respect to time-dependent volatility holds. Moreover, we proved a global regularity for the free boundary.


Introduction
It is well-known that, for the American option-pricing model, there is an optimal holding region for contracts holders (see [1][2][3][4][5]).The part of the boundary for the region is unknown (free boundary), which is often referred as the optimal excising boundary for option traders.This free boundary has to be calculated along with the option price of the security.The mathematical model for the problem is highly nonlinear and there is no explicit solution representation even when volatility and interest rate are assumed to be constants (see [4]).On the other hand, for the financial world as well as for the intrinsic interest itself, it is extremely important to find the location of the free boundary along with the option price of the security.Particularly, people would like to know how the price of a security changes near the option expiry time since it may change dramatically [6,7].
During the past few decades, there are many research papers concerning for various option-pricing models.There are several Monographs devoted to this topic (see, for examples, [1,3,4,8]).For the American option model as well as its generalization, the existence and uniqueness are studied by many researchers ( here just a few examples, [2,5,9-12]).A basic fact is that the American option-pricing model can be reformulated as a variational inequality of parabolic type.Hence, many known results about existence and uniqueness can be applied to the model.However, the disadvantage of the method is that there is no information about the free boundary.To overcome the shortcoming, several authors employed other methods to establish the existence and uniqueness for the problem (see [7,[13][14][15][16][17]).Because of the practical importance, many researchers paid a special attention to the asymptotic behavior for the free boundary near the expiration time(see [6,[18][19][20][21][22][23][24][25]).Moreover, various nu-merical computations for the location of free boundary are also carried out by many people (see, for examples, [14,[25][26][27][28] and the references therein).More recently, some global property of the free boundary attracts some interest.The authors of [29,30] proved that the free boundary is convex if the volatility in the model is assumed to be a constant.However, this global property is not valid in the real financial market since the volatility depends on time and other economical factors.When the volatility depends on time and the security, the problem becomes much more challenging.In this paper we would like to study some global property of the free boundary.We want to find how the optimal exercising boundary changes when the volatility changes during the life-time of the option contract.This question is very important for structured products in the financial world.
We first recall the classical model for the American option-pricing model with one security or one type of asset.Let   , V s t be the option price for a security such as a stock with price s at time .Then it is well- known that satisfies the Black-Scholes equation with no dividend [31,32]: where is the interest rate and r  represents the market volatility of the stock, is the region defined below.

 
, V s t  0, called the optimal exercising boundary.
On the free boundary   = s S t , we know from the continuity of the option price that satisfies: where K is the striking price.We also know the payoff value at the terminal time once the striking price is given: For later use, we introduce : where In financial markets, the volatility  plays a major role for the option pricing model.Option price often changes dramatically when the stock market is in a chaotic movement.This was the case when the flash-crash happened on May 6, 2010 as well as the case on Oct. 19, 1987.On the other hand, for a relatively stable market, the volatility mainly depends on time.This is particularly true for an index fund such as S&P500 index in the U.S. market.Hence, we assume that throughout this paper.Our question is how the free boundary changes when the volatility   t  changes during the life-span of the option contract.We show that there is a global comparison principle for the free boundary with respect to the change of volatility .Moreover, a global existence result is also established as a by-product.Our proof is based on the line method (see [15]), which is different from existing literature (see [21,13] and the references therein).Although the existence of a solution for the problem is already known, our method does have several advantages.One of them is that the free boundary is determined along with the option price at each discrete time simultaneously.Moreover, a global regularity for the free boundary is also obtained.To author's knowledge, this regularity result is new and optimal (see [19,21,12]).

t  
The paper is organized as follows.In Section 2, we construct a sequence of approximation solutions by using the line method.After deriving some uniform estimates, a global existence is established.Moreover, an optimal global regularity for the free boundary is also obtained.In Section 3, we first derive some comparison properties for the approximation solution and then show that the limit solution preserves the same property.Some concluding remarks are given in Section 4.
Remark 1.1:After this paper is completed, the author le

Existence and Uniqueness
ased on the discrete owing conditions are always assumed throughou arned that E. Ekströn proved a result in [33] (2004) about the monotonicity of option price with respect to volatility.However, there is no result about the comparison result for the free boundary.Moreover, the method in [33] is totally different from ours here.In addition, we also present a regularity result for the free boundary.
Since our argument in Section 3 is b problem, we give the complete details about the construction of the approximation solution sequence.We also show that the approximation sequence is convergent to the solution of the original problem (1.1)-(1.5).As a byproduct, an optimal regularity of the free boundary is obtained.
The foll t this paper.
Now we construct an approximate solution sequence by teger.Divide using the line method.
Let N be a positive in If we use difference quotient to approximate and This leads us to define the approximate solution   n V s and n S as follows: m the terminal conditio Fro n, we know So we define .
Suppose we have obtained and where w exte d n 

It see th e above free boun ary problem (2 erpolation to define the free bo
for each n .Actually, since the problem is ional one n find the solution   n V s and n S explicitly (see [4] for detailed calculation Now we use the int undary   N S t as follows:   We also is convergent to the solution of the dary problem (1.1)-(1.5).To this end, we need to derive some uniform estimates.

Lemma 2.1:
Proof the definition, we see if T. Suppose we have shown that then at this minimum point, we see which contradicts the right-hand side of the Equation (2.1).It follows that other hand, we We assume that   n, suppose that .The

 
V s attains at an interior poin where depends only on known data, but not on his estimate is similar to the energy esti te r a parabolic equation.Indeed, we introduce new variables: = ln , = .
Then the original free boundary problem .1)-(1.5) is eq (1 uivalent to the following one: where

Lemma 2.4:
There exists a constant su that On the other hand, by the definition we know It follows that Thus, .

Now we can extend
, we use the continuity of where if (2.9) a classical parabolic equation (see [34], estimate (5.15) 137) a n the desired energy estimate.By the definition, we see clearly that where depends only on known da , but not on .Proof: Note that is uniformly Lipschitz con- . We may assume that

 
It follows that satisfies the following equations: (2.10) (2.11) The maximum principle yields that where depends only on the known data an From the boundary condition (2.5), we see From the Equation (2.4) and the boundary conditions (2.5) and (2.6), we see which is uniformly bounded.By differentiating Equation (2.4) with respect to x   1 = 0, , .
where depends only on known data and the 138).The uniq s llows from the variational inequality.Moreover, regularity theory for parabolic equation implies that Moreover, since the coefficients of the Equation (2.4) depends only on  , we use the interior re ularity of para g bolic equations to conclude that To see the regularity of the free boundary, we use Lemma 2.5 to see Hence, by Ascoli-Arzela's lemma, we can extract a subsequence, still denoted by , such converges to a function, denoted by   S t .Moreover, where C depends only on kn ata   own d , and p  .Now we convert back to the origin variables to conclude that al To see more regularity for , we use the boundary condition (2.5)-(2.6).Ind the condition (2.5)-(2.6),we see From the Equation (2.4) we obtain Now we consider the free boundary problem for It is easy to see that a unique solution exists with oblem (1 To prove the theorem, we show that the c mparison property holds for the discrete solution und r certain co o e ndition.

Lemma 3.1:
Proof: If necessary, we may use an approximation to replace by a smooth c vex function on Without loss of generality, we may simply assume     .Th from the regularity theory, we know th From the maximum principle, we see that  .
We differen = tiate Equation (2.1) for with respect to = n N n to obtain: The maximum principle implies that , , t S (1.5)

 a
When the volatility is a constant, it has been known for a long time that the option price is b e volatility is bigger.However, when the of time,   = t   nor the optimal excise latility changes for the whole time period   0,T .In this paper we answered such a question.We show that a comparison property for option price and the optimal excising bound (Theorem 3.1) when the volatility . This result is important for option traders.Moreover, we proved a global regularity result for the free boundary by using a very different method from the existing literature.ledgements Some results in this paper were reported at the international conference "Problems and Challenges in Financial Engineering and Ri ary hold

Acknow
sk Management" held in Tongji 1.The author would like to an, Professor Xinfu Chen, all, New York, 2008.
which are exactly the same as for on page and the bound depends on known data and  .Now we can m is uniformly b nded.One can also use the same argument for W to conclude the es- After a f r of steps, we obtain the desired result of Lemma 3.1.Since we are interested in the relation between Q.E.D. Now we are ready to prove the main theorem in this section.

Properties of Free Bo dary
.