Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
ThA15.3
Pricing and optimal conversion strategy of convertible bonds Bing Yang and Hua Xiao
Abstract— This paper develop a method based on the reflected Backward Stochastic Differential Equations (BSDEs for short) to solve the pricing and the optimal conversion strategy of noncallable American Style convertible bonds. We characterize the value functions of the noncallable convertible bonds in terms of the reflected Backward Stochastic Differential equation, and provide the optimal conversion strategy for bondholders. Some numerical Simulation methods for the pricing and the optimal conversion strategy of noncallable American Style convertible bonds are given.
I. INTRODUCTION A convertible bond is a structured security that consists of two securities, a long position in a hypothetical straight bond and a short position in a call option. Corporations would issue putable bonds when they need long-term financing in spite of the general expectation that interest rates may rise soon. In order to convince investors to buy their long-term debt they include a put provision that allows the investor to sell the bond back to the issuing corporation at par at a date, or a series of dates in the future. Of course, the corporation is paid for giving this option to the investor. As with the call option embedded in the callable bond, the price of the put option is implicitly included in the bond covenants of the putable bond. For example, the coupon rate on a putable bond is typically lower than otherwise identical but straight bonds. Based on the firm value approach originated by Black and Scholes (1973) and Merton (1974), Ingersoll (1977) first introduced theoretical model of the value of convertible bond, and given the optimal conversion strategy for investors and the optimal call policy for the issuer. Brennan and Schwartz (1977) considered pricing a convertible bond in the firm’s value framework for more general cases. A finite difference scheme was introduced to solve the pricing Partial Differential Equation. Brennan and Schwartz(1980) extended their pricing method by including stochastic interest rates. McConnell and Schwartz(1986) studied liquid yield option notes and develop a pricing model based on a finitedifference method with the stock price as stochastic variable. Lewis (1991) developed a formula for convertible bonds that accounts for more complex capital structures, i.e. multiple issues. Arzac (1997) discussed the rationale for mandatory convertibles from the point of view of issuers as well as Bing Yang is with Faculty School of Mathematics and Statistics, Shandong University at Weihai, 264209 Weihai, China
[email protected] Hua Xiao is with Faculty of School of Mathematics and Statistics, Shandong University at Weihai, Weihai 264209, China, and a Ph.D. student in School of mathematics, Shandong University, Jinan 250100, China
xiao
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
investors. He noted that mandatory convertibles allow highly leveraged (or temporarily troubled) companies to restructure their balance sheet by helping to control for the asymmetric information problem. Tsiveriotis and Fernandes (1998) proposed an approach that splits the value of a convertible bond into a stock component and a straight bond component. Ammann et. al. (2003) given an approach by accounting for call features with various trigger conditions. The convertible bonds with the special features were studied by Arzac(1997, mandatory convertibles), Hillion and Vermaelen(2001, spiral convertibles), and Yigitbasioglu(2001, crosscurrency convertibles). Hung and Wang (2002) proposed a tree-based model that accounts for both stochastic interest rates and default probabilities but looses its recombining property. A further tree-based model is presented by Carayannopoulos and Kalimipalli(2003). Sirbu, Pikovsky and shreve (2006) discussed perpetual convertible bonds and given a full description of the Perpetual convertible bond price. Ammann, Kind and Wilde (2008)proposed a pricing model for convertible bonds on Monte Carlo simulation and present an empirical pricing study of the US market. The nonlinear Backward Stochastic Differential Equations were first introduced by Pardoux and Peng [1991], who proved the existence and uniqueness of a solution under suitable hypotheses on the coefficient and the terminal value of the BSDE. Since then, EI Karoui, Kapoudjian, Pardoux, Peng and Quenez [1997] have introduced the notion of a reflected BSDE with one lower barrier. Their motivations are linked the pricing of American options and viscosity solutions of Partial Differential Equations with an obstacle. Cvitanic and Karatzas [1996] studied the reflected BSDEs with two continuous barriers and established the existence and uniqueness results for adapted solutions of BSDE‘s with two reflecting barriers. The theory of BSDE is by now a primary tool in financial mathematics, stochastic optimal control and partial differential equations. In this paper, we would like to provide a method based on the reflected Backward Stochastic Differential Equations to solve the pricing and the optimal conversion strategy of noncallable American Style convertible bonds. We characterize the value functions of the noncallable convertible bonds in terms of the reflected Backward Stochastic Differential equation, and provide the optimal conversion strategy for bondholders. Some numerical Simulation methods for the pricing and the optimal conversion strategy of noncallable American Style convertible bonds are given. The paper is organized as follows: In section 2, we introduce some preliminaries and describe our model. In section 3, we employe the reflected BSDEs with one
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ThA15.3 lower barrier to describe the value of the noncallable convertible bond. Section 4 provide the optimal conversion strategy for the bondholders, and section 5 give Some numerical Simulation methods for the pricing and the optimal conversion strategy of noncallable American style convertible bonds. Section 5 give some conclusions.
II. M ARKET MODEL Let (Ω, F , {Ft } , P) is a complete, filtered probability space on which a standard one-dimensional Brownian motion W = {Wt , 0 ≤ t ≤ T } is defined. The filtration {Ft , 0 ≤ t ≤ T } is generated by the Brownian motion W , with usual augmentation. Thus it satisfies the usual hypotheses. Let us consider a market in which three financial instruments are traded continuously. Three financial instruments include the stock, the convertible bond, and a money market account with risk-free rate of interest r > 0. We assume throughout this section that there is a fixed time horizon T, 0 < T < ∞. Denote the value process of the firm at each time t by Xt , and assume the value of the firm consists of equity and debt. The debt Dt is due to a single convertible bond. This assumption of a single bond means that all debt is converted simultaneously. Denote by St the total value of equity. For noncallable American Style convertible bonds, we assume that: •
•
• •
•
•
The convertible bond is noncallable. Equity owners receive dividends paid continuously over time at a rate δ St , δ ≥ 0. The bondholder receives coupons paid continuously over time at a rate α > 0. If there is no conversion prior to maturity T , then at maturity the bondholder receives the par value L from the firm, provided XT ≥ L. Otherwise, the bondholder receives β XT , 0 < β < 1. The bondholder may convert the bond to stock at any time t ∈ [0, T ], thereby immediately receiving stock valued at the conversion factor γ ∈ (0, 1) times the firm value Xt . The firm value is not affected by this conversion. Miller-Modigliani theorem holds: Xt = St + Dt .
Assume that there exists an equivalent probability measure Q ∼ P such that both the discounted stock priced plus the cumulative discounted dividend payments and the discounted convertible bond price plus the cumulative discounted coupons are local martingales under the risk-neutral probability measure Q. Under the risk-neutral probability measure Q, the value of the firm evolves according to the stochastic differential equation dXt = rXt dt − α dt − δ St dt + σ Xt dWt∗ , X0 = x − − − − − − (2 − 1)
where volatility σ > 0 is constant and {Wt∗ , 0 ≤ t ≤ T } is a Brownian motion under the risk-neutral probability measure Q. For any time t ∈ [0, T ], we shall price the convertible bond under the assumption that Xt = x. Given these initial conditions, we denote by Xst,x the solution to (2.1) at time s ∈ [t, T ]. Denote the filtration by (Fst , s ∈ [t, T ]), which generated by (Ws∗ −Wt∗ , s ∈ [t, T ]), and denote the set of all (Fst , s ∈ [t, T ])stopping times with values in [t, T ] by Π[t, T ]. Once the conversion strategies τ ∈ Π[t, T ] is chosen by bondholder, we use the fact that both the discounted stock price plus the cumulative discounted dividend payments and the discounted convertible bond price plus the cumulative discounted coupons are local martingale under some riskneutral probability measure Q to write the value of the bond Dt at time t as J(t, x, τ ) hZ τ i = E∗ e−r(u−t) α du + e−r(u−t)Dτ t hZ τ ∗ e−r(u−t) α du = E
t i +e−r(u−t) (1τ 0 such that for any ′ ′ y, y , z, z ∈ R, ′
′
′
′
| f (ω ,t, y, z) − f (ω ,t, y , z ) |≤ c(| y − y | + | z − z |). − − − − − − (3 − 1) (A3):The barrier process {St ; 0 ≤ t ≤ T } is a continuous progressively measurable real-valued process satisfying
xt ≤ x0 e(r− 2 σ
′
′′
f (s, ys , zs ) + KT − Kt −
yt ≥ St (a.s.),
Z T 0
Z T t
zs dws
yt ≥ St (a.s.),
Z T 0
Z T t
yt,x u
Lemma 3.1 Under the assumption (A.1), (A.2) and (A.3) on f , ξ and (St ) are satisfied, if {(Yt , Zt , KT ) , 0 ≤ t ≤ T } of Ft is a solution of the reflected BSDE (3 − 1), then there ′ exists a constant c which depends only on c such that
Z T 0
0
, 0 ≤ t ≤ T.
=ξ+
Z T t
e
−r(u−t)
α du + KTt,x − Kut,x −
Z T t
zt,x u dwu ,
t≤u≤T
After these preliminaries, we are going to show existence and uniqueness of the solution for the reflected BSDE with a lower barrier (3 − 1) under the above assumptions on f , ξ and S.
0≤t≤T
2 )t+σ W ∗ t
Z ∈ H 2 ,Y ∈ L 2 , K ∈ C 2 .
zs dws + KT − Kt
(yt − St ) dKt = 0
Z T
1
Theorem 3.4 The optimal stopping problem about noncallable American Style convertible bonds (2 − 3) has a unique solution, that is, for any (t, x) ∈ [0, T ] × R+ , there are Ft − progressively measurable processes that value function { Yut,x , Zut,x , Kut,x , 0 ≤ t ≤ T } such V (t, x) = Ytt,x and { Yut,x , Zut,x , Kut,x ,t ≤ u ≤ T } satisfied following the reflected BSDE with a lower barrier:
− − − − − − (3 − 3)
E[ sup (Yt2 ) +
′′
Xtt,x = x − − − − − −(3 − 6)
Z ∈ H 2 ,Y ∈ L 2 , K ∈ C 2 . f (s, ys , zs ) −
′
For any (t, x) ∈ [0, T ] × R+ , we denote the solution of the following stochastic differential equation by (Xut,x ,t ≤ u ≤ T )
(yt − St ) dKt = 0
For the data {ξ , {S}, f }, the adapted solution to reflected BSDE with a lower barrier is a triple {(Yt , Zt , Kt ) , 0 ≤ t ≤ T } of Ft progressively measurable processes taking values in R, R and R+ , respectively, and satisfying: Z T
′′
xt = xt + xt ≤ xt = x0 e(r− 2 σ
, 0 ≤ t ≤ T.
dXut,x = rXut,x du − α du − δ Sudu + σ Xut,xWu∗ ,t ≤ u ≤ T
− − − − − − (3 − 2)
′
2 )t+σ W ∗ t
Using the fact that xt ≤ 0, we got
t
≤ c E[ξ 2 +
1
′
Z T
t
′′
By Ito’s formula we obtain, ′′
Consider the following reflected BSDE with a lower barrier:
yt = ξ +
′′
dxt = rxt dt + σ xt dWt∗ , x0 = x0 .
xt = x0 e(r− 2 ]σ
∈ S 2.
yt = ξ +
′′
Proof. Let the processes xt , xt satisfied the following the equations ′ ′ dxt = −α dt − δ St dt, x0 = 0,
0≤t≤T
, i.e.,
,0 ≤ t ≤ T
− − − − − − (3 − 5)
E( sup (St+ )2 ) < ∞ {St+ }
2 )t+σ W ∗ t
yt,x u ≥ Su (a.s.),
Z T t
t,x t,x yt,x u − Su dKu = 0
− − − − − − (3 − 7)
where ξ = XTt,x ∧ L, Su = e−r(u−t) γ Xut,x ,t ≤ u ≤ T . Proof. Fist, from ξ 2 = (XTt,x ∧ L)2 ≤ L2 , it follows that ξ ∈ L 2 . Moreover, we have that the function e−r(u−t) α satisfies (3 − 1). Because (Su )2 = (e−r(u−t) γ Xut,x )2 ≤ γ 2 (Xut,x )2 , by Lemma 3.3, we have 1
(Su )2 ≤ γ 2 (Xut,x )2 ≤ γ 2 x2 e2(r− 2 σ
| Zt |2 dt + Kt2 ]
2 )t+2σ W ∗ t
Hence, | f (t, 0, 0) |2 dt + sup (St+ )2 ]
E ∗ ( sup (Su )2 ) ≤ E ∗ (γ 2 (Xut,x )2 )
0≤t≤T
t≤u≤T
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ThA15.3 1
≤ E ∗ (γ 2 x2 e2(r− 2 σ
2 )t+2σ W ∗ t
). Yut,x
we get 1
E ∗ ( sup (Su )2 ) ≤ γ 2 x2 e2|r− 2 σ
2 |T
∗
E ∗ ( sup (e2σ Wt ))
2σ E ∗ [ sup |Wt∗ |] t≤u≤T
= C1 e
√ 2σ T 1
−
t≤u≤T
t≤u≤T
≤ C1 e
≤ C1 e2σ
√
Yτt,x ∗ =ξ + u
< ∞, − − −(3 − 8)
2
Z ∈ H 2 ,Y ∈ L 2 , K ∈ C 2 . Z T
=ξ+
e
u
−r(u−t)
α du + KTt,x − Kut,x −
Z T u
t,x yt,x u ≥ Su (a.s.),
yt,x u =ξ+
Z T u
=E
Z τ
= Ytt,x .
∗ t,x yt,x u = E [yu | Fu ]
e
u
α ds + Kτt,x − Kut,x −
≥ E ∗ [yt,x τ +
Z τ u
Z T u
zt,x u dwu
Z τ u
≥ E [1τ 0. ii The price St of stock at t will descend to dSt at t + 1 during the interval ∆t, where d < 0. Let the probability of the price rising is P, then the descending is 1 − P. where,with the neutral risk probability P provided, we have: P=
√ √ er∆t − d 1 , µ = eσ ∆t , d = = e−σ ∆t. µ −d µ
· · · · · · (5.2)
B. Simulation Based on LSM [6] How to generate the set of possible stopping time points is the difficulty of our stimulation. The numerical way to the expectation E (J (t, x, τ ) |Ft ) is the key to solve the problem. Least square regression, by which E (J (t, x, τ ) |Ft ) is computed with S(t) and e−rV (t + 1), is applied for the approximation of E (J (t, x, τ ) |Ft ) .
4 0
20
40
60
80
100
120
140
Fig. 1. The paths are generated based on the GARCH model .Green points:The set consisting of all the possible stopping time points in different tracks,Red points:The set consisting of all the optimal stopping time in different tracks.Blue trend curves:the stimulation tracks of one equity.The price of the bond is 899.3918 yuan
Compute the arithmetic mean of all the discounted value gained in step 4 as the price of the bond. 2) Result of Stimulation: We show the numeral results of stimulation in 1, 2, where, the parameter are: • The par value of the bond f is 1000 yuan. • The duration of the noncallable convertible bond T is 252 days(off-days not considered). • The constant free-risk rate r is 5%/252 per day. • The conversion constant K the number of equities converted by the bond is 90 • the price of the equity at t St is 10. • the fluctuate of equities at t σt is 0.4 • w=0.001 • α =0.002 • β =0.004
1) Process of Simulation: • Generate all the paths of the price for one equity in the assumed stock market. • Find out the set of possible stopping points in paths through the idea of Backward Pass. • Find out the one and the only optimal stopping time point in every path with the guide of the proved in section 4. • Discount the exercise price of the bond atτ ∗ to t in every path.
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•
VI. CONCLUSION AND RESULTS •
•
•
•
This paper develop a method based on the reflected Backward Stochastic Differential Equations to solve the pricing and the optimal conversion strategy of noncallable American Style convertible bonds. Based on the reflected BSDE approach, we obtain the existence of the value function of the noncallable American Style convertible bonds. Based on the reflected BSDE approach, we give the optimal conversion strategy of noncallable American Style convertible bonds. we propose some numerical Simulation methods for
ThA15.3
Fig. 2. The paths are generated based on the Binomial Tree Model. Yellow shadow area consists of all the convertible points.There are two figures in each crunode,and the upper of the two is the price of the stock, the lower is the price of the convertible bond.
the pricing and the optimal conversion strategy of noncallable American style convertible bonds .
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