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Speical Convertible Bond A convertible bond issuer pays periodic coupons to the convertible bond holder. The bond holder can convert the bond into the underlying stock within the period(s) of time specified by the conversion schedule. The bond issuer can call the bond and the holder can put it according to the call and put provisions. The Exchangeable feature assumes that the convertible bond and the underlying stock are issued by different parties. There are two possible cases with respect to stock conversion We assume that the short term interest rate process follows a stochastic differential equation (SDE) of the Hull-White form, and that the stock’s price follows geometric Brownian motion with drift. For tractability, however, the stock’s price SDE is modified so that its drift does not depend on the stochastic short-term interest rate. Based on the Hull-White single-factor tree building approach, respective trinomial trees are constructed for the short-term interest rate and stock’s price processes. Using the Hull-White two-factor tree building procedure, a combined tree is constructed by matching the mean, variance and correlation corresponding to each combined tree node. The convertible bond price is given from the combined tree by backward induction. Here the issue time refers to the coupon payment immediately prior to, or including, the valuation time; otherwise it corresponds to the bond’s issuance. Since the valuation time is taken to be zero, the issue time must be less than or equal to zero. We build respective trinomial trees for the approximate stock’s price and short-term interest rate processes above. This construction is based on Hull and White’s single factor tree-building technique, but is generalized to accommodate non-uniformly spaced partitions of the interval to bond maturity. The separate trinomial trees are then combined into a joint tree using Hull and White’s two-factor tree building procedure.
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