Proving put-call parity with two portfolios

Question

Write down put-call parity for European options on a stock that pays no dividends, and prove it.

Short answer

$C+Ke^{-rT}=P+S_0$ . Portfolio 1 is a call plus a zero-coupon bond paying $K$ at $T$ ; portfolio 2 is a put plus one share. At expiry both are worth $\max(S_T,K)$ in every state. Since they pay the same, no-arbitrage requires equal prices today; otherwise sell the expensive portfolio and buy the cheap one for riskless profit.

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 6.1 Option Pricing, 'Put-call parity', p.137; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

No-Arbitrage Bounds and Put-Call Parity

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