= Solution
For strike $K>0$ and maturity $T$, a <European call option> and a <European put option> have <boundary conditions>
$$
C(T,S)=(S-K)^+,\qquad P(T,S)=(K-S)^+.
$$
The <Black-Scholes equation> is solved backward from these conditions. At zero <stock> price, the zero state is absorbing in <geometric Brownian motion>, so
$$
C(t,0)=0,\qquad P(t,0)=Ke^{-r(T-t)}.
$$
At the other boundary, with $\tau=T-t>0$,
$$
C(t,S)-[S-Ke^{-r\tau}]\longrightarrow0,\qquad P(t,S)\longrightarrow0
\quad(S\to\infty).
$$
These conditions and a suitable growth bound select the financial solution and provide boundary data for numerical truncations. In the constant-coefficient model, under the <risk-neutral measure>,
$$
S_T=S\exp\{(r-\sigma^2/2)\tau+\sigma\sqrt{\tau}\,Z\},\qquad Z\sim N(0,1).
$$
For $d_1=[\log(S/K)+(r+\sigma^2/2)\tau]/(\sigma\sqrt{\tau})$ and $d_2=d_1-\sigma\sqrt{\tau}$, the exercise event is $Z>-d_2$. Completing the square in $e^{\sigma\sqrt{\tau}Z}$ times the normal density gives the truncated first moment and hence
$$
\boxed{C=S\Phi(d_1)-Ke^{-r\tau}\Phi(d_2),\qquad
P=Ke^{-r\tau}\Phi(-d_2)-S\Phi(-d_1),}
$$
where $\Phi$ is the standard normal <cumulative distribution function>. These are the <Black-Scholes formula> values and satisfy the specified terminal and <boundary conditions>.
<Put-call parity> is the relation
$$
\boxed{C-P=S-Ke^{-r(T-t)}.}
$$
A call minus a put has terminal <financial payoff> $S_T-K$. A <portfolio> of one share minus a <zero-coupon bond> paying $K$ has the same <financial payoff>, so absence of <arbitrage> and <law of one price> equate their current values. The formula here assumes the same strike and maturity and no dividends; known dividends modify the stock-side value.
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