A Study of Convertible Bond: Optimal Strategies and Pricing.

摘要

This dissertation contains two parts: a non-zero-sum game approach of convertible bond and exotic options pricing under exponential-type jump-diffusion model.;In the first part, we propose a non-zero-sum stochastic game approach of pricing convertible bond under the framework that the capital structure of the firm involves tax rebate and endogenous default policy. Convertible bond is a hybrid security which embodies characteristics of both straight bond and equity. Beyond the bond provisions, it endows a conversion option for the bondholder to convert the bond for the equity of the issuing firm and a call option for the firm to buy the debt back. The conflict of interests between bondholder and shareholder affects the security prices significantly. In Chapter 2, we investigate how to use a non-zero-sum game framework to model their interaction and to evaluate the convertible bond accordingly. Mathematically, this problem can be reduced down to a system of variational inequalities. After we clarify the structure of the optimal exercise region of both parties, we manage to explicitly derive a unique Nash equilibrium to the constraint game and specify the associated optimal exercise strategies. Our model shows that tax benefit and credit risk can produce considerable impact on the optimal strategies of both parties. The firm may issue a call when the debt is out-of-the-money or in-the-money. This is consistent with the empirical findings of "late and early calls" (Ingersoll (1977), Mikkelson (1981), Cowan et al. (1993) and Ederington et al. (1997)) . In addition, the optimal call policy under our model offers an explanation to some stylized patterns related to the returns of the company value as well.;In the second part, we use Laplace transform to study the pricing problems of various path-dependent exotic options with the underlying asset following an exponentially distributed jump diffusion process. These exotic options include double-barrier option and some occupation-time-related derivatives such as step options, corridor options, and quantile options. The result about double barrier options is presented in Chapter 3, where we prove non-singularity of a related high-dimensional matrix, which guarantees the existence and uniqueness of the solution. Chapter 4 is our work on occupation-time-related options, which presents an extension of the Black-Scholes setting to Kou's double-exponential jump diffusion model. We derive the closed-form Laplace transform of the joint distribution of the occupation time and the terminal value of the double-exponential jump diffusion process, and apply the result to price various occupation-time-related derivatives. This is done by solving the associated two correlated ordinary integro-differential equations, thanks to the special property of the exponential. All the Laplace transform-based analytical solutions can be inverted easily via Euler Laplace inversion algorithm, and the numerical results illustrate that our pricing methods are accurate and efficient.;Key words. Convertible Bond; Non-zero-sum Differential Game; Tax Benefit; Credit Risk; Early/Late Calls; Positive/Negative Stock Return; Double-barrier Options; Step Options; Corridor Options; Quantile Options; Occupation-Time; Jump-Diffusion Process.