What are the lower bounds on European call and put prices?

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AcadiFi · Accepted Answer

For a European call on non-dividend stock: c >= max(S - K*e^(-rT), 0). For a European put: p >= max(K*e^(-rT) - S, 0). Both must also be >= 0 (options never have negative value). Derivation for call: consider two portfolios. Portfolio A: one European call + cash K*e^(-rT) invested at risk-free rate. Portfolio B: one share of stock. At expiry, A is worth max(S_T, K) and B is worth S_T. Since A >= B always, today A >= B, so c + K*e^(-rT) >= S, giving c >= S - K*e^(-rT). Example: Hollowpoint Mining stock at $72, K=$65, T=9 months, r=4%. Lower bound for call = 72 - 65*e^(-0.04*0.75) = 72 - 63.079 = $8.921. Any market price below $8.92 creates arbitrage: buy call ($8.80 say), short stock ($72 in), buy T-bill at $63.08. At expiry if S_T > 65, exercise call for K=$65 using T-bill proceeds, return stock, keep $0.12 plus T-bill interest remnant. If S_T <= 65, call lapses, buy stock at market <=65, return, keep remainder. With dividends D (known): c >= S - D - K*e^(-rT). These bounds are model-free — they hold regardless of volatility assumptions, only requiring no-arbitrage and rational exercise.

What are the lower bounds on European call and put prices?

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