Boundary Condition

The temporal boundary condition on the option price is that at maturity at time t^*, if the stock has risen above c, the call option is worth w(x,t^*)=x-c, so that a caller could buy the option at time t^* at the strike price c, making a profit x-c, which would equal the cost or value of the call option then, w(x,t^*). However, if the stock price x has fallen below the strike price c, then the call option is not exercised since it would result in a loss, and the value of the option is worthless, or w(x,t^*) = 0 if $x \leq c$. The boundary condition is continuous at x=c.