Pricing barrier stock options with discrete dividends by approximating analytical formulae

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 approximating 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 <inline-formula id="ILM0001"><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="rquf_a_853319_ilm0001.gif"/></inline-formula> is approximated by an accumulated price decrement due to a continuous dividend yield up to time <inline-formula id="ILM0002"><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="rquf_a_853319_ilm0002.gif"/></inline-formula>. Thus, the stock price process prior to time <inline-formula id="ILM0003"><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="rquf_a_853319_ilm0003.gif"/></inline-formula> and after time <inline-formula id="ILM0004"><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="rquf_a_853319_ilm0004.gif"/></inline-formula> 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.