看涨-看跌平价

Put-call parity for European options:
$$c + K e^{-r\tau} \;=\; p + S - D,$$ where $c$ and $p$ are European call/put on the same underlying with the same maturity $T$ and strike $K$. ($D$ is the PV of future dividends.)

• Because $p \ge 0$, we get a lower bound for the call: $$S - D - K e^{-r\tau} \;\le\; c \;\le\; S.$$

For American options, a strict put-call parity formula does not hold. However, for a non-dividend-paying stock, we have two inequalities: $$S - K \;\le\; C - P \;\le\; S - K e^{-r\tau}.$$

• $C - P \;\le\; S - K e^{-r\tau}$:
  • On a non-dividend stock, never optimal to early-exercise an American call, so $C=c$. From the European put-call parity, $c + K e^{-r\tau} = p + S$. Also $P \ge p$. Hence $C - P \le S - K e^{-r\tau}$.
• $S - K \;\le\; C - P$:
  • By constructing two portfolios that replicate or dominate each other, we get $S-K \le C - P$.

Question: Can you write down the put-call parity for a European option on a nondividend-paying stock and prove it?