Deriving accurate analytical formulas for pricing stock options with discrete dividend payouts is a hard problem even for the simplest vanilla options. This is because the falls in the stock price process due to discrete dividend payouts will significantly increase the mathematical difficulty in pricing the option. On the other hand, much literature uses other dividend settings to simplify the difficulty, but these settings may produce inconsistent pricing results. This paper derives accurate

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... proximating formulae for pricing a popular path-dependent option, the barrier stock option, with discrete dividend payouts. The fall in stock price due to dividend payout at an exdividend date t is approximated by an accumulated price decrement due to a continuous dividend yield up to time t. Thus, the stock price process prior to time t and after time t can be separately modelled by two different lognormal-diffusive stock processes which help us to easily derive analytical pricing formulae. Numerical experiments suggest that our formulae provide more accurate and coherent pricing results than other approximation formulae. Our formulae are also robust under extreme cases, like the high volatility (of the stock price) case. Besides, our formulae also extend the applicability of the first-passage model (a type of structural credit risk model) to measure how the firm's payout influences its financial status and the credit qualities of other outstanding debts. Pricing stock options with discrete dividend payouts has drawn a lot of attention in the literature. Frishling (2002) shows that the underlying stock price processes are usually modelled in the following three different ways. Model 1 suggests that the stock price minus the present value of future dividends over the life of the option follows a lognormal diffusion process (see Roll 1977 , Geske 1979 . Model 2 suggests that the stock price plus the forward values of the dividends paid from today up to the option's maturity follows a lognormal diffusion process (see Heath and Jarrow 1988, Musiela and Rutkowski 1997) . Model 3 suggests that the stock price falls by the amount of the dividend paid at the exdividend date, and follows the lognormal diffusion process between two adjacent exdividend dates. For pricing vanilla options, Frishling (2002) argues that these three models are incompatible with each other and generate very different prices. In addition, Frishling (2002) , Bender and Vorst (2001), and Bos and Vandermark (2002) argue that only Model 3 can reflect the reality and generate consistent option prices. Apart from the three aforementioned models, Chiras and Manaster (1978) suggest that the discrete dividends can be transformed into a fixed continuous dividend yield. The vanilla