Abstract
The model of pricing American-style convertible bond is formulated as a zero-sum Dynkin game, which can be transformed into a parabolic variational inequality (PVI). The fundamental variable in this model is the stock price of the firm which issued the bond, and the differential operator in PVI is linear. The optimal call and conversion strategies correspond to the free boundaries of PVI. Some properties of the free boundaries are studied in this paper. We show that the bondholder should convert the bond if and only if the price of the stock is equal to a fixed value, and the firm should call the bond back if and only if the price is equal to a strictly decreasing function of time. Moreover, we prove that the free boundaries are smooth and bounded. Eventually we give some numerical results.
Highlights
Firms raise capital by issuing debt bonds and equity shares of stock
The convertible bond is intermediate between these two instruments, which entitles its owner to receive coupons plus the return of principle at maturity
After issuing a convertible bond, the bondholder will find a proper time to exercise the conversion option in order to maximize the value of the bond, and the firm will choose its optimal time to exercise its call option to maximize the value of shareholder’s equity
Summary
Firms raise capital by issuing debt bonds and equity shares of stock. The convertible bond is intermediate between these two instruments, which entitles its owner to receive coupons plus the return of principle at maturity. After issuing a convertible bond, the bondholder will find a proper time to exercise the conversion option in order to maximize the value of the bond, and the firm will choose its optimal time to exercise its call option to maximize the value of shareholder’s equity. The free boundary S t between CT and CV means the optimal conversion strategy, which is dependent on the time t and more than K/γ In this model, their exist three domains: the continuation domain CT, the callable domain CL, and the conversion domain CV {x ≥ 0}. We provide numerical result applying the binomial method
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