A Variational Inequality from Pricing Convertible Bond

Abstract

The model of pricing American-style convertible bond is formulated as a zero-sum Dynkin game, which can be transformed into a parabolic variational inequality (PVI). The fundamental variable in this model is the stock price of the firm which issued the bond, and the differential operator in PVI is linear. The optimal call and conversion strategies correspond to the free boundaries of PVI. Some properties of the free boundaries are studied in this paper. We show that the bondholder should convert the bond if and only if the price of the stock is equal to a fixed value, and the firm should call the bond back if and only if the price is equal to a strictly decreasing function of time. Moreover, we prove that the free boundaries are smooth and bounded. Eventually we give some numerical results.

1. Introduction

Firms raise capital by issuing debt (bonds) and equity (shares of stock). The convertible bond is intermediate between these two instruments, which entitles its owner to receive coupons plus the return of principle at maturity. However, prior to maturity, the holder may convert the bond into the stock of the firm, surrendering it for a preset number of shares of stock. On the other hand, prior to maturity, the firm may call the bond forcing the bondholder to either surrender it to the firm for a previously agreed price or convert it into stock as before.

After issuing a convertible bond, the bondholder will find a proper time to exercise the conversion option in order to maximize the value of the bond, and the firm will choose its optimal time to exercise its call option to maximize the value of shareholder's equity. This situation was called "two-person" game (see [1, 2]). Because the firm must pay coupons to the bondholder, it may call the bond if it can subsequently reissue a bond with a lower coupon rate. This happens as the firm's fortunes improve, then the risk of default has diminished and investors will accept a lower coupon rate on the firm's bonds.

In [2] the authors assume that a firm's value is comprised of one equity and one convertible bond, the value of the issuing firm has constant volatility, the bond continuously pays coupons at a fixed rate, and the firm continuously pays dividends at a rate that is a fixed fraction of equity. Default occurs if the coupon payments cause the firm's value to fall to zero, in which case the bond has zero value. In their model, both the bond price and the stock price are functions of the underlying of the firm value. Because the stock price is the difference between firm value and bond price and dividends are paid proportionally to the stock price, a nonlinear differential equation was established for describing the bond price as a function of the firm value and time.

As we know, it is difficult to obtain the value of the firm. However, it is easier to get its stock price. So we choose the bond price as a function of the stock price of the firm and time (see Chapter 36 in [3] or [4–7]).

In Section 2, we formulate the model and deduce that in the domain and is governed by the following variational inequality in the domain :

(1.1)

where , ,  , and are positive constants. is the coupon rate, is the conversion ratio for converting the bond into the stock of the firm, is the call price of the firm, is the face value of the bond with , and is just B-S operator (see [8]),

(1.2)

where , , and are positive constants and represent the risk-free interest rate, the volatility, and the dividend rate of the firm stock, respectively. In this paper, we suppose that and . From a financial point of view, the assumption provides a possibility of calling the bond back from the firm (see Section 2 or [2]). Furthermore, we suppose that . Otherwise, the firm should call the bond back before maturity and the value makes no sense (see Section 2). It is clear that is the unique solution if . So we only consider the problem in the case of .

Since (1.1) is a degenerate backward problem, we transform it into a familiar forward nondegenerate parabolic variational inequality problem; so letting

(1.3)

we have that

(1.4)

where

(1.5)

There are many papers on the convertible bond, such as [1, 2, 9]. But as we know, there are seldom results on the properties of the free boundaries—the optimal call and conversion strategies in the existing literature. The main aim of this paper is to analyze some properties of the free boundaries.

The pricing model of the convertible bond without call is considered in [9], where there exist two domains: the continuation domain CT and the conversion domain CV. The free boundary between CT and CV means the optimal conversion strategy, which is dependent on the time and more than .

But in this model, their exist three domains: the continuation domain CT, the callable domain CL, and the conversion domain . The boundary between CV and is , which means the call strategy. The free boundary is the curve between CT and CL (see Figure 1), which means the optimal call strategy. And there exist such that

(1.6)

and is strictly decreasing in .

Figure 1

The free boundary .

It means that the bondholder should convert the bond if and only if the stock price of the firm is no less than , whereas, in the model without call, the bondholder may not convert the bond even if . More precisely, the optimal conversion strategy without call is more than that in this paper (see [9] or Section 2). When the time to the expiry date is more than , the firm should call the bond back if . Neither the bondholder nor the firm should exercise their option if the time to the expiry date is less than and . Moreover, when the time to the maturity lies in , the bondholder should call the bond back if .

In Section 2, we formulate and simplify the model. In Section 3, we will prove the existence and uniqueness of the strong solution of the parabolic variational inequality (1.4) and establish some estimations, which are important to analyze the property of the free boundary.

In Section 4, we show some behaviors of the free boundary , such as its starting point and monotonicity. Particularly, we obtain the regularity of the free boundary . As we know, the proof of the smoothness is trivial by the method in [10] if the difference between and the upper obstacle is decreasing with respect to . But the proof is difficult if the condition is false (see [11–14]). In this problem, , which does not match the condition. Moreover, , and the starting point of the free boundary is not on the initial boundary, but the side boundary in this problem. Those make the proof of more complicated. The key idea is to construct cone locally containing the local free boundary and prove ; then the proof of is trivial. Moreover, we show that there is a lower bound of and converges to as converges to in Theorem 4.4.

In the last section, we provide numerical result applying the binomial method.

2. Formulation of the Model

In this section, we derive the mathematical model of pricing the convertible bond.

The firm issues the convertible bond, and the bondholder buys the bond. The firm has an obligation to continuously serve the coupon payment to the bondholder at the rate of . In the life time of the bond, the bondholder has the right to convert it into the firm's stock with the conversion factor and obtains from the firm after converting, and the firm can call it back at a preset price of . The bondholder's right is superior to the firm's, which means that the bondholder has the right to convert the bond, but the firm has no right to call it if both sides hope to exercise their rights at the same time. If neither the bondholder nor the firm exercises their right before maturity, the bondholder must sell the bond to the firm at a preset value or convert it into the firm's stock at expiry date. So, the bondholder receives from the firm at maturity. It is reasonable that both of them wish to maximize the values of their respective holdings.

Suppose that under the risk neutral probability space ; the stock price of the firm follows

(2.1)

where ,  , and are positive constants, representing risk free interest rate, the dividend rate, and volatility of the stock, respectively. is a standard Brown motion on the probability space . Usually, the dividend rate is smaller than the risk free interest rate . So, we suppose that .

Denote by the natural filtration generated by and augmented by all the -null sets in . Let be the set of all -stopping times taking values in .

The model can be expressed as a zero-sum Dynkin game. The payoff of the bondholder is

(2.2)

where . The stopping time is the firm's strategy, and is the bondholder's strategy.

The bondholder chooses his strategy to maximize ; meanwhile, the firm chooses its strategy to minimize .

Denote the upper value and the lower value as

(2.3)

If , then it is called the value of the Dynkin game and denoted as .

As we know, if the Dynkin game has a saddlepoint , that is,

(2.4)

then the value of the Dynkin game exists and

(2.5)

If , then we deduce that, for any ,

(2.6)

So, in this case, is a saddlepoint, and the value of the Dynkin game is

(2.7)

In the case of , applying the standard method in [15], we see that the strong solution of the following variational inequality is the value of the Dynkin game:

(2.8)

If , then the firm is bound to call the bond back before the maturity because the firm pays after calling, but more than without calling. In this case, the value makes no sense. So, we suppose that .

If , then the firm is bound to abandon its call right. From a financial point of view, the firm would pay to the bondholder at time after calling the bond, whereas, if the firm does not call in the time interval , then he would pay the coupon payment and at most of the face value of the convertible bond at time . So, the discounted value of the bond without call is at most . Hence, the firm should not call the bond back at time .

From a stochastic point of view, we can denote a stopping time

(2.9)

If ,  , then , and, for any , we have

(2.10)

Moreover, . So, for any such that , it is clear that in the domain

(2.11)

which means that is not the optimal call strategy, and the firm should not call in the domain .

From a variational inequality point of view, since

(2.12)

provided that , which contradicts with the third inequality in (2.8), so, if , then in the domain .

To remain the call strategy, we suppose that . We will consider the other case in another paper because the two problems are fully different.

Since we suppose that and , then

(2.13)

Hence, is empty in problem (2.8). So, problem (2.8) is reduced into problem (1.1).

The model of pricing the bond without call is an optimal stopping problem

(2.14)

It is clear that

(2.15)

Since , then

(2.16)

where CV* is the conversion domain in the model without call and CV is that in this paper.

3. The Existence and Uniqueness of Solution of Problem (1.4)

Since problem (1.4) lies in the unbounded domain , we need the following problem in the bounded domain to approximate to problem (1.4):

(3.1)

where and .

Following the idea in [10, 16], we construct a penalty function (see Figure 2), which satisfies

(3.2)

Figure 2

The function .

Consider the following penalty problem of (3.1):

(3.3)

where is a smoothing function because the initial value is not smooth. It satisfies (see Figure 3)

(3.4)

Figure 3

The function .

Lemma 3.1.

For any fixed  , problem (3.3) has a unique solution    for any   and

(3.5)

(3.6)

Proof.

We apply the Schauder fixed point theorem [17] to prove the existence of nonlinear problem (3.3).

Denote and . Then is a closed convex set in . Defining a mapping by is the solution of the following linear problem:

(3.7)

Furthermore, we can compute

(3.8)

where is the parabolic boundary of . Thus is a supersolution of the problem (3.7), and . Hence . On the other hand,

(3.9)

which is bounded for fixed . So, it is not difficult to prove that is compact in and is continuous. Owing to the Schauder fixed point theorem, we know that problem (3.3) has a solution . The proof of the uniqueness follows by the comparison principle. Here, we omit the details.

Now, we prove (3.5). Since

(3.10)

Therefore, is a supersolution of problem (3.3), and in . Moreover,

(3.11)

Hence, is a subsolution of problem (3.3). On the other hand,

(3.12)

Thus, is a subsolution of problem (3.3) as well, and we deduce .

In the following, we prove (3.6).

Indeed, and imply that . Furthermore, and that imply . Differentiating (3.3) with respect to and denoting , we obtain

(3.13)

Then the comparison principle implies (3.6).

Theorem 3.2.

For any fixed  , , problem (3.1) admits a unique solution     for any  , where , . Moreover, if is large enough, one has that

(3.14)

(3.15)

(3.16)

Proof.

From (3.5) and the properties of , we have that

(3.17)

By and () estimates of the parabolic problem [18], we conclude that

(3.18)

where is independent of . It implies that there exists a and a subsequence of (still denoted by ), such that as ,

(3.19)

Employing the method in [16] or [19], it is not difficult to derive that is the solution of problem (3.1). And (3.14), (3.15) are the consequence of (3.5), (3.6) as .

In the following, we will prove (3.16). For any small , satisfies, by (3.1),

(3.20)

Applying the comparison principle with respect to the initial value of the variational inequality (see [16]), we obtain

(3.21)

Thus (3.16) follows.

At last, we prove the uniqueness of the solution. Suppose that and are two solutions to problem (3.1), and denote

(3.22)

Assume that is not empty, and then, in the domain ,

(3.23)

Denoting , we have that

(3.24)

Applying the A-B-P maximum principle (see [20]), we have that in , which contradicts the definition of .

Theorem 3.3.

Problem (1.4) has a unique solution     for any  , , and . And . Moreover,

(3.25)

(3.26)

(3.27)

Proof.

Rewrite Problem (3.1) as follows:

(3.28)

where implies that and

(3.29)

where denotes the indicator function of the set .

Hence, for any fixed , if , combining (3.14), we have the following and uniform estimates [18]:

(3.30)

here depends on and depends on , but they are independent of . Then, we have that there is a and a subsequence of (still denoted by ), such that for any , ,

(3.31)

Moreover, (3.30) and imbedding theorem imply that

(3.32)

It is not difficult to deduce that is the solution of problem (1.4). Furthermore, (3.32) implies that . And (3.25)–(3.27) are the consequence of (3.14)–(3.16). The proof of the uniqueness is similar to the proof in Theorem 3.2.

4. Behaviors of the Free Boundary

Denote

(4.1)

Thanks to (3.26), we can define the free boundary of problem (1.4), at which it is optimal for the firm to call the bond, where

(4.2)

(see Figure 1). It is clear that

(4.3)

Theorem 4.1.

Denote  . If  , then  , which means that

(4.4)

Proof.

Define

(4.5)

We claim that possess the following four properties:

• (i) ,

• (ii) , for all ,

• (iii) , for all ,

• (iv) , a.e. in .

In fact, from the definition of , we have that

(4.6)

then property (i) is obvious.

Moreover, if , then we deduce

(4.7)

Combining , we have property (ii). It is easy to check property (iii) from the definition of . Next, we manifest property (iv) according to the following two cases. In the case of ,

(4.8)

In the other case of ,

(4.9)

So, we testify properties (i)–(iv). In the following, we utilize the properties to prove .

Otherwise, is nonempty; then we have that

(4.10)

Moreover, on the parabolic boundary of . According to the A-B-P maximum principle (see [20]), we have that

(4.11)

which contradicts the definition of . So, we achieve that .

Combining for any , it is clear that

(4.12)

which means that , , and for any

Theorem 4.2.

The free boundary   is decreasing in the interval . Moreover, . And .

Proof.

(3.26) and (3.27) imply that

(4.13)

Hence, for any unit vector satisfying , the directional derivative of function along admits

(4.14)

that is, is increasing along the director . Combining the condition in , we know that is monotonically decreasing. Hence, exists, and we can define

(4.15)

Since , so . On the other hand, if , then

(4.16)

It is impossible because is continuous on .

In the following, we prove that is continuous in . If it is false, then there exists ,   such that (see Figure 4)

(4.17)

Moreover,

(4.18)

Differentiating (4.18) with respect to , then

(4.19)

On the other hand, for any in this case, and we know that by (3.26). Applying the strong maximum principle to (4.19), we obtain

(4.20)

So, we can define in . Considering and , we see that in , which contradicts that for any . Therefore .

Figure 4

Discontinuous free boundary .

Theorem 4.3.

There exists some such that for any and    is strictly decreasing on .

Proof.

Define . In the first, we prove that . Otherwise, and for .

Recalling the initial value, we see that

(4.21)

Meanwhile, in the domain implies that for any (see Figure 4); then is not continuous at the point , which contradicts .

In the second, we prove that . In fact, according to Lemma 3.1, for any , hence, . If , then the free boundary includes a horizontal line ,  . Repeating the method in the proof of Theorem 4.2, then we can obtain a contradiction. So, .

At last, we prove that is strictly decreasing on . Otherwise, has a vertical part. Suppose that the vertical line is , then for any . Since is continuous across the free boundary, then for any . In this case, we infer that

(4.22)

On the other hand, in the domain , and satisfy, respectively,

(4.23)

Then the strong maximum principle implies that , which contradicts the second equality in (4.22).

Theorem 4.4.

for any with (see Figure 1), where

(4.24)

where is the positive characteristic root of , that is, the positive root of the algebraic equation

(4.25)

Proof.

Define

(4.26)

We claim that and possess the following three properties:

(i)  for   and     for ,

(ii)  in ,

(iii)  in .

In fact, if we notice that , then we have that

(4.27)

It is obvious that . So, we obtain property (i).

Moreover, we compute

(4.28)

Hence, we have property (ii).

It is not difficult to check that, for any ,

(4.29)

Then we show property (iii).

Repeating the method in the proof of Theorem 3.2, we can derive that in from properties (i)-(ii). And property (iii) implies that in the domain , which means that for any .

Next, we prove that . Otherwise, ; then the free boundary includes a horizontal line , . Repeating the method in the proof of Theorem 4.2, then we can obtain a contradiction. So, .

Theorem 4.5.

The free boundary  .

Proof.

Fix and , and denote . According to Theorem 4.4, the free boundary while lies in the domain (see Figure 5).

In the first, we prove that there exists an such that

(4.30)

In fact, satisfy the equations

(4.31)

then the interior estimate of the parabolic equation implies that there exists a positive constant such that

(4.32)

here

(4.33)

On the other hand, we see that in from (3.27), and . Applying the strong maximum principle to , we deduce that

(4.34)

It means that is strictly decreasing on . It follows that, by ,

(4.35)

Moreover,

(4.36)

Employing the strong maximum principle, we see that there is a , such that

(4.37)

Provided that is small enough. Combining (4.32), there exists a positive such that

(4.38)

Next, we concentrate on problem (1.4) in the domain . It is clear that satisfies

(4.39)

And we can use the following problem to approximate the above problem:

(4.40)

Recalling (4.38), we see that, if is small enough, on the parabolic boundary of . Moreover, satisfies

(4.41)

Applying the comparison principle, we obtain

(4.42)

As the method in the proof of Theorem 3.3, we can show that weakly converges to in and (4.30) is obvious.

On the other hand, we see that in for any positive number from (3.26) and (3.27). So,

(4.43)

which means that there exists a uniform cone such that the free boundary should lies in the cone. As the method in [9], it is easy to derive that . Moreover can be deduced by the bootstrap method. Since is arbitrary and the free boundary is a vertical line while , then .

Figure 5

The free boundary .

5. Numerical Results

Applying the binomial tree method to problem (1.4), we achieve the following numerical results—Figure 6:

Figure 6

The free boundary.

Plot of the optimal exercise boundary is a function of . The parameter values used in the calculations are , , , , , , , and . In this case, the free boundary is increasing with . The numerical result is coincided with that of our proof (see Figure 6).