Valuation of Convertible Bonds

Abstract

Convertible bonds are hybrid financial instruments with complex features. They have characteristics of both debts and equities, and usually several options are embedded in this kind of contracts. The optimality of the conversion decision depends on equity price, interest rate and default probability of the issuer. The decision making can be further complicated by the fact that most convertible bonds have call provisions allowing the bond issuer to call back the bond at a predetermined call price. <br />In this thesis, we first adopt a structural approach where the Vasicek-model is applied to incorporate interest rate risk into the firm's value process which follows a geometric Brownian motion. Default is triggered when the firm's value hits a lower boundary. The complex nature of the firm's capital structure and information asymmetry may make it hard to model the firm's value and the capital structure. In this case, reduced-form models are applied for the study of convertible bonds. We adopt a parsimonious, intensity-based default model, in which the default intensity is modeled as a function of the pre-default stock price. <br />We first analyze the contract features of the convertible bonds and show that callable and convertible bonds can be decomposed into a straight bond and a game option component. Then the no-arbitrage prices of the European- and American-style callable and convertible bonds are derived. In American-style contracts, the focus is on the analysis of the strategic optimal behavior. The bondholder and issuer choose their stopping times to maximize or minimize the expected payoff respectively. For the bondholder it is optimal to select the stopping time which maximizes the expected payoff given the minimizing strategy of the issuer, while the issuer will choose the stopping time that minimizes the expected payoff given the maximizing strategy of the bondholder. The no-arbitrage price can be approximated numerically by means of backward induction. In the structural model, the recursion is carried out alongside a recombining binomial tree. Whereas in the reduced-form approach, the optimization problem is formulated and solved with the help of the theory of doubly reflected backward stochastic differential equations. <br />In practice, it is often a difficult problem to calibrate a given model to the available data. Determining the volatility of the firm's value process or stock price process is not a trivial problem. We therefore assume that the volatility of the firm's value process/stock price process lies between two extreme values, and combine it with the results on game option in incomplete market to derive certain pricing bounds for callable and convertible bonds. The maximizing strategy of the bondholder and the choice of the most pessimistic pricing measure from his perspective determine the lower bound of the no-arbitrage price. Whereas the minimizing strategy of the issuer and the most pessimistic expectation from his aspect construct the upper bound of the no-arbitrage price. Numerically, to make the computation tractable a constant interest rate is assumed. The pricing bounds can be calculated with recursion alongside a recombining trinomial tree or with the finite-difference method.