A Variational Inequality from Pricing Convertible Bond
© Huiwen Yan and Fahuai Yi. 2011
Received: 30 December 2010
Accepted: 11 February 2011
Published: 10 March 2011
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.
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  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  or [4–7]).
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 ). 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 .
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 , 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 .
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  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  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.
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 .
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 .
where CV* is the conversion domain in the model without call and CV is that in this paper.
We apply the Schauder fixed point theorem  to prove the existence of nonlinear problem (3.3).
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.
In the following, we prove (3.6).
Then the comparison principle implies (3.6).
Thus (3.16) follows.
Applying the A-B-P maximum principle (see ), we have that in , which contradicts the definition of .
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
then property (i) is obvious.
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, .
Hence, we have property (ii).
Then we show property (iii).
Fix and , and denote . According to Theorem 4.4, the free boundary while lies in the domain (see Figure 5).
5. Numerical Results
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).
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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