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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
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
. The boundary condition is continuous at x=c.