What are the upper bounds on call and put prices?

Question

AcadiFi · Accepted Answer

European and American calls: c <= S and C <= S. A call gives the right to buy at K, so it can never be worth more than simply owning the stock (which costs S now and gives you the same upside plus avoids paying K later).

European put: p <= K*e^(-rT). The best case for a put is the stock going to zero, paying off K at expiry, worth K*e^(-rT) today. American put: P <= K, because it can be exercised immediately for max K.

Example: Caldera Chemicals stock at $38, European call K=$30, T=1 year. Upper bound c <= $38. If market call traded at $40, arbitrage: short call ($40 in), buy stock ($38 out), pocket $2 risk-free. If exercised, deliver stock for K=$30, total = 40 - 38 + 30 = $32. If not exercised, keep stock worth S_T plus $2, always profitable.

European put example: Tidewater Realty K=$100, T=6 months, r=3%. Upper bound p <= 100*e^(-0.03*0.5) = $98.51. If put traded at $99, sell put ($99 in), invest at rf. Maximum payout at expiry is K=$100, and $99*e^(0.03*0.5) = $100.49 > $100, guaranteeing profit.

Bounds tighten options pricing substantially before volatility-based models are even applied, and serve as sanity checks when models produce questionable outputs.

What are the upper bounds on call and put prices?

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