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A Variational Inequality from Pricing Convertible Bond
Advances in Difference Equations volume 2011, Article number: 309678 (2011)
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
:

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]),

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

we have that

where

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

and is strictly decreasing in
.
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

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

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

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,

then the value of the Dynkin game exists and

If , then we deduce that, for any
,

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

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:

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

If ,  
, then
, and, for any
, we have

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

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

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

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

It is clear that

Since , then

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):

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

Consider the following penalty problem of (3.1):

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

Lemma 3.1.
For any fixed  , problem (3.3) has a unique solution  
  for any
  and


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:

Furthermore, we can compute

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

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

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

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

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

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



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

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

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

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),

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

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

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

Denoting , we have that

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,



Proof.
Rewrite Problem (3.1) as follows:

where implies that
and

where denotes the indicator function of the set
.
Hence, for any fixed , if
, combining (3.14), we have the following
and
uniform estimates [18]:

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
,
,

Moreover, (3.30) and imbedding theorem imply that

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

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

(see Figure 1). It is clear that

Theorem 4.1.
Denote  . If  
, then  
, which means that

Proof.
Define

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

then property (i) is obvious.
Moreover, if , then we deduce

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
,

In the other case of ,

So, we testify properties (i)–(iv). In the following, we utilize the properties to prove .
Otherwise, is nonempty; then we have that

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

which contradicts the definition of . So, we achieve that
.
Combining for any
, it is clear that

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

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

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

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

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)

Moreover,

Differentiating (4.18) with respect to , then

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

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

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

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

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

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

Proof.
Define

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

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

Hence, we have property (ii).
It is not difficult to check that, for any ,

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

In fact, satisfy the equations

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

here

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

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

Moreover,

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

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

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

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

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

Applying the comparison principle, we obtain

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,

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
.
5. Numerical Results
Applying the binomial tree method to problem (1.4), we achieve the following numerical results—Figure 6:
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).
References
Sîrbu M, Pikovsky I, Shreve SE: Perpetual convertible bonds. SIAM Journal on Control and Optimization 2004,43(1):58-85. 10.1137/S0363012902412458
Sîrbu M, Shreve SE: A two-person game for pricing convertible bonds. SIAM Journal on Control and Optimization 2006,45(4):1508-1539. 10.1137/050630222
Wilmott P: Derivatives, the Theory and Practice of Financial Engineering. John Wiley & Sons, New York, NY, USA; 1998.
Kifer Y: Game options. Finance and Stochastics 2000,4(4):443-463. 10.1007/PL00013527
Kifer Y: Error estimates for binomial approximations of game options. The Annals of Applied Probability 2006,16(2):984-1033. 10.1214/105051606000000088
Kühn C, Kyprianou AE: Callable puts as composite exotic options. Mathematical Finance 2007,17(4):487-502. 10.1111/j.1467-9965.2007.00313.x
Kyprianou AE: Some calculations for Israeli options. Finance and Stochastics 2004,8(1):73-86. 10.1007/s00780-003-0104-5
Black F, Scholes M: The pricing of options and coperate liabilities. Journal of Political Economy 1973, 81: 637-659. 10.1086/260062
Yang Z, Yi F: A free boundary problem arising from pricing convertible bond. Applicable Analysis 2010,89(3):307-323. 10.1080/00036810903517563
Friedman A: Parabolic variational inequalities in one space dimension and smoothness of the free boundary. Journal of Functional Analysis 1975, 18: 151-176. 10.1016/0022-1236(75)90022-1
Blanchet A: On the regularity of the free boundary in the parabolic obstacle problem. Application to American options. Nonlinear Analysis: Theory, Methods & Applications 2006,65(7):1362-1378. 10.1016/j.na.2005.10.009
Blanchet A, Dolbeault J, Monneau R: On the continuity of the time derivative of the solution to the parabolic obstacle problem with variable coefficients. Journal de Mathématiques Pures et Appliquées. Neuvième Série 2006,85(3):371-414.
Caffarelli L, Petrosyan A, Shahgholian H: Regularity of a free boundary in parabolic potential theory. Journal of the American Mathematical Society 2004,17(4):827-869. 10.1090/S0894-0347-04-00466-7
Petrosyan A, Shahgholian H: Parabolic obstacle problems applied to finance. In Recent Developments in Nonlinear Partial Differential Equations, Contemporary Mathematics. Volume 439. American Mathematical Society, Providence, RI, USA; 2007:117-133.
Friedman A: Stochastic games and variational inequalities. Archive for Rational Mechanics and Analysis 1973, 51: 321-346.
Friedman A: Variational Principles and Free-Boundary Problems, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1982:ix+710.
Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften. Volume 224. 2nd edition. Springer, Berlin, Germany; 1983:xiii+513.
Ladyženskaja OA, Solonnikov VA, Uralceva NN: Linear and Quasi-linear Equations of Parabolic Type. American Mathematical Society, Providence, RI, USA; 1968:736.
Yi F, Yang Z, Wang X: A variational inequality arising from European installment call options pricing. SIAM Journal on Mathematical Analysis 2008,40(1):306-326. 10.1137/060670353
Tso K: On Aleksandrov, Bakel'man type maximum principle for second order parabolic equations. Communications in Partial Differential Equations 1985,10(5):543-553. 10.1080/03605308508820388
Acknowledgments
The project is supported by NNSF of China (nos. 10971073, 10901060, and 11071085) and NNSF of Guang Dong province (no. 9451063101002091).
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Yan, H., Yi, F. A Variational Inequality from Pricing Convertible Bond. Adv Differ Equ 2011, 309678 (2011). https://doi.org/10.1155/2011/309678
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DOI: https://doi.org/10.1155/2011/309678
Keywords
- Variational Inequality
- Free Boundary
- Stock Price
- Comparison Principle
- Bond Price