65. Accurate Formulas for Evaluating Barrier Options with Dividends Payout and the Application in Credit Risk Valuation

To price the stock options with discrete dividend payout reasonably and consistently, the stock price falls due to dividend payout must be faithfully modeled. However, this will significantly increase the mathematical difficulty since the post-dividend stock price process, the stock price process after the price falls due to dividend payout, no longer follows the lognormal diffusion process. Analytical pricing formulas are hard to be derived even for the simplest vanilla options. This chapter approximates the discrete dividend payout by a stochastic continuous dividend yield, so the post-dividend stock price process can be approximated by another lognormally diffusive stock process with a stochastic continuous payout ratio up to the ex dividend date. Accurate approximation analytical pricing formulas for barrier options are derived by repeatedly applying the reflection principle. Besides, our formulas can be applied to extend the applicability of the first passage model — a branch of structural credit risk model. The stock price falls due to the dividend payout in the option pricing problem is analog to selling the firm’s asset to finance the loan repayment or dividend payout in the first passage model. Thus, our formulas can evaluate vulnerable bonds or the equity values given that the firm’s future loan/dividend payments are known.