Perpetual Options and the Leland Model

Perpetual options are time‐independent, so the fundamental PDE is actually an ODE. The optimal exercise boundary can be found by directly optimizing over the boundary or by using smooth pasting. The chapter explains the pricing of perpetual calls, perpetual puts, securities that pay a given amount at a hitting time, securities that pay at a hitting time but are knocked out if another boundary is hit first, and securities that pay cash flows continuously prior to a hitting time. The valuation results are applied to analyze the optimal bankruptcy time of a firm with a given debt burden, the optimal amount of debt for the firm, and the optimal time to take on more debt when debt is perpetual (Leland’s model of the trade‐off theory). Finite maturity debt is briey discussed.