Table of Contents
ISRN Probability and Statistics
Volume 2014, Article ID 743030, 26 pages
Research Article

Error Estimates for Binomial Approximations of Game Put Options

Institute of Mathematics, Hebrew University, 91904 Jerusalem, Israel

Received 17 October 2013; Accepted 21 November 2013; Published 30 January 2014

Academic Editors: P. E. Jorgensen and M. Montero

Copyright © 2014 Yonatan Iron and Yuri Kifer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A game or Israeli option is an American style option where both the writer and the holder have the right to terminate the contract before the expiration time. Kifer (2000) shows that the fair price for this option can be expressed as the value of a Dynkin game. In general, there are no explicit formulas for fair prices of American and game options and approximations are used for their computations. The paper by Lamberton (1998) provides error estimates for binomial approximation of American put options and here we extend the approach of Lamberton (1998) in order to obtain error estimates for binomial approximations of game put options which is more complicated as it requires us to deal with two free boundaries corresponding to the writer and to the holder of the game option.

1. Introduction

A put option on a stock can be interpreted as a contract between a holder and a writer which allows the former to claim from the latter at an exercise time the amount , where is a fixed amount called the option’s strike, is the stock price at time , and . In the American options case its holder has the right to choose any exercise time before the contract matures, while in the game options case the contract writer also has the right to terminate it at any time before its maturity, but then he is required to pay a cancellation fee in addition to the payoff above.

The fair price of American options and of game options is defined as the minimal amount the writer needs to construct a self-financing portfolio which covers his obligation to pay according to the option’s contract. It is well known that in the American options case the fair price can be obtained as a value of an appropriate optimal stopping problem, while for game options we have to deal with an optimal stopping (Dynkin) game (see [1]). For more information about results on Dynkin games and game options we refer the reader to the survey [2]. In general, both for American options and, even more so, for game options with finite maturity explicit formulas for their price are not available and approximation methods come into the picture, while estimates of their errors become important. One of the most easily implemented methods is the binomial approximation of stock prices modelled by the geometric Brownian motion and [3] provided corresponding error estimates for American put options. In the present paper we extend this approach in order to provide error estimates of binomial approximations for game put options. We observe that for perpetual game options some explicit formulas can be obtained (see [4]), but the finite maturity case studied here seems to be more realistic.

Approximating the Brownian motion by appropriately normalized sums of Bernoulli random variables the paper [3] provided (error) estimates const and const for the difference between the price of an American put option and the price of its corresponding th binomial model approximation. Using again the binomial approximation of the Brownian motion as above we construct in this paper two approximating procedures such that the difference between the price of a game put option and its th approximation in the first procedure is between const and const and in the second procedure is between const and const. The error estimates here are somewhat worse than in the case of American put options which is due to the lack of a smooth fit on the boundary of the writer’s stopping region which causes substantial difficulties in the study of regularity of payoff functions.

We observe that specific properties of game put options had to be used in order to obtain error estimates with the above precision. For instance, when payoffs are path dependent (and not only dependent on the present value of the stock) [5] provides error estimates of similar binomial approximations only of order . Since price functions of game options can be represented as solutions of doubly reflected backward stochastic differential equations the results of [6] are also related to game options approximations. Nevertheless, approximations in [6] are not by binomial models, where computations can be done by means of the effective dynamical programming algorithm (see [5]), but by time discretizations, and so relevant probability space and -algebras remain infinite which prevents effective computations. Furthermore, error estimates in [6] applied to our situation are of order ; that is, they are worse than for binomial approximations which we construct here for the specific case of game put options.

Our exposition proceeds as follows. In Section 2 we provide basic results concerning game put option price functions, introduce our approximation processes, and formulate our main result Theorem 1. In Section 3 we show that the price function can be represented as a solution of a variational inequality problem closely related to the Stefan problem (see [7]). We then use this representation to study regularity properties of the price function near the free boundary of the option’s holder exercise region. In Section 4 we study the price function near the boundary of the exercise region of the writer. We use the information about this region from [8] in order to represent the price function as an explicit solution of the heat equation. This representation enables us to understand better the behavior of the price function near the boundary. We estimate also the rate of decay of the price function when the initial stock price tends to infinity. Section 5 is devoted to the proof of Theorem 1. Finally, in Section 6 we exhibit some computations of the price functions and of the free boundaries.

2. Preliminaries and Main Results

The Black-Scholes (BS) model of a financial market consists of two assets among which one is nonrisky and the other one is risky. A nonrisky asset is called a bond and its price at time is given by the formula , where is interpreted as the interest rate. A risky asset is called a stock and its price at time is determined by a geometric Brownian motion where is called volatility and is a standard Brownian motion defined on a complete probability space . If we write also for . The fair price of an American put option at time with a strike (price) and a maturity (horizon) time can now be written as a function of time and the current stock price having the form (see, e.g., [9]) where denotes the set of all stopping times of the Brownian filtration with values in the interval and is the expectation with respect to the measure . If we set , , and then we can rewrite (2) in the form

Relying on [1] (see also [4, 8, 10]) we can also write the fair price of a game put option at time with a strike price , a maturity time , and a constant penalty as a function of time and the current stock price in the form where and is the indicator of an event . Using the functions and as above we can rewrite this formula in the form It follows also (see [1, 8, 10, 11]) that the saddle point (optimal) stopping times for the game value expressions (4) and (5) are given by

Next, we introduce our binomial approximations of the Brownian motion where are independent identically distributed (i.i.d.) random variables taking on values 1 and −1 with probability and denotes the integral part of a number . It is convenient to view as defined on the sequence space by the formula if . Then will be defined on the probability space , where is the product measure and is generated by cylinder sets.

Now set which is the price of the American put option with a maturity and a strike provided the initial stock price is . It is easy to see that if the penalty then it does not make sense for the writer to cancel the corresponding game put option (see Lemma 3.1 in [10]), and so in this case the prices of American and game options are the same; that is, . Since approximations of American options were studied in [3] we assume in this paper that . Observe that is continuous in and it is strictly decreasing to 0 as increases to , and so for each there exists a unique such that . Furthermore, we can define to be the minimal such that and set . In order to define two sequences of functions and which will approximate we set , and introduce stopping times where if the infimum above is taken over the empty set and we set . Introduce a filtration , where is the trivial -algebra and is generated by . Denote by the set of all stopping times with respect to the filtration taking on value in the set . Then, clearly, . Now, for we define and for we set The second approximation function is defined for all by Setting we formulate now our main result.

Theorem 1. For each there exists such that for all ,

Observe that appearing in Theorem 1 is defined via and which can be obtained only knowing precise price function of the American put option with the initial stock price equal to . But from the computational point of view we can obtain only approximately using, for instance, the algorithm from [3]. One of the ways to overcome this difficulty is to proceed as follows. Let denote the th binomial approximation of obtained in [3] which uniformly in satisfies for some . Denote by the minimal such that taking if this inequality does not hold true for all . Set which unlike can be computed employing [3]. It is well known that exists (see, e.g., [3]) and, clearly, this derivative is nonpositive. In fact, it is possible to show that This together with (13) yields that From the definitions (9)–(11) it follows that for each there exists independent of such that Now we obtain from Theorem 1 together with (15) and (16) the following.

Corollary 2. For each there exists such that for all ,

In the following sections we will analyze regularity properties of the price function of game put options and will complete the proof of Theorem 1 in Section 5 providing some computations in Section 6. The general strategy of the proof resembles that of [3], but the study of the price function of game put options is more complicated than in the American options case, in particular, because of appearance of two exercise boundaries (holder’s and writer’s) having different properties. Our proof will be based on regularity properties of solutions of parabolic partial differential equations with free boundary and of the corresponding variational inequalities and we will rely also on some prior results from [3, 4, 8].

3. Price Function Near the Holder’s Exercise Boundary

3.1. Some Previous Results

First, we state the following result from [8] (see also [10]) which we will use later on.

Proposition 3. (i) There exists an increasing function such that and for all satisfying .
(ii) There exists such that for every there is a so that for and for one has for all .
(iii) Furthermore, for all
In particular, is of class , that is, continuously differentiable once with respect to and twice with respect to , and so, in fact, it is a smooth function there.
(iv) Finally, is convex and strictly decreasing in and nonincreasing in .

Next, we introduce an operator which acts on Borel functions on by Clearly, can be viewed as a discretization of the differential operator . We will rely on the following results from [3] concerning the operator .

Proposition 4. For each Borel function on there exists a martingale with respect to the filtration such that and for every ,

Proposition 5. Let and . Assume that is a function on . Then where

We will need also the following result concerning the free boundary of the holder exercise region of our game put option which in the case of American options appears as Proposition 1 in [3] and it can be proved for game options in the same way.

Proposition 6. Let and let ; then .

We also observe that it follows from the Berry-Esseen estimate (see [12]) that, for some constant independent of , and ,

We will also rely on the following standard bounds on derivatives of solutions of 2nd-order parabolic equations with constant coefficients (see, e.g., [13, 14]).

Proposition 7. Let and let be a solution in of the following parabolic equation: Suppose that for all and that there exists such that for all . Then for every and there exists such that

3.2. Price Function and Variational Inequalities

Next, we will show that the price function of the game put option can be represented as a solution of a variational inequality (v.i.) problem which is a generalization of the Stefan problem (see [7], VIII). This will enable us to derive certain regularity properties of this price function which we will use later on. Details of some of the proofs concerning the solutions of the v.i. problem below which are similar to the proofs in the case of the Stefan problem will not be given here. For the corresponding results in the American put option case we refer the reader to [3, 9] and to references there in.

Let be such that and set Using the maximum principle, properties of price functions of American and game put options, and the fact that after time the price functions of the game and American option are the same we obtain that for every the time derivative is strictly negative and we can find satisfying such that, for some constant , Relying on Proposition 3(iii) we also observe that for all ,

Let be such that . Introduce the domain and for all in the closure of define the functions We obtain that and from the definition of it follows that for any , Since and are bounded we obtain that the integrability properties of the first- and second-order derivatives of and are the same in . Now set Then by (28) and (32), It follows from (29) and (30)-(31) that on the set , and on the set we obtain Hence we arrive at the following (see [7]).

Lemma 8. The function is the unique solution of the following variational inequality problem.
v.i. Problem 1. Find such that(i), (ii) a.s for every ,(iii) for ,(iv) for ,(v) for .

Proof. We will prove uniqueness and the fact that is a solution to v.i. Problem 1 follows from (30)–(36). Assume that and are two solutions of v.i. Problem 1. Since (property (i)) we can use the property (ii) of and replace by . Since both of them are solutions we obtain that Define the parabolic boundary as the boundary of without the interval and let . Note that is zero on the parabolic boundary and the sum of the two inequalities (37) is Integrating both sides of (38) on we obtain four terms on the left side. For the first term we have Integration by parts of the second term and the fact that on the parabolic boundary yield For the third term note that and that for every , and so The last term satisfies since . We conclude that the left side of (38) cannot be negative and so it must be zero. Since all terms in the left-hand side of (38) are nonnegative and their sum is equal to 0 we obtain that , and so almost everywhere (a.e.). Hence, a.e., and so there is only one continuous solution.

Denote parts of the boundary of by and set Thus, is a parabolic boundary of . For every we define following functions.(1)A smooth function on such that for and for , where satisfies and for .(2)A smooth function satisfying (3) with defined in (33).(4)A smooth function such that and for some , Set which is a Lipschitz continuous function and for every constant there is such that whenever and . Let be a function on satisfying and, moreover, relying on Chapter 3 in [14] we can choose so that(1) for some (in fact for each ) and we refer the reader to Chapter 3 in [14] for the definition of and for conditions yielding that a function defined only on the boundary can be extended to a function from ;(2) at the points and .

By the theory of semilinear parabolic equations (see [14]) there exists a function for some such that In particular are continuous on .

Let . By differentiating with respect to (47) and taking into account (33), (47), and the properties of we obtain that We see that in the function is a solution to a parabolic equation and since we can use the maximum principle Therefore in order to bound the function we only need to bound its values on the parabolic boundary. First, we estimate the left hand side of (49). For we have that , and so . In view of (28), (31), and the definition of above there exists such that, for every , On the interval we have and since we see that . Since on this interval we obtain Hence, on . We obtain next that It follows that .

Next, we estimate the right-hand side of (49). On we have that where is a constant independent of , and so We conclude that there are some constants and such that for every , Since and for we deduce that and because is uniformly bounded it follows that is also uniformly bounded. By the properties of we see that Let be an upper subrectangle of , where is the same as in the definition of the function in (4). From the definition we have in and since is nonnegative, we obtain that , and so .

This means that on the function satisfies the parabolic equation For and we also have that

Next, let be a function on such that and all of its first- and second-order derivatives are bounded there. Such a function exists since we can choose a smooth function on the remaining part of the parabolic boundary of which extends as a smooth function to the whole and then use Theorem 12 from Chapter 3 in [7]. For each we define in the domain , where . Then for every . Fix ; then by Proposition 4.5 from Section 4.1 of [7] we obtain that for every , where a constant is independent of . Since we assume that for it follows that (for every ). Let ; then by Theorem in Chapter 3 of [14] we obtain that for every (and in fact every ) and there is a constant independent of such that For the definition of norms in (60) see Chapter 3 in [14].

In particular, we get Considering again the whole region we have Hence, Since all terms in the right-hand side are uniformly bounded there is a constant independent of such that for every . Now we see that in the equation all terms in the right hand side are uniformly bounded and therefore the term in the left is uniformly bounded, as well.

We summarize this in the following lemma.

Lemma 9. There are constants such that, for every ,

We now obtain the following (see [7]).

Proposition 10. For any and , as weakly in . Furthermore, uniformly on and also uniformly in for each . The function is the unique solution of v.i. Problem 1.

Next, we analyze properties of second order derivatives starting with the following result.

Lemma 11. There is a constant such that, for any ,

Proof. Set , and . Multiply (48) by to obtain Integrating this equation over and recalling that and are nonnegative we obtain that, for any , By (47) and (55) we estimate the third term in (68): For the second term in (68) we see that Since and the function is uniformly bounded in we see in view of (61) that is uniformly bounded near the boundary and , while for some constant independent of . Thus, we conclude from (68) that for some independent of . Integrating the last equation over we obtain Since the function is uniformly bounded it follows that there is independent of such that

We will now deal with the properties of the function .

Lemma 12. There is a constant such that for any and every ,

Proof. Set , and . Multiplying (48) by the function we have and an integration with respect to over yields Fix some . Since we see that From (61) it follows that for some constant independent of , and so we obtain
Now we deal with the last term in (76). Since and we obtain that We plug this inequality into (78) and obtain Integrate the last inequality with respect to over the interval to obtain Next, integrating in over the interval for some and taking into account that by the property (2) of we obtain that Now, by (55), (56), and Lemma 11 together with the Cauchy–Schwarz inequality we estimate the right hand side of (82) by a constant independent of . Hence, and Lemma 12 follows.

As a corollary of previous results we obtain the following.

Proposition 13. Let and . Define . Then where by definition is the set of all the functions in with an weak second order derivatives. Also there exists such that for every ,

Proof. From Lemmas 12, 11, and 9 we obtain that are uniformly bounded in and so they have a weak limit . Since uniformly we must have that , and so . Since is the solution of (36) we can apply Proposition 7 and using the fact that the constant in (26) does not depend on we can obtain in a similar way that for a fixed there is a constant such that, for every , From (32) we can deduce the same result for the function .

Corollary 14. For each the function is Hölder continuous with the exponent .

Proof. For every Proposition 13 gives us that . Hence, the result is a consequence of the Sobolev inequality.

Corollary 15. For every the functions and as functions of are continuous in the closed interval .

Proof. For the function the result follows from (32) and the previous corollary. Since is a solution of (29) in the interval and since the functions and are continuous in the interval we obtain the result for , as well.

Corollary 16. Let and . Define ,  and . Then and there exists such that, for every ,

Proof. The assertion (87) follows from Proposition 13 and the definition of . For (88) note that then use (85) and the fact that for the functions and are bounded.

4. Price Function Near the Writer’s Exercise Boundary

4.1. Regularity Properties of Price Function

Let be the price function of the put game option (see Section 2). We begin this section by showing that near the writer’s exercise region the function is continuous. Let which is a non homogeneous in time Markov process in , where and . Let which is the infinitesimal generator of when considered on the space of all functions. This is a parabolic operator with bounded smooth coefficients in the domain where . Let and be the probability and the corresponding expectation for the Markov process starting at the point . We will first show that for any , where and for any closed set we set to be the arrival time at the set for a Markov process under consideration which is here. Indeed, choosing an appropriate nonnegative function on the boundary and relying on Chapter 3 in [14] we can choose which solves the equation in and equals 1 on the boundary part for while decaying smoothly to when grows to . Then and so

Next let and . Recall that the price of a put game option with an expiration time and a constant penalty can be written in the form where and for any bounded Borel functions and we write