Pricing Convertible Bonds with Credit Risk under Regime Switching and Numerical Solutions

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 381943, 13 pages http://dx.doi.org/10.1155/2014/381943 Research Article Pricing Convertible Bonds with Credit Risk under Regime Switching and Numerical Solutions Wei-Guo Zhang and Ping-Kang Liao School of Business Administration, South China University of Technology, Guangzhou 510640, China Correspondence should be addressed to Wei-Guo Zhang; Received 26 December 2013; Accepted 9 April 2014; Published 19 May 2014 Academic Editor: Fenghua Wen Copyright © 2014 W.-G. Zhang and P.-K. Liao. 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. This paper discusses the convertible bonds pricing problem with regime switching and credit risk in the convertible bond market. We derive a Black-Scholes-type partial differential equation of convertible bonds and propose a convertible bond pricing model with boundary conditions. We explore the impact of dilution effect and debt leverage on the value of the convertible bond and also give an adjustment method. Furthermore, we present two numerical solutions for the convertible bond pricing model and prove their consistency. Finally, the pricing results by comparing the finite difference method with the trinomial tree show that the strength of the effect of regime switching on the convertible bond depends on the generator matrix or the regime switching strength. 1. Introduction Convertible bond is a kind of the most important financing instruments, so the convertible bond market occupies an important position in the international financial market. American convertible bond market is the largest market in the world, and it has issued more than 400 trillion dollars in total from 1980 to 2011. Hong Kong is an international financial center; there are about 70 trillion yuan of convertible bonds in 2012. Although the amount of convertible bonds issued in developing countries is much less than that in developed countries, the convertible bonds markets of some countries are developing rapidly. For example, China issued more than 60 trillion yuan of new convertible bonds in 2010, which is almost three times the level of four years ago. Because of the importance of the convertible bonds in the financial market, the convertible bond pricing problem is a hot topic. Taking the corporate value as a basic variable, Ingersoll [1] constructed a structural model for pricing convertible bonds by deriving a Black-Scholes-type partial differential equation based on Black-Scholes’ theory. Following the work of Ingersoll [1], Brennan and Schwartz [2] explored the valuation of the convertible bonds with dividends and callable provision. However, the structural approach has a shortcoming which is the fact that the data of the corporate value is difficult to measure and observe. To overcome this shortcoming, McConnell and Schwartz [3] firstly selected the stock price as the basic variable, which becomes a mainstream later. With further research, the clauses of convertible bond also are indepth studied. Kimura and Shinohara [4] and Yang et al. [5] explored the effect of reset clause on noncallable convertible bonds; they also derived an exact solution on the valuation of the convertible bonds with and without dilution effect. The numerical solutions are also adapted to price the convertible bonds. Brennan and Schwartz [6] took stochastic interest rate into account firstly and built the so-called two-factor model for convertible bond pricing that is popular since it is available. Then, a three-factor model was established by Davis and Lischka [7] to value the convertible bonds with stochastic credit risk. To solve the complex multifactor models, characteristics/finite elements and trinomial tree model are applied to the valuation of the convertible bonds such as Barone-Adesi et al. [8] and Xu [9]. Some other researches consider special convertible bonds or other factors of the convertible bonds. Yagi and Sawaki [10] proposed a valuation model of callable-puttable reverse convertible bonds, which are issued by a company to be exchanged for the shares of another company. Instead of the popular geometric 2 Brownian motion model, Labuschagne and Offwood [11] valued the convertible bonds with the CGMY stock price process. Lee and Yang [12] presented the unexplained portion of the valuation model of convertible bonds based on MCDM. Credit risk is another important factor that affects the value of convertible bonds. As early as the structural approach is available, the credit risk is considered by comparing the corporate value and the convertible bond value. However, this idea is not popular for the shortcoming of the structural approach. There are two main ways to deal with the credit risk in the pricing models now. One way measures the credit risk by the credit spread of bonds. McConnell and Schwartz [3] applied this idea to the valuation of the convertible bonds. Tsiveriotis and Fernandes [13] innovatively defined the “cash only part of the convertible bonds,” whose discount rate contains credit spread and is different from the rest of the convertible bonds. This idea is also applied in binomial tree, trinomial tree, and multifactor model of the convertible bond pricing with credit risk, such as the work of Bardhan et al. [14]. The other way measures the credit risk by the default intensity. The default intensity to price convertible bonds with credit risk was introduced to measure the credit risk in the early work of Duffie and Singleton [15] and Takahashi et al. [16]. However, these works are not reasonable enough for they cannot measure the credit risk accurately so that they need to be improved. Thus, Ayache et al. [17] proposed a new model for the convertible bonds pricing with credit risk (AFV model). Different from previous work, they adopt default intensity to measure the credit risk and build a model by deriving the PDE of the convertible bonds. This idea is also introduced to trinomial tree and multifactor model, such as the work of Chambers and Liu [18]. Milanov et al. [19] set a binomial tree model for the valuation of convertible bond and prove that their model converges in continuous time to the AFV model. Since the early 20th century, the global economy interchanges between the boom and the bust frequently. Similarly, the security market also interchanges between the “bull market” and the “bear market” frequently. The stock price processes have different characteristics in different states of economy. Considering the regime switching, some options pricing models with regime switching are proposed. The options pricing with regime switching can be traced back to the work of Naik [20], who establishes a formula of option pricing with only two states. Then, Buffington and Elliot [21] built pricing mo (...truncated)