A novel state-transition forest: pricing corporate securities with intertemporal exercise policies and corresponding capital structure changes

While ordinary options are wholly exercised at a chosen time to maximize holders' benefits, empirical evidence favors the intertemporal exercise of corporate securities such as convertible bonds and executive/employee stock options. This is because the exercises of these securities can dilute the issuers' equity, inject capital into the issuers' assets, and change the capital structures, which influences the exercise payoffs and the remaining security values. However, the literature simplifies these impacts on exercise policies, leading to biased and even irrational pricing results. We address this via a state-transition forest that partitions the securities into k units and the time span of the forest into n steps. A representative holder can exercise an integral multiple of units at different steps to maximize her benefit, given that the lump sum of the exercise units does not exceed k. It can be further extended to model intertemporal exercises conducted by competitive investors who hold too-small amounts of securities to influence prices. The forest is then composed of k + 1 trees to capture transitions among capital structures due to different lump sums of option exercise units. The intertemporal exercise decision at each forest node is optimally determined by recursively solving the Bellman equation. This carefully designed forest structure reduces the running time complexity from $ O(n^{2k}) $ O(n2k) to $ O(k^2n^2) $ O(k2n2). We show that ignoring or simplifying intertemporal exercises and corresponding dilutions/injections significantly influences pricing results.