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Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 309678, 21 pages doi:10.1155/2011/309678 Research Article A Variational Inequality from Pricing Convertible Bond Huiwen Yan and Fahuai Yi School of Mathematics, South China Normal University, Guangzhou 510631, China Correspondence should be addressed to Fahuai Yi, fhyi@scnu.edu.cn Received 30 December 2010; Accepted 11 February 2011 Academic Editor: Jin Liang Copyright q 2011 H Yan and F Yi 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 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 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, 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 2 Advances in Difference Equations 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 V S, t as a function of the stock price S of the firm and time t see Chapter 36 in or 4–7 In Section 2, we formulate the model and deduce that V S, t γ S in the domain {S ≥ K/γ } and V S, t is governed by the following variational inequality in the domain {0 ≤ S ≤ K/γ }: −∂t V − L0 V if V < K, S, t ∈ DT c, −∂t V − L0 V ≤ c, V V S, T K ,t γ Δ 0, K γ × 0, T , K, S, t ∈ DT , if V 1.1 ≤ t ≤ T, K, max L, γ S , 0≤S≤ K , γ where c, γ , K, and L are positive constants c is the coupon rate, γ is the conversion ratio for converting the bond into the stock of the firm, K is the call price of the firm, L is the face value of the bond with < L ≤ K, and L0 is just B-S operator see , L0 V σ2 S ∂SS V r − q S∂S V − rV, 1.2 where r, σ, and q 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 c > rK and r ≥ q From a financial point of view, the assumption provides a possibility of calling the bond back from the firm see Section or Furthermore, we suppose that L ≤ K Otherwise, the firm should call the bond back before maturity and the value L makes no sense see Section It is clear that V K is the unique solution if L K So we only consider the problem in the case of L < K Since 1.1 is a degenerate backward problem, we transform it into a familiar forward nondegenerate parabolic variational inequality problem; so letting u x, t V S, T − t , x ln S − ln K ln γ, 1.3 Advances in Difference Equations we have that ∂t u − Lu if u < K, x, t ∈ ΩT c, ∂t u − Lu ≤ c, u 0, t u x, if u K, Δ −∞, × 0, T , K, x, t ∈ ΩT , 1.4 ≤ t ≤ T, max{L, Kex }, x ≤ 0, where Lu σ2 ∂xx u r−q− σ2 ∂x u − ru 1.5 There are many papers on the convertible bond, such as 1, 2, 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 S t between CT and CV means the optimal conversion strategy, which is dependent on the time t and more than K/γ But in this model, their exist three domains: the continuation domain CT, the callable domain CL, and the conversion domain CV {x ≥ 0} The boundary between CV and CT ∪ CL is x 0, which means the call strategy The free boundary h t is the curve between CT and CL see Figure , which means the optimal call strategy And there exist t0 , T0 such that < t0 < T0 Δ c − rL ln , r c − rK h t ∈ C t0 , T0 ∩ C∞ t0 , T0 , lim h t − t → T0 −∞, 1.6 and h t is strictly decreasing in t0 , T0 It means that the bondholder should convert the bond if and only if the stock price S of the firm is no less than K/γ , whereas, in the model without call, the bondholder may not convert the bond even if S > K/γ More precisely, the optimal conversion strategy S t without call is more than that K/γ in this paper see or Section When the time to the expiry date is more than T0 , the firm should call the bond back if S < K/γ Neither the bondholder nor the firm should exercise their option if the time to the expiry date is less than t0 and S < Keh t Moreover, when the time to the maturity lies in t0 , T0 , the bondholder should call the bond back if Keh t ≤ S < K/γ 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 h t , such as its starting point and monotonicity Particularly, we obtain the regularity of the free boundary h t ∈ C0,1 t0 , T0 ∩ C∞ t0 , T0 As we know, the proof of the smoothness is trivial by the method Advances in Difference Equations t u=K CL T0 h(t) CT Kex < u < K CV u = Kex t0 x Figure 1: The free boundary h t in 10 if the difference between u and the upper obstacle K is decreasing with respect to t But the proof is difficult if the condition is false see 11–14 In this problem, ∂t u − K ≥ 0, which does not match the condition Moreover, ∂xx u ln L − ln K, ∞, and the starting point 0, t0 of the free boundary h t is not on the initial boundary, but the side boundary in this problem Those make the proof of h t ∈ C∞ t0 , T0 more complicated The key idea is to construct cone locally containing the local free boundary and prove h t ∈ C0,1 t0 , T0 ; then the proof of C∞ t0 , T0 is trivial Moreover, we show that there is a lower bound h∗ t of h t − and h t converges to −∞ as t converges to T0 in Theorem 4.4 In the last section, we provide numerical result applying the binomial method 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 c In the life time of the bond, the bondholder has the right to convert it into the firm’s stock with the conversion factor γ and obtains γ S from the firm after converting, and the firm can call it back at a preset price of K 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 L or convert it into the firm’s stock at expiry date So, the bondholder receives max{L, γ S} 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 Ω, F, È ; the stock price of the firm Ss follows St,S s s S t r − q St,S du u s σSt,S dWu , u S ∈ 0, ∞ , s ∈ t, T , t ∈ 0, T , 2.1 t where r, q, and σ are positive constants, representing risk free interest rate, the dividend rate, and volatility of the stock, respectively Wt is a standard Brown motion on the probability space Ω, F, È Usually, the dividend rate q is smaller than the risk free interest rate r So, we suppose that q ≤ r Advances in Difference Equations Denote by Ft the natural filtration generated by Wt and augmented by all the È-null sets in F Let Ut,T be the set of all Ft -stopping times taking values in t, T The model can be expressed as a zero-sum Dynkin game The payoff of the bondholder is τ∧θ cert−ru du R S, t; τ, θ ert−rθ γ St,S I{θ≤τ, θ rK and r ≥ q, then −∂t γ S − L0 γ S qγ S ≤ rK < c 2.13 Hence, {V γ S} 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 U S, t Δ ess sup Q S, t; θ | Ft , θ∈Ut,T 2.14 θ ce Q S, t; θ rt−ru du e rt−rθ t γ St,S I{θ ln K − ln L L, if un < K, x, t ∈ Ωn , T if un un 0, t max{L, Kex }, K, x, t ∈ Ωn , T K, ≤ t ≤ T, −n ≤ x ≤ 0, 3.1 Advances in Difference Equations Following the idea in 10, 16 , we construct a penalty function βε s which satisfies βε s ∈ C∞ −∞, ∞ , ε > and small enough, βε s 0, if s ≤ −ε, βε s ≥ 0, lim βε s ε→0 see Figure , βε C0 Δ βε s ≥ 0, ⎧ ⎨0, s < 0, ⎩ ∞, c − rK > 0, βε s ≥ 0, s > 3.2 Consider the following penalty problem of 3.1 : ∂t uε,n − Luε,n uε,n −n, t L, βε uε,n − K uε,n 0, t πε Kex − L uε,n x, c, in Ωn , T ≤ t ≤ T, K, 3.3 −n ≤ x ≤ 0, L, where πε s is a smoothing function because the initial value max{L, Kex } is not smooth It satisfies see Figure πε s πε s ∈ C∞ IR , πε s ≥ s, ⎧ ⎨s, s ≥ ε, ⎩0, s ≤ −ε, ≤ πε s ≤ 1, 3.4 πε s ≥ 0, lim πε s ε→0 s 2,1 Lemma 3.1 For any fixed ε > 0, problem 3.3 has a unique solution uε,n ∈ Wp Ωn ∩ C Ωn for T T any < p < ∞ and max{L, Kex } ≤ uε,n ≤ K in Ωn , T ∂x uε,n ≥ in Ωn T 3.5 3.6 Proof We apply the Schauder fixed point theorem 17 to prove the existence of nonlinear problem 3.3 Denote B C Ωn and D {w ∈ B : w ≤ c/r} Then D is a closed convex set in B T Defining a mapping F by F w uε,n is the solution of the following linear problem: ∂t uε,n − Luε,n uε,n −n, t uε,n x, L, βε w − K uε,n 0, t πε Kex − L c K, L, in Ωn , T ≤ t ≤ T, −n ≤ x ≤ 3.7 Advances in Difference Equations C0 s ε Figure 2: The function βε s s ε −ε Figure 3: The function πε Furthermore, we can compute ∂t c r −L c r βε w − K c > K ≥ uε,n r r on c r βε w − K ≥ c, 3.8 ∂p Ωn , T where ∂p Ωn is the parabolic boundary of Ωn Thus c/r is a supersolution of the problem 3.7 , T T and uε,n ≤ c/r Hence F D ⊂ D On the other hand, ≤ βε w − K ≤ βε c −K , r 3.9 which is bounded for fixed ε > So, it is not difficult to prove that F D is compact in B and F is continuous Owing to the Schauder fixed point theorem, we know that problem 3.3 has 2,1 a solution uε,n ∈ Wp Ωn The proof of the uniqueness follows by the comparison principle T Here, we omit the details Now, we prove 3.5 Since ∂t K − LK βε K − K K ≥ uε,n rK on ∂p Ωn T βε c, 3.10 10 Advances in Difference Equations Therefore, K is a supersolution of problem 3.3 , and uε,n ≤ K in Ωn Moreover, T ∂t Kex − L Kex βε Kex − K Kex |x −n βε Kex − K ≤ qKex qKex Ke−n ≤ L uε,n −n, t , Kex |x Kex ≤ max{Kex , L} ≤ πε Kex − L K L βε qKex c − rK ≤ c, uε,n 0, t , uε,n x, 3.11 Hence, Kex is a subsolution of problem 3.3 On the other hand, ∂t L − LL βε L − K rL uε,n −n, t , L βε L − K ≤ rL L ln K − ln L, problem 3.1 admits a unique solution un ∈ 2,1 for any < p < ∞, < δ < n, where P0 − ln K ln L, , C Ωn ∩ Wp Ωn \ Bδ P0 T T Bδ P0 { x, t : x ln K − ln L t2 ≤ δ2 } Moreover, if n is large enough, one has that max{L, Kex } ≤ un ≤ K ∂x un ≥ in Ωn , T in Ωn , T 3.14 3.15 ∂t un ≥ a.e in Ωn T 3.16 Proof From 3.5 and the properties of βε s , we have that ≤ βε uε,n − K ≤ βε c − rK 3.17 2,1 By Wp and Cα,α/2 < α < estimates of the parabolic problem 18 , we conclude that uε,n 2,1 Wp Ωn \Bδ P0 T uε,n Cα,α/2 Ωn T ≤ C, 3.18 Advances in Difference Equations 11 2,1 where C is independent of ε It implies that there exists a un ∈ Wp Ωn \ Bδ P0 T a subsequence of {uε,n } still denoted by {uε,n } , such that as ε → , uε,n un 2,1 in Wp Ωn \ Bδ P0 T weakly, uε,n −→ un ∩ C Ωn and T in C Ωn T 3.19 Employing the method in 16 or 19 , it is not difficult to derive that un is the solution of problem 3.1 And 3.14 , 3.15 are the consequence of 3.5 , 3.6 as ε → In the following, we will prove 3.16 For any small δ > 0, w x, t satisfies, by 3.1 , ∂t w − Lw L δ K, x, t ∈ −n, × 0, T − δ , if w un −n, t , w 0, δ un x, δ ≥ max{L, Kex } w x, un x, t if w < K, x, t ∈ −n, × 0, T − δ , c, ∂t w − Lw ≤ c, w −n, t Δ K ≤ t ≤ T − δ, un 0, t , un x, , 3.20 −n ≤ x ≤ Applying the comparison principle with respect to the initial value of the variational inequality see 16 , we obtain un x, t w x, t ≥ un x, t , δ x, t ∈ −n, × 0, T − δ 3.21 Thus 3.16 follows At last, we prove the uniqueness of the solution Suppose that u1 and u2 are two n n 2,1 Wp,loc Ωn ∩ C Ωn solutions to problem 3.1 , and denote T T N Δ x, t ∈ Ωn : u1 x, t < u2 x, t n n T 3.22 Assume that N is not empty, and then, in the domain N, u1 x, t < u2 x, t ≤ K, n n Denoting W ∂t u1 − Lu1 n n c, ∂t u1 − u2 − L u1 − u2 ≥ n n n n 3.23 u1 − u2 , we have that n n ∂t W − LW ≥ in N, W on ∂p N 3.24 Applying the A-B-P maximum principle see 20 , we have that W ≥ in N, which contradicts the definition of N 12 Advances in Difference Equations 2,1 Theorem 3.3 Problem 1.4 has a unique solution u ∈ C ΩT ∩Wp ΩR \Bδ P0 T ∞, R > 0, and δ > And ∂x u ∈ C ΩT \ Bδ P0 Moreover, for any < p < in ΩT , max{L, Kex } ≤ u ≤ K 3.25 ∂x u ≥ a.e in ΩT , 3.26 ∂t u ≥ a.e in ΩT 3.27 Proof Rewrite Problem 3.1 as follows: ∂t un − Lun un −n, t L, un x, 2,1 where un ∈ Wp Ωn \ Bδ P0 T x, t ∈ Ωn , T f x, t , un 0, t ≤ t ≤ T, K, 3.28 −n ≤ x ≤ 0, max{L, Kex }, p implies that f x, t ∈ Lloc Ωn and T f x, t cI{un δ > 0, if n > R, combining 3.14 , we have the following Wp and Cα,α/2 uniform estimates 18 : un 2,1 Wp ΩR \Bδ P0 T ≤ CR,δ , un R Cα,α/2 ΩT ≤ CR , 3.30 here CR,δ depends on R and δ, CR depends on R, but they are independent of n Then, we 2,1 have that there is a u ∈ Wp,loc ΩT ∩ C Ω T and a subsequence of {un } still denoted by {un } , such that for any R > δ > 0, p > 1, 2,1 u in Wp ΩR \ Bδ P0 T un weakly as n −→ ∞ 3.31 Moreover, 3.30 and imbedding theorem imply that un −→ u R in C ΩT , ∂x un −→ ∂x u R in C ΩT \ Bδ P0 as n −→ ∞ 3.32 It is not difficult to deduce that u is the solution of problem 1.4 Furthermore, 3.32 implies that ∂x u ∈ C ΩT \ Bδ P0 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 Advances in Difference Equations 13 Behaviors of the Free Boundary Denote { x, t : u x, t < K} CT { x, t : u x, t CL continuation region , 4.1 callable region K} Thanks to 3.26 , we can define the free boundary h t of problem 1.4 , at which it is optimal for the firm to call the bond, where inf{x ≤ : u x, t ht 0 u} is nonempty; then we have that u x, t < w x, t ≤ K, ∂t u − Lu c, ∂t u − w − L u − w ≥ 0, in N 4.10 Moreover, u − w ≥ on the parabolic boundary of N According to the A-B-P maximum principle see 20 , we have that u−w ≥0 in N, 4.11 which contradicts the definition of N So, we achieve that w ≤ u Combining w x, t K for any t ≥ T0 , it is clear that K w x, t ≤ u x, t ≤ K, for any t ≥ T0 , which means that CT ⊂ {0 < t < T0 , x < 0}, CL ⊃ {t ≥ T0 , x < 0}, and h t t ≥ T0 4.12 −∞ for any Theorem 4.2 The free boundary h t is decreasing in the interval 0, T0 Moreover, h limt → h t And h t ∈ C 0, T0 Δ Proof 3.26 and 3.27 imply that ∂x u − K ≥ 0, Hence, for any unit vector n of function u − K along n admits ∂t u − K ≥ a.e in ΩT 4.13 n1 , n2 satisfying n1 , n2 > 0, the directional derivative ∂n u − K ≥ a.e in ΩT , 4.14 that is, u − K is increasing along the director n Combining the condition u − K ≤ in ΩT , we know that x h t is monotonically decreasing Hence, limt → h t exists, and we can define h lim h t t→0 4.15 Advances in Difference Equations K, so h ≤ On the other hand, if h < 0, then Since u 0, t u x, t K, 15 ∀ x, t ∈ h , × 0, T , max{L, Kex } < K, u x, ∀x ∈ h , 4.16 It is impossible because u is continuous on ΩT In the following, we prove that h t is continuous in 0, T0 If it is false, then there exists x1 < x2 < 0, < t1 < T0 such that see Figure lim h t x1 , t → t− lim h t x2 t → t1 4.17 Moreover, ∂t u − Lu c in M Δ { x, t : x2 < x < h t , < t ≤ t1 } 4.18 Differentiating 4.18 with respect to x, then ∂t ∂x u − L ∂x u in M 4.19 On the other hand, ∂x u x, t1 for any x ∈ x1 , x2 in this case, and we know that ∂x u ≥ by 3.26 Applying the strong maximum principle to 4.19 , we obtain ∂x u x, t 0, in M 4.20 So, we can define u x, t g t in M Considering u h t , t K and u ∈ C ΩT , we see that u x, t ≡ K in M, which contradicts that u x, t < K for any x < h t Therefore h t ∈ C 0, T0 Theorem 4.3 There exists some t0 ∈ 0, T0 such that h t decreasing on t0 , T0 for any t ∈ 0, t0 and h t is strictly Proof Define t0 sup{t : t ≥ 0, h t 0} In the first, we prove that t0 > Otherwise, h and h t < for t > Recalling the initial value, we see that ∂x u x, Kex for any x ∈ ln L − ln K, , lim ∂x u x, x → 0− K 4.21 Meanwhile, u x, t K in the domain { x, t : h t < x < 0, < t < T0 } implies that ∂x u 0, t for any t > see Figure ; then ∂x u is not continuous at the point 0, , which contradicts ∂x u ∈ C ΩT \ Bδ P0 −∞ for In the second, we prove that t0 < T0 In fact, according to Lemma 3.1, h t T0 , then the free boundary includes a horizontal line t any t ≥ T0 , hence, t0 ≤ T0 If t0 T0 , x ∈ −∞, Repeating the method in the proof of Theorem 4.2, then we can obtain a contradiction So, t0 < T0 16 Advances in Difference Equations t T0 CL h(t) CT t1 x x1 x2 Figure 4: Discontinuous free boundary h t At last, we prove that h t is strictly decreasing on t0 , T0 Otherwise, x h t has K for any a vertical part Suppose that the vertical line is x x1 , t ∈ t1 , t2 , then u x, t x, t ∈ −∞, x1 × t1 , t2 Since ∂x u is continuous across the free boundary, then ∂x u x1 , t for any t ∈ t1 , t2 In this case, we infer that ∂t u x1 , t 0, On the other hand, in the domain N ∂t u − Lu c for any t ∈ t1 , t2 ∂t ∂x u x1 , t in N, −∞, x1 × t1 , t2 , u and ∂t u satisfy, respectively, u x1 , t ∂t ∂t u − L ∂t u ∂t u x1 , t 4.22 K for any t ∈ t1 , t2 , ∂t u ≥ 0, in N, 0, 4.23 for any t ∈ t1 , t2 Then the strong maximum principle implies that ∂x ∂t u x1 , t < 0, which contradicts the second equality in 4.22 Theorem 4.4 h t > h∗ t for any t ∈ 0, T0 with limt → T0− h t h∗ t ln L K −∞ (see Figure 1), where c − rL e−rt − c − rK ln , α rK where α is the positive characteristic root of Lw σ2 α ≤ t < T0 , 4.24 0, that is, the positive root of the algebraic equation r−q− σ2 α−r 4.25 Proof Define W x, t c − r c − L e−rt r K α αx e , Lα x, t ∈ ΩT0 4.26 Advances in Difference Equations 17 We claim that W x, t ∈ C2 ΩT0 and possess the following three properties i W x, ≥ u x, for −∞ < x < and W 0, t ≥ K for < t ≤ T0 , ii ∂t W − LW c in ΩT0 , iii W x, t < K in { x, t : x < h∗ t , ≤ t < T0 } In fact, if we notice that α > 0, then we have that ⎧ ⎪L ⎪ ⎨ W x, L max{L, Kex } K α αx e ≥ ⎪ Lα ⎪L ⎩ Kα Lα L K α if x ≤ ln L − ln K, u x, ≥ K ≥ u x, if ln L − ln K ≤ x ≤ 4.27 It is obvious that W 0, t ≥ K So, we obtain property i Moreover, we compute ∂t W − LW c −L r r e−rt r c − r c −L r e−rt c 4.28 Hence, we have property ii It is not difficult to check that, for any t ∈ 0, T0 , W h∗ t , t K, ∂x W αK α αx e > Lα 4.29 Then we show property iii Repeating the method in the proof of Theorem 3.2, we can derive that u ≤ W in ΩT0 from properties i - ii And property iii implies that u < K in the domain { x, t : x < h∗ t , ≤ t < T0 }, which means that h t ≥ h∗ t for any t ∈ 0, T0 −∞ Otherwise, limt → T0− h t x1 > −∞; then the Next, we prove that limt → T0− h t free boundary includes a horizontal line t T0 , x ∈ −∞, x1 Repeating the method in the −∞ proof of Theorem 4.2, then we can obtain a contradiction So, limt → T0− h t Theorem 4.5 The free boundary h t ∈ C0,1 0, T0 ∩ C∞ t0 , T0 Proof Fix t1 ∈ 0, t0 and t2 ∈ t0 , T0 , and denote X h∗ t2 − According to Theorem 4.4, Δ the free boundary h t while t ∈ t0 , t2 lies in the domain N { x, t : X < x < 0, t1 < t ≤ t2 } see Figure In the first, we prove that there exists an M0 > such that M0 ∂x u − ∂t u ≥ in N 4.30 In fact, u, ∂t u satisfy the equations ∂t u − Lu c, ∂t ∂t u − L ∂t u 0, x, t ∈ CT, 4.31 18 Advances in Difference Equations t CL CT t2 h(t) Γ1 t0 t1 Γ2 x X Figure 5: The free boundary h t then the interior estimate of the parabolic equation implies that there exists a positive constant C such that ∂t u x, t ≤ C on Γ1 ∪ Γ2 , 4.32 here Γ1 Δ {x X, t1 ≤ t ≤ t2 }, Γ2 Δ {X ≤ x ≤ 0, t t1 } On the other hand, we see that ∂t u ≥ in ΩT from 3.27 , and ∂t u 0, t the strong maximum principle to ∂t u x, t , we deduce that ∂tx u 0, t < 0, 4.33 Applying t ∈ 0, t0 4.34 It means that ∂x u 0, t is strictly decreasing on 0, t0 It follows that, by ∂x u 0, t0 ∂x u 0, t1 > 0, 4.35 Moreover, ∂x u ≥ 0, ∂t ∂x u − L ∂x u 0, x, t ∈ CT 4.36 Employing the strong maximum principle, we see that there is a δ > 0, such that ∂x u x, t ≥ δ on Γ1 ∪ Γ2 , Provided that δ is small enough Combining 4.32 , there exists a positive M0 such that M0 ∂x u − ∂t u ≥ δ on Γ1 ∪ Γ2 4.37 C/δ 4.38 Advances in Difference Equations 19 t −0.8 −0.6 −0.4 −0.2 x h(t) Figure 6: The free boundary Next, we concentrate on problem 1.4 in the domain N It is clear that u satisfies ∂t u − Lu if u < K, x, t ∈ N, c, ∂t u − Lu ≤ c, u X, t u X, t , u x, t1 if u K, x, t ∈ N, u 0, t u x, t1 , K, t1 ≤ t ≤ t2 , 4.39 X ≤ x ≤ And we can use the following problem to approximate the above problem: ∂t uε − Luε uε X, t βε uε − K u X, t , uε x, t1 uε 0, t u x, t1 , c, K, in N, t1 ≤ t ≤ t2 , 4.40 X ≤ x ≤ Recalling 4.38 , we see that, if ε is small enough, M0 ∂x uε −∂t uε ≥ on the parabolic boundary of N Moreover, w Δ M0 ∂x uε − ∂t uε satisfies ∂t w − Lw βε uε − K w 4.41 20 Advances in Difference Equations Applying the comparison principle, we obtain M0 ∂x uε − ∂t uε w ≥ in N 4.42 As the method in the proof of Theorem 3.3, we can show that uε weakly converges to u in 2,1 Wp N and 4.30 is obvious On the other hand, we see that M∂x u ∂t u ≥ in N for any positive number M from 3.26 and 3.27 So, M0 ∂x u ± ∂t u ≥ in N, 4.43 which means that there exists a uniform cone such that the free boundary should lies in the cone As the method in , it is easy to derive that h t ∈ C0,1 t1 , t2 Moreover h t ∈ C∞ t0 , t2 can be deduced by the bootstrap method Since t2 is arbitrary and the free boundary is a vertical line while t ∈ 0, t0 , then h t ∈ C0,1 0, T0 ∩ C∞ t0 , T0 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 h t is a function of t The parameter values used in the calculations are r 0.2, q 0.1, σ 0.3, L 1, K 1.5, c 0.5, T 2, and n 3000 In this case, the free boundary is increasing with x 0 The numerical result is coincided with that of our proof see Figure Acknowledgments The project is supported by NNSF of China nos 10971073, 11071085, and 10901060 and NNSF of Guang Dong province no 9451063101002091 References M Sˆrbu, I Pikovsky, and S E Shreve, “Perpetual convertible bonds,” SIAM Journal on Control and ı Optimization, vol 43, no 1, pp 58–85, 2004 M Sˆrbu and S E Shreve, “A two-person game for pricing convertible bonds,” SIAM Journal on ı Control and Optimization, vol 45, no 4, pp 1508–1539, 2006 P Wilmott, Derivatives, the Theory and Practice of Financial Engineering, John Wiley & Sons, New York, NY, USA, 1998 Y Kifer, “Game options,” Finance and Stochastics, vol 4, no 4, pp 443–463, 2000 Y Kifer, “Error estimates for binomial approximations of game options,” The Annals of Applied Probability, vol 16, no 2, pp 984–1033, 2006 C Kuhn and A E Kyprianou, “Callable puts as composite exotic options,” Mathematical Finance, vol ¨ 17, no 4, pp 487–502, 2007 A E Kyprianou, “Some calculations for Israeli options,” Finance and Stochastics, vol 8, no 1, pp 73–86, 2004 F Black and M Scholes, “The pricing of options and coperate liabilities,” Journal of Political Economy, vol 81, pp 637–659, 1973 Advances in Difference Equations 21 Z Yang and F Yi, “A free boundary problem arising from pricing convertible bond,” Applicable Analysis, vol 89, no 3, pp 307–323, 2010 10 A Friedman, “Parabolic variational inequalities in one space dimension and smoothness of the free boundary,” Journal of Functional Analysis, vol 18, pp 151–176, 1975 11 A Blanchet, “On the regularity of the free boundary in the parabolic obstacle problem Application to American options,” Nonlinear Analysis: Theory, Methods & Applications, vol 65, no 7, pp 1362–1378, 2006 12 A Blanchet, J Dolbeault, and R Monneau, “On the continuity of the time derivative of the solution to the parabolic obstacle problem with variable coefficients,” Journal de Math´matiques Pures et Appliqu´es e e Neuvi`me S´rie, vol 85, no 3, pp 371–414, 2006 e e 13 L Caffarelli, A Petrosyan, and H Shahgholian, “Regularity of a free boundary in parabolic potential theory,” Journal of the American Mathematical Society, vol 17, no 4, pp 827–869, 2004 14 A Petrosyan and H Shahgholian, “Parabolic obstacle problems applied to finance,” in Recent Developments in Nonlinear Partial Differential Equations, vol 439 of Contemporary Mathematics, pp 117– 133, American Mathematical Society, Providence, RI, USA, 2007 15 A Friedman, “Stochastic games and variational inequalities,” Archive for Rational Mechanics and Analysis, vol 51, pp 321–346, 1973 16 A Friedman, Variational Principles and Free-Boundary Problems, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1982 17 D Gilbarg and N S Trudinger, Elliptic Partial Differential Equations of Second Order, vol 224 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 2nd edition, 1983 18 O A Ladyˇ enskaja, V A Solonnikov, and N N Uralceva, Linear and Quasi-linear Equations of Parabolic z Type, American Mathematical Society, Providence, RI, USA, 1968 19 F Yi, Z Yang, and X Wang, “A variational inequality arising from European installment call options pricing,” SIAM Journal on Mathematical Analysis, vol 40, no 1, pp 306–326, 2008 20 K Tso, “On Aleksandrov, Bakel’man type maximum principle for second order parabolic equations,” Communications in Partial Differential Equations, vol 10, no 5, pp 543–553, 1985  ... Quasi-linear Equations of Parabolic z Type, American Mathematical Society, Providence, RI, USA, 1968 19 F Yi, Z Yang, and X Wang, ? ?A variational inequality arising from European installment call... American Mathematical Society, Providence, RI, USA, 2007 15 A Friedman, “Stochastic games and variational inequalities,” Archive for Rational Mechanics and Analysis, vol 51, pp 321–346, 1973 16 A. .. Equations 21 Z Yang and F Yi, ? ?A free boundary problem arising from pricing convertible bond,” Applicable Analysis, vol 89, no 3, pp 307–323, 2010 10 A Friedman, “Parabolic variational inequalities