Option Pricing Under Dividend Barrier Strategies

This paper investigates risk-neutral price of European option under dividend barrier strategy when cumulative log-return during time interval [0,t] of the underlying stock in the absence of dividends follows a Brownian motion with drift. Such a dividend barrier strategy means that in the presence of dividends, there are no dividend payments when cumulative log-return is below a positive constant barrier b, and dividends in the case of cumulative log-return larger than barrier b are paid to ensure that cumulative log-return is reflected and bounded by barrier b. Overflow over dividend barrier b can be expressed in terms of running maximum of a Brownian motion with drift. Under dividend barrier strategy and risk neutral probability measure, we use joint probability density function of Brownian motion with drift and its running maximum to derive put-call parity and closed-form European call and put option pricing formula, the latter of which will be reduced to classical Black-Scholes-Merton option pricing formula in some sense as dividend barrier approaches positive infinity