Convertible bonds are hybrid instruments for raising capital on financial markets. They offer investors the right to give up a bond in exchange to a specified number of shares of common stock. In 2007, total convertible bond issuance in the US was $56 billion while the initial equity offering in the same year was $35 billion. Despite the significant influence on financial markets, pricing convertible bonds remains a challenge due to the complex embedded options, i.e., convertible bonds normally allow the bond issuer (holder) to call (sell) back the bonds at a pre-decided call (put) price only if the underlying stock price is above (below) the trigger price for a certain prescribed and consecutive time. Such options are named Parisian options. As a hybrid product with equity and fixed income characteristics, convertible bonds are also subject to default risk, interest rate risk, and market risk of stock price.
This paper is therefore motivated to answer two questions. First, how can we solve the pricing model that incorporates the possibility of early conversion, callability by the issuer and putability by the holder with Parisian features, stochastic interest rate, and credit risk? Second, do Parisian options matter when we empirically price the convertible bond?
We propose a contingent claim model with two factors: stock price and interest rate. Following Barone-Adesi et al. (2003), we value a convertible bond with Parisian options by a finite element method. This method gives convergent deterministic approximations under realistic and low smoothness assumptions on the payoff function and in particular allows a higher rate of convergence compared with finite difference method. We model the underlying stock process under Black-Scholes Geometric Brownian Motion (GBM) framework and stochastic interest rates process as Cox–Ingersoll–Ross (CIR). In addition, we allow an instantaneous correlation between the dynamics of stock and interest rate. A new state variable ? is used to model the barrier time for a Parisian option following Haber et al. (1999). Pricing partial differential equations (PDE) are then derived and solved numerically.
This paper contributes to the literature in the following aspects. First, we solve the theoretical price for convertible bonds with embedded Parisian options and stochastic interest rate in a stock-based model. The literature on the theoretical modeling of convertible bond prices can be divided into two branches based on the underlying asset: firm value or stock. Ingersoll (1977) finds a closed-form solution to the convertible bond price depending on the firm value as the underlying state variable. The pricing formula is further developed by Lewis (1991) and Buehler and Koziol (2002) accounting for more complex capital structures. The first numerical solution for the firm-value-based models is found in Brennan and Schwarts (1977). Brennan and Schwarts (1980) extend their pricing method by including stochastic interest rate. Their model is further extended by Buchan (1998) to allow senior debt. Because the Parisian options are written on the stock price and firm values are not observable, firm value models have difficulties in modeling these path-dependent contingent claims.
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Pricing Convertible Bonds with Embedded Parisian Options: Theory and Evidence
