CONVERTIBLE BONDS: VALUATION AND OPTIMAL STRATEGIES FOR CALL AND CONVERSION

Abstract

THE THEORY OF OPTION and warrant pricing has only of late been placed on a sound theoretical basis in a context of security market equilibrium [1, 6]; closed form expressions have been derived by Black-Scholes [1] and Merton [6] for the value of an option when the underlying stock pays no dividend or the option is protected against dividends, and when the stock pays a continuous dividend which is proportional to the market value of the stock. Further research has extended this option pricing model to take account of jumps in security returns [3, 8], and the basic option pricing model has been shown to obtain under certain assumptions, even in the absence of continuous trading opportunities [ 11]. More recently, algorithms have been developed [12] to solve the relevant dynamic programming problem when the stock does pay dividends and the option is not protected against dividend payments, so that the possibility of exercise prior to maturity must be considered for an American type option. As yet however, little attempt has been made to apply the principles of the option pricing model to the most common type of convertible security, namely the convertible bond.1 This security is considerably more complex than the warrant, not only because it pays a periodic coupon, but also because it involves a dual option: on the one hand, the bondholder possesses the option to convert the bond into common stock at his discretion, and on the other hand, the firm possesses the option to call the bond for redemption, the bondholder retaining the right to convert the bond or to redeem it. This call option is usually subject to some kind of restriction, a common one being that the bond may not be called for five years. The investor's optimal conversion strategy then depends on the firm's call strategy, and it appears at first sight that the optimal call strategy must also depend on the investor's conversion strategy, so that both optimal strategies must be solved for simultaneously. Fortunately, as we shall show below, the optimal call strategy is simply to call the bond as soon as the value of the bond if called is equal to the value if not called, so that the problem is considerably simplified.